Constacyclic codes as invariant subspaces

^^^^'V/ Soi'öJ^^

Constacyclic Codes as Invariant

Subspaces

Constacyclic Codes as Invariant

Subspaces

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof.dr.ir. J. T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 26 januari 2009 om 12.30 uur

door

Diana Ilieva RADKOVA

Sofia University. Bulgarije

geboren te Yambol, Bulgarije

TU Delft Library \

Prometheusplem 1

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr. A.J.van Zanten en

prof.dr. S.M. Dodunekov

Samenstelling promotiecommissie: Rector Magnificus, voorzitter

Prof.dr. A.J. van Zanten, Universiteit Maastricht, fac. Algemene Wetenschappen, promotor

Prof.dr. S.M. Dodunekov. Bulgarian Acad, of Sciences, Inst.Math.. Sofia, promotor Prof.dr. H.T. Koehnk. Radboud Universiteit Nijmegen

Prof.dr. J.M.A.M. van Neerven. TU Delft. EWI Prof.dr. P. Charpin, INRIA, Fi-ankrijk

Prof.dr. W. Willems, Universiteit Magdeburg. Duitsland Prof.dr.ir. C. Roos, TU Delft, EWI

Prof.dr. C. Witteveen, TU Delft, EWI, reservelid

Table of Contents

Table of Contents v

Preface vii 1 Introduction 1

1.1 Outline of the Next Chapters 2 2 Some Classes of Codes as Invariant Subspaces 5

2.1 Linear Cyclic Codes as Invariant Subspaces 5 2.2 Linear Constacyclic Codes as Invariant Subspaces 17 2.3 Linear Quasi-Twisted Codes as Invariant Subspaces 20 3 The Case W h e n the Field Characteristic Divides the Length 35

3.1 Linear Cyclic Codes of Length Divisible by the Field Characteristic . . 35 3.2 Linear Constacyclic Codes of Length Divisible by The Field

Charac-teristic 44 4 Idempotent Matrices 49

4.1 Idempotent Matrices for Linear Cyclic Codes 49 4.2 Idempotent Matrices for Linear Constacyclic Codes 57

4.3 Computation of Idempotent Matrices 61 5 Bounds on Minimum Distance for Constacyclic Codes 65

5.1 BCH Bound for Constacyclic Codes 65 5.2 Hartmann-Tzeng Bound for Constacyc;lic Codes 70

5.3 Roos Boimd for Constacyclic Codes 74 5.4 Van Lint-Wilson Bounds for Constacyclic Codes 80

6 B C H Constacyclic Codes 87 6.1 BCH Constacyclic Codes 87 6.2 On the Mininunn Distance of Composite-Length BCH Constacyclic

Codes 88

vi

Bibliography

Acknowledgements

Summary

Samenvatting

Curriculum Vitae

i

Preface

This thesis presents the results of the author's research during the period 2004-2008. She started as a Ph.D. Student at Bulgarian Academy of Sciences (2003) and contin-ued her work at Delft University of Technology (2007).

The subject of study are constacyclic codes over finite fields and a description of these codes in terms of linear algebra. Most of the results included in this thesis are based on the following publications:

1. Cyclic Codes as Invariant Subspaces, to appear in Annuaire Univ. Sofia. Fac. Math. Inform. 98, 2005 (with A. Bojilov).

2. Cyclic Codes and Quasi-Twisted Codes: An Algebraic Approach, Report MICC 07-08, Universiteit Maastricht, 2007 (with A. Bojilov and A.J. van Zanten).

3. Idempotent Matrices and Bounds for the Minimum Distance for Constacyclic Codes, Report MICC 08-01, Universiteit Maastricht. 2008 (with A.J. van Zanten).

4. More Bounds for the Minimum Distance of Constacyclic Codes and the Case when the Field Characteristic Divides the Length, Report MICC 08-02, Universiteit Maastricht, 2008 (with A.J. van Zanten).

5. Bounds on Minimum Distance in Constacyclic Codes, Proc. Eleventh In-ternational Workshop on Algebraic and Combinatorial Coding Theory, Pamporovo, Bulgaria, 16-22 June, pp. 236-242, 2008 (with A.J. van Zanten).

6. Constacyclic Codes as Invariant Subspaces, Linear Algebra and its Applications, vol. 430. pp. 855-864, 2009 (with A.J. van Zanten).

In [1] some properties of cyclic codes are regarded that are closely related to the structure of these codes as invariant subspaces of F " with respect to the cyclic shift map. In [2] quasi-twisted codes are considered as invariant subspaces of F " with respect to an a—constacyclic shift map over k positions, where A; is a divisor of the length n and 0 ^ a G F. The important classes of constacyclic and cyclic codes are realized as special cases of quasi-twisted codes. This approach enables the authors to derive some properties for the corresponding idempotent matrices of constacyclic codes in [3] and to obtain lower bounds for the minimum distance of constacyclic codes in [3], [4], [5] and [6], that are generalizations of some well-known bounds for cyclic codes.

Chapter 1

Introduction

The basic idea of noisy coding is to add some redundancy symbols to the information symbols in order to be able to detect or correct errors that occur during a transmission over a noisy channel. In this thesis we will focus our attention on the most important class among error-correcting block codes - the linear codes (cf. [7]). In general a code is a sot of words of fixed length, say n, over a certain alphabet F. Often in coding theory, a code's alphabet is taken to be a finite field. Then the set F " is a vector space over F, and a linear code C over F of block length n is defined as a subspace of F " . The dimension fc of C as a vector space over F is called a dimension of C. By virtue of the fact that the code is a subspace of F " , a linear combination aci + /?C2 of two codewords in C with coefficients in F is again a codeword in C. Thus the entire code can be represented as the span of a minimal set of codewords known as a basis in linear algebra terms. The subspace definition also gives rise to the important property that the minimum Hamming distance between codewords is simply the minimum Hamming weight of all codewords. Though not sufficient for uiuque classification, a linear code's block length, dimension, and minimimi distance are three crucial parameters in determining the strength of the code. A linear code with block length n, dimension k. and minimum distance d is referred to as an [n, k, d]—code. Thus linear codes have an accessible mathematical structure which leads to effective coding and decoding methods and allows a relatively easy analysis of their performance. On the other hand, linear codes are strongly connected to many classical mathematical objects and their study is therefore also of purely theoretical interest.

Cyclic codes, which form a special class of linear codes, are among the most used linear block codes (cf. [8 14], and also the references cited in [8]). Examples of cycUc codes such as the Reed-Solomon codes [15] and the Hamming codes [16] are landmarks in the field of comnmnication and information management. Cyclic codes also play a central role in coding theory from a theoretical standpoint, and they serve as building blocks for other codes, such as the Kerdock [17], Preparata [18] and Justesen [19] codes. For further information on these codes we refer to [20 23]. Related to cyclic

2 Chapter 1. Introduction

codes are negacychc codes (cf. [24-26]) and constacyclic codes (cf. [27 30]). Both families can be regarded as generalizations of cyclic codes, playing a similar role.

This thesis is about linear constacyclic codes, which were first introduced in [31] as generalizations of linear cyclic codes. A g—ary constacyclic code of length n can be defined by an n x n—generator matrix with the property that each row (apart from the last one) (CQ, c i , . . . ,c„_i),Ci e GF(g), defines the next row as (ac„_i,co,... ,c„_2), where a is some fixed element from GF(5) \ {0}. Recent interest in constacyclic codes was sparked by the discovery of a relationship between constacyclic codes over integer residue rings Zpk+i and cyclic codes over finite fields GF(p) via the Gray map and its generalizations [32]. It has also been shown that many cyclic codes of composite length can be decomposed into constacyclic codes via matrix products [33]. Constacyclic codes are used to obtain a construction of good nonlinear (cf. [34]) and cyclic (cf. [35]) codes. They have a simple encoder and can be decoded using standard equipment (cf. [34] and [36]).

Some important classes of codes are realized as special cases of constacyclic codes. The case a = 1 gives cyclic codes, while a = —I yields negacyclic codes.

Cyclic codes are traditionally described by using methods of commutative algebra (cf. e.g. [37] and [38]). In this approach a codeword (co,Ci.... ,c„_i) corresponds to a polynomial CQ + CiX + • • • + Cn~ix^^^ which is in Rn[x\, the ring of polynomials in X mod x" — 1. A cyclic shift of a codeword then corresponds to multiplication of the polynomial by x, and hence the theory of linear cyclic codes comes down to studying principal ideals in R„[x] generated by some generator polynomial. In [39] an alternative point of view is taken by regarding constacyclic codes as a certain kind of contractions of cyclic codes.

The above mentioned approach of cyclic codes seems not very appropriate for generalization to constacyclic codes in general. Since linear codes of length n over the field GF(g) have the structure of linear subspaces of GFicff, an alternative de-scription of constacyclic codes in terms of linear algebra appears to be another quite natural setting. In this thesis we develop such an approach. We observe that the con-stacyclic shift map is a linear operator in GF{q)"'. Our approach is to consider linear constacyclic codes as invariant subspaces of GF(g)" with respect to this operator and thus obtain a description of constacyclic codes.

1.1 Outline of the Next Chapters

The content of this thesis is spread among the various chapters as follows. Since our approach is closely connected with the interpretation of linear codes as linear subspaces of F " , where n is the word length and F = GF(g) is the alphabet, we start with the description of some classes of linear codes as invariant subspaces of F " with respect to suitable linear operators. Throughout Chapter 2 we require that (n, q) = \, which is a common practice in coding theory. In sections 2.1 and 2.2 linear cyclic and constacyclic codes are treated in this way. The description given in Section 2.2 is our

1.1. Outline of the N e x t Chapters 3

major tool in the remaining part of the thesis. Another starting point for the next chapters will be the polynomial

where the numerator is the characteristic polynomial of the matrix which represents the constacyclic transformation tpa with respect to a, 0 ^ a E F, and g{x) = f^alci-'')-In Section 2.3 we give generalization of the results of the previous sections for the wider class of quasi-twisted codes. For further information on the use of these codes we refer to [40-42].

Much of the theory in Chapter 2 also holds without the restriction on n and q. The subjects of Chapter 3 are linear cyclic and constacyclic codes of length divisible by the field characteristic. Again we consider these classes of codes as invariant subspaces of F " and thus obtain a description of these codes in a more general setting.

The main purpose of Chapter 4 is to study some properties of the idempotent matrices, defined by polynomials h{x). corresponding to cyclic and constacyclic codes. Idempotent matrices that correspond to minimal invariant subspaces of F" are called primitive. We show that every idempotent matrix is a sum of primitive idempotent matrices and that its columns generate the corresponding code. Also the idempotent matrix of a dual code is determined. In Section 4.3 we discuss a straightforward method to determine the idempotent matrices for given polynomials h{x) and g{x).

In Chapter 5 we derive some lower bounds for the minimum distance of consta-cyclic codes. There are several well-known lower bounds for the minimum distance of a cyclic code. In Section 5.1 we give a generalization for constacyclic codes of the oldest and best known of these bounds, the Bose-Chaudhuri-Hocquenghem (BCH) bound [37]. This generalization states that if the polynomial h{x) has a string of (5 — 1 consecutive zeros, then the minimum distance of the corresponding constacyclic code C is at least ö. In Section 5.2 we show that the Hartmann-Tzeng bound for cyclic code (cf. [43] and [44]) can also be generalized for the wider class of constacyclic codes. The generalized theorem says that if there are s+1 consecutive sets of 5 — 1 consecutive ze-ros of h{x), the minimum distance of C is at least S + s. Next, in Section 5.3 we derive an even more general bound for the minimum distance of constacyclic codes, which is a generalization of Roos' bound for cyclic codes [45]. It applies to constacyclic codes for which the set of zeros of h{x) is the union of (not necessarily consecutive) sets of consecutive zeros. In Section 5.4 we give two improvements of the BCH bound for constacyclic codes which in turn are generalizations of the van Lint-Wilson bounds [46]. These bounds imply the bounds derived in the previous sections of this chapter.

In Chapter 6 we define BCH constacyclic codes. In Section 6.2 we derive a theorem which generalizes a theorem of Dianwu Yue and Hongbo Zhu in [47]. Due to this theorem, we are able to present a class of BCH constacyclic codes of composite length with the property that their minimum distance equals the BCH bound.

4

Chapter 1. Introduction

Chapter 2

Some Classes of Codes as

Invariant Subspaces

2.1 Linear Cyclic Codes as Invariant Subspaces

Let

r pn y pn

' I / ' T - -_{• \(xi,X2,..-.Xn) 1-» {Xn,Xi,...,Xn-l)} ^f^ 'y I I ^ I-y ' » * , 'Ï» . 1 ^ '

T h e n (p € Honi [F", F") and it has t h e following m a t r i x ^ 0 0 0 . . . IN 1 0 0 . . . 0 0 1 0 . . . 0

w

with respect to t h e basis e = ( e i . 62 e „ ) . Note t h a t t h e relations A' = A^^ and A " = E hold. T h e characteristic polynomial of A is

fA{x) -X Q 0 . . I -X Q .. 0 1 - X . . 0 0 0 . . . 1 . 0 . 0 . —X

= ( - l ) " ( x " - l ) .

In t h e next wo shall d e n o t e (2.3) by f{x). For our purposes we need t h e following well-known fact.

Chapter 2. Some Cleisses of Codes £is Invariant Subspaces

Proposition 2.1.1. Let U be a ifi-invariant subspace of V and dim pV = n. Then f^\i,{x) divides f^{x). In particular, ifV = U®W and W is a ip-invariant subspace ofF'' then f^{x) = /^i„(x)/^|^^.(x).

For the proof of Proposition 2.1.1 we refer to [48].

Let f{x) = { — l)"'fi{x)... ft{x) be the factorization of f{x) into irreducible factors over F. According to the Theorem of Cayley-Hamilton (cf. e.g. [49] and [50]) the matrix A of (2.2) satisfies

f {A) = O. (2.4) We assume that (n.q) = 1. In that case f{x) has distinct factors fi{x), i = I,... ,t,

which are monic. Furthermore, we consider the homogeneous set of equations

fi(A)x = 0, X G F " (2.5)

for i = l,...,t. li Ui stands for the solution space of (2.5), then we may write [/, = Ker/,(<p).

Theorem 2.1.1. The subspaces Ui of F'^ satisfy the following conditions: 1) Ui is a ip-invariant subspace of F";

2) if W is a ip—invariant subspace of F " and Wi = W D Ut for i = i,... ,t. then

Wi is ip—invariant and W = Wi® • • • ®Wt\

3) F-^ = Ui®---®Ut;

4) dinipUt = degfiix) = h;

5 ; ^ | „ . ( x ) = (-l)'=-/.(x);

6) Ui is a minimal tp-invariant subspace of F"^.

Proof 1) Let u e C/„ i.e., f,{A)u = 0. Then f^{A)^{u) = fi{A)Au = Afi{A)n = 0,

so that iy3(u) € Ui.

2) Let fi{x) = - ^ for i = 1 , . . . , i. Since ( / i ( x ) . . . . , ft{x)) = 1, by the Euclidean algorithm there are polynomials ai{x),... ,at{x) 6 F[x] such that

2.1. Linear Cyclic Codes as Invariant Subspaces 7

Then for every vector w & W the equality w = a i ( ^ ) / i ( ^ ) w + • • • + at{A)ft(A)w holds. Let w, = ai{A)fi{A)-w e W. Then /i(/4)w,: = ai{A)f{A)yf = 0 because of (2.4), and so wt e VitlW = W^. Hence,

Assume that w € Wir\J2j^, Wj, then /i(A)w = 0. / i ( ^ ) w = 0. Since (/i(a:), fi{x)) = 1, there are polynomials a{x),b{x) € -f [a;], such that a{x)fi{x) + b{x)fi{x) = 1. Hence, a{A)fi{A)\f + &(yl)/,(yl)w = w = 0. so that Wi n Xi^y, W^i = {0}- Thus

W = Wi®---®Wf

3) This follows from 2) with H^ = F " .

4) Let g e f7i be an arbitrary nonzero vector and let k>\ be the smallest natural number with the property that the vectors g, 'f>{g),.... ¥''^(g) are linearly dependent. Then there are elements CQ, .. •, Ck-i 6 F, at least one of which is nonzero, such that

¥'''(g) = Cog + cMg) + ••• + Cfc_iv3''-l(g).

Consider the polynomial t{x) = x^ — Ck-ix'^^^ — • • • — Co € F[x\. Since (i(v5))(g) = (/t(</5))(g) = 0, it follows that [{t{x).U{x)){ip)]{g) = 0. But {t(x).fi{x)) is equal to 1 or to fi{x). Hence, {t{x), fi{x)) = fi{x) and fi(x) divides t{x). Thus fc; = deg/t(x) < dGgi(z) = k. On the other hand, the vectors g.',3(g) <(?*'(g) are linearly dependent, since ifiX'P)){s) = 0. and from the minimality of k we obtain k = ki. Then dinif-f/,- > fc,. Therefore

t t t

n = dim p F " = ^ dim ff/,- ^ X^ '^i = X^ ^''-S /i(^) = '^«^g ƒ (a;) = n

i = l i—1 Ï —1

and dim pUi = ki.

5) Let (g^ g^ ) be a basis of f/, over F, i = 1, i, and let At be the matrix of (/3|c/, with respect to that basis. Let fi{x) = /^i^ (x). Suppose that {fi{x), fi{x)) =

8 Chapter 2. Some Classes of Codes as Invariant Subspaces

1. Hence, there are polynomials a{x).b{x) € F[x]. such that a{x)fj{x) + b{x)fi{x) = 1. Then a{Ai)fi{Ai) + b{Ai)fi{Ai) = E. Therefore b{Ai)fi{A,) = E. We will show that fi{Ai) = O, which contradicts the last equation.

M)

g^ ) is a basis of By property 3) we obtain that g = (gj^' si^i'- • • •.

si*'-F " and ip is represented by the following matrix

A2 A'

V

A J

with respect to that basis. We also have A' = T^^AT, where T is the transformation

matrix from the standard basis of F'" to the basis g. Then

MA2)

fi{A') UT-'AT)=T-'fi{A)T.

\ MAt) I

Let g^"' = Ajj ei + h AJ„e„, j = 1 . fc,-. Since g j ' ' € t/j, we obtain that /OX ^o^^ ^A«\

/,(^')

_T-V.(^)

Vfn)

1

Vo7 Vo7

where 1 is on the (/ci + • • • + fcj-i + j) —th position. According to the last equation /i(Ai) = O. Therefore {fi{x), fi{x)) ^ 1. Since fi{x) and fi{x) are polynomials of the same degree /c, and fi{x) is monic and irreducible, we obtain that fi{x) = ( —l)*^'/,:(x). 6) Let U be (^-invariant subspace of F " and let {0} ^V'Z.JJi. Then by Propo-sition 2.1.1 we obtain that /^]j,(x) divides fi{x). Since the polynomial fi{x) is irre-ducible, dim pU = dim pUi and f/ = t/j. •

Proposition 2.1.2. Let U be a ip—invariant subspace of F'^. Then U is a direct sum of some of the minimal tp—invariant subspaces Ui of F".

2.1. Linear Cyclic Codes as Invariant Subspaces 9

Proof. This follows immediately from property 2) of Theorem 2.1.1. 0

Definition 2.1.1. A linear code of length n and rank k is a linear subspace C with dimension k of the vector space F".

Definition 2.1.2. A code C of length n over F is called cyclic, if whenever x = (ci, C2,..., c„) is in C, so is its cyclic shift y = {c„, C i , . . . , c„_i).

The following statement is clear from the definition.

Proposition 2.1.3. A linear code C of length n over F is cyclic iffC is a if—invariant subspace of F^.

Theorem 2.1.2. Let C be a linear cyclic code of length n over F. Then the following facts hold:

1) C = Ui^ ® • • • ® Ui^ for some minimal ip—invariant subspaces Ui^. of F" and k := dimpC = ki^ + • • • -I- A:,,, where kr is the dimension ofUi^;

2) /^icW = ( - I ) ' 7 M ( ^ ) • • • fi.i^) = s W ;

3) c e C iff g{A)c = 0:

4) the polynomial g{x) has the smallest degree with respect to property 3);

5) rank{g{A)) = n — k.

Proof. 1) This follows from Proposition 2.1.2.

2) Let (gi % . . . ,ëk ) be a basis of Ui^ over F, r = 1 , . . . , s, and let Ai^ be the matrix of ip\u^ with respect to that basis. Let fi{x) — f^\^, ix). Then (gi , • • •, gfc 1 • • •! gi ! • • •! gi' ) is a basis of C over F and ip\c is represented by the following matrix

10 Chapter 2. Some Classes of Codes £is Invariant Subspaces

MM

\

\ A, J

with respect to that basis. Hence,

/ ^ l , ( x ) = / , , ( x ) . . . 4 ( x ) = (-!)'=•.+•••+'=.3/._(3:).../.^(a:).

Note that Ai^ and fi^{x) are defined as in the proof of Theorem 2.1.1.

3) Let c e C. Then c = Ujj + • • • + U;^ for some Uj^ € t/,,., r = 1 , . . . . s, and

giA)c = {-in{f,,...fiMA)u,, +••• + {fi,...fJ{A)uü = 0.

Conversely, suppose that g{A)c = 0 for some c G F " . According to Theorem 2.LI we have that c = Ui + h U(. Uj € f/,. Then ff(A)c = (-l)''"[(/ii ... fi,){A)ui + • • • + (/ii • • • fi,){A)ut] = 0, so that g{A)[uj, + • • • + u^,] = 0 . where { j i , . ..ji} = {l,...,t}\{ii,. ...is}- Let V = Uj, 4 h Uj, and

Since {h{x),g{x)) = 1, there are polynomials a{x), b{x) e F[x] so that a{x)h{x) + b{x)g{x) = 1. Hence v = a{A)h{A)-v + b{A)g(A)\ = 0 and c = u,, H h Uj^ e C.

4) Suppose that 6(x) e F[x] is a nonzero polynomial of smallest degree such that b{A)c = 0 for all c G C By the division algorithm in F[x] there are polynomials

q{x),r(x) such that g{x) = b{x)q{x) +r{x), where degr(x) < deg6(x). Then for each

vector c e C we have g{A)c = q{A)b{A)c + r{A)c and hence, r{A)c = 0. But this contradicts the choice of 6(x) unless r(x) is identically zero. Thus, b{x) divides g{x). If degfe{x) < deg5(x), then b{x) is a product of some of the irreducible factors of p(x) and without loss of generaUty we may assume that b{x) = ( —1)*'" •'"•••^'^•'"/ii . •. fi„^ and m < s. Let us consider the code C" = t/j; © • • • ® Ui^ C C. Then b{x) = f^p^^, (x)

2.1. Linear Cyclic Codes as Invariant Subspaces 11

and by the equation g{A)c = 0 for all c G C we obtain that C C C'. This contradiction proves the statement.

5) By property 3) C is the solution space of the homogeneous set of equations 5(A)x = 0. Then dimpC = k = n — Tank{g{A)). which proves the statement. D

Definition 2.1.3. Lei x = ( a ; i , . . . , x „ ) and y = {yi....y„) be two vectors in F". We define an inner product over F btj (x, y) = xiyi + • • • + XnPn- If (x, y) = 0, we say that x and y are orthogonal to each other.

Definition 2.1.4. Let C be a linear code over F. We define the dual of C (which is denoted by C-^) to be the set of all vectors which are orthogonal to all codewords in

C, i.e.,

C-L = { v e F " | ( v , c ) = 0 , V c e C } .

It is well known that if C is /c—dimensional, then C''- is an (n — fc)—dimensional subspace of F", so C-^ is a linear code again.

Proposition 2.1.4. The dual of a linear cyclic code is also cyclic.

Proof. Let h = {hi.... ^h^) € C-^ and c = ( c i , . . . , c „ ) € C. We shall show that

VP(h) = (hn-hi,.. . , / i „ - i ) e C^. We have

(</3(h),c) = cih„ + • • • + cnhn^i = {h.<p-\c)) = (h, yp""! (c)) = 0 ,

which proves the statement. D

Proposition 2.1.5. The matrix H, the rows of which constitute an arbitrary set of n — k linearly independent rows of g{A), is a parity check matrix of C.

Proof. The proof follows from the equation g{A)c = 0 for every vector c e C and the fact that rank {g{A)) = n — k. D

12 Chapter 2. Some Classes of Codes as Invariant Subspaces

Let g(j ëi„-k ^'^ * basis of C"*", where g/,, is a /,. —th vector row of g{A). By the equation g(A)h{A) = O we obtain that (g(,.. h,;) = 0 for each i = 1 , . . . , n, r = 1 , . . . , n — /c. The last equation gives us that the columns h; of h{A) are codewords in C.

We show that rank(/i(yl)) = k. By the inequaUty of Sylvester (cf. e.g. [51]) we obtain that rank(O) = 0 > ra.nk{g{A)) + rank(/i(v4)) — n. Since ta,nk{h{A)) < n — rank(g(j4)) = n — (n — k) = k. On the other hand the inequality of Sylvester, applied to the product h(A) = (-l)""*^/j, (A)... fj^ {A), gives us that rank {h{A)) >

'"'h + • • • + Tji — n{l — 1) = nl ~ kj^ — • • • — kjj — nl + n = n — {kj^ + • • • + kj^) = n — (n — fcj, — • • • — fcj,) = n — (n — /c) = A:. Therefore rank {h{A)) = k. Thus we have proved the following proposition.

Proposition 2.1.6. The matrix G, the rows of which constitute an arbitrary set of k linearly independent rows of {h{A)) , is a generator matrix of the code C.

Lemma 2.1.1. If g(x) e F[x], then g{A^^) = ^(A*) = {g{A)f. In particular, if n divides degg(x), then g*{A) = {g(A)) , where g*(x) is the reciprocal polynomial of

9{x).

Proof Letg{x) =gox'' + gix'' ^+ •• • + gk^ix + gk. then g{A) = goA''+giA'' ^ + --- +

gk-iA + gkE- Transposing both sides of the last equation, we obtain that {g{A))' =

goiA'')* +gi{A''~^)' + • • •+gk-iA* + g,E = go{A*f +gi{A*)'-' + • • • + gk^,A* + gkE =

9{A*).

In particular, if degg{x) = ns for some s € N, then g*{A) = A^^g{A~^) = A-^g{A*)=g{A') = {g{A)f.U

Let /i^i j_ [x) = h{x). By Theorem 2.1.2 it follows that h{x) is the polynomial of the smallest degree such that h{A)u = 0 for every u 6 C"*-. Let h*{x) = h{x)q{x) + r{x), where degr(x) < deg/i(x). Then by Lemma 2.1.1 h*{A) = A"-'^(/i(^))' = h{A)q{A)+ r{A), hence for every vector u € C-^ the assertion A^~^{h{A)) u = q{A)h{A)u'\-r{A)\i holds, so that r(x) = 0. Thus h{x) divides h*{x). Since both are polynomials of the same degree, h*{x) = ah{x), where a G F is the leading coefficient of the product

2.1. Linear Cyclic Codes as Invariant Subspaces 13

Ux) = U*ix) = {-iy'-'-f;^{x)...fi{x) =

' 1

where Oj^ is the leading coefficient of ft{x). Note that the polynomials fs^{x) = •^fj^ix) are rnonic irreducible and divide f{x) = (—l)"(x" — 1).

Now we show that C""- = C/s, ® • • • ® t/^,. By Theorem 2.1.2 C^ is the solution space of the homogeneous system with matrix h{A). Let u e C/ = t/,,, ® • • • ® t/j, and let u = Usj + • • • + Us, for u,,^ e Ug^, r = 1,.. .,1. Then

MA)u = ( - l ) " - ' = [ { / , , . . . / „ ) ( A K . + • • • + ( ƒ „ . . . A , ) ( A ) u , , ] = 0 . Hence U < C-^. Since dim pU = d i m f C ^ , then

Thus we have proved the following theorem.

Theorem 2.1.3. Let C = Ui^ ® • • • ® Ui^ he a linear cyclic code of length n over F, and {ji. • • • .ji} = {1 t}\{ii,... ,is}- Then the dual code of C is given by C-L = { / „ ® . . . ® f / , „ andf,^{x) = (-l)'=.r/^^(a:) = {-i)k,.^j*^(x), where f^ix) is the reciprocal polynomial of fj^{x) with leading coefficient equal to aj^., r = 1,... ,1.

Example 2.1.1. Consider the matrix A of (2.2) for n = 7 and q = 2. Then we have

f{x):=fA{x) = x'' + l.

Factorizing f{x) into irreducible factors over GF(2) yields

fix) = fi{x)f2{x)f3{x) = (x + l){x^ + X + l)(a;3 + x2 + 1).

The factors fi{x) define minimal if—invariant spaces Ui, for i = 1.2,3. We define the cyclic linear code C

14 C h a p t e r 2. S o m e Classes of C o d e s as Invariant S u b s p a c e s

According to Theorem 2.1.2, we have dimC = 4 and

gix) ••= Uicix) = (x+ l){x^ + x'^ + 1) = X* + x'^ + X + I. It follows that ^1 0 0 1 0 1 l>y 1 1 0 0 10 1 1 1 1 0 0 10 S(A) = 0 1 1 1 0 0 1 10 1 1 1 0 0 0 10 1 1 1 0 \ 0 0 1 0 1 1 1/

The rank of this matrix is rank{g{A)) = 7 — 4 = 3. Taking 3 independent rows

yields by Proposition 2.1.5 a parity check matrix for the code C, i.e.,

He

a 0 1 1 1 0 ON, 0 1 0 1 1 1 0 I C: 0.

VO 0 1 0 1 1 F

Notice that the columns of H represent integers 1,2,..., 7 in binary. So the code C

is equivalent to the Hamming code W3.

Furthermore, the polynomial h{x) = ^p• is equal to x^ + x + I, and therefore we

have h{A) / I 0 0 0 1 0 1 \ 1 1 0 0 0 1 0 0 1 1 0 0 0 1 1 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 Vo 0 0 1 0 1 1/

2.1. Linear Cyclic Codes as Invariant Subspaces 15

We can immediately verify that g{A)h*{A) = O and also that rank(h{A)) = 4. Taking

4 independent columns of h{A) yields a generator matrix for C, i.e.,

/ I 0 0 0 1 1 0\ 0 1 1 0 1 0 0 0 0 1 1 0 1 0 Vo 0 0 1 1 0 1/

Example 2.1.2. Consider the matrix A of (2.2) for n = 8 and q = 3 (so (n, q) = 1 again). Then

f{x):=fA{x)=x^-l.

Factorizing f{x) into irreducible factors over GF{S) yields

fix) = fl{x)f2{x)f3{x)fi{x)f5{x) = {X+ 1){X - l){x'' + l){x'' +X- l){x'' - X - I).

Next, we define

c := f/2 e t/s e t/4 © t^5,

corresponding to the polynomial

g{x) := f^iaix) = f2ix)f3ix)fi{x)f5{x) = I ^ = x' -x^+ x'- -x' +x' -x^ + x-l.

Jl\X)

It follows immediately that

g{A) = {-l 1 - 1 1 - 1 1 - 1 1)^,

where the matrix g{A) is represented by its fir.st row. The other rows can be obtained by

cyclic permutations of the first row, as is indicated by the subindex c. It will be obvious that rank(g{A)) = 1, and hence that dimC = 8—1 = 7 (cf. also Proposition 2.1.5).

16 C h a p t e r 2. S o m e Classes of C o d e s as Invariant S u b s p a c e s

The parity check matrix H for C is a (1, 8) —matrix which consists of the first row of

g{A). A generator matrix for C is obtained from h{x) = x + l, which provides us with

h{A) = ( 1 0 0 0 0 0 0 1 ) ^ .

Any [7,8) — submatrix of h*{A) is a generator matrix for C.

Another possible choice for a liner cyclic code would be

C' := U2 e Ui,

with

g{x) = {x~ l)(x^ + X - I) = x^ + x + I,

and

h{x) = {x + l){x'^ + l)(x2 - a; - 1) = x'^ • x^ + x - 1.

Consequently, we have dimC' = 3. ^ parity check matrix for C' can be obtained by

taking 5 independent rows from the matrix

g{A) = {l 0 0 0 0 1 0 1)^, e.g. / I 0 1 1 0 0 0 0\ 0 1 0 1 1 0 0 0 H = 0 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 Vo 0 0 0 1 0 1 1/

A generator matrix can be obtained by taking 3 independent columns from

h{A) = ( 1 0 - 1 - 1 1 - 1 0 0)^,

2.2. Linear Constacyclic C o d e s £is Invarijint S u b s p a c e s 17

G =

/ I 0 0 - 1 1 - 1 - 1 0 \ 0 1 0 0 - 1 1 - 1 - 1 \ 0 0 - 1 1 - 1 - 1 0 1 /

Let C C -F" be an arbitrary, not necessary linear, cyclic code. Let us consider the action of the group G = {<p) — {id, </?,..., </'"~^} — Cn over F " . Then the following theorem holds.

Theorem 2.1.4. C = Qi U . . . U Qs, where Q, are G-orbits and k, = |Qj| is a divisor

s

of \G\ = n. In particular, \C\ = ^ ki. 1 = 1

2.2 Linear Constacyclic Codes as Invariant Subspaces

Definition 2.2.1. Let a be a nonzero element of F. A code C of length n over F is called constacyclic with respect to a, if whenever x = (ci. C2, • • •, c„) is in C. so is

y = (ac„,ci, c„_i).

Let a be a nonzero clement of F and let

' p n _> pn

(2.6)

" • \{Xl,X2,---.Xn) 1-^ (aX„.Xl....,Xn-l)

Then il^a € Hom ( F " , F " ) and it has the following matrix ' 0 0 0 . 1 0 0 . 0 1 0 . ^0 0 0 ax 0 ' . 0 (2.7) 0

with respect to the basis e = (ei,e2, • •. ,e„). Note that the relations Bn{a)~^ = Bn{-^Y and B" = aE hold. The characteristic polynomial of S„ is fs^ix) = (—l)"(a;" — a). We assume that (n,q) = 1. The polynomial fB„{x) has no multiple roots and splits into distinct irreducible monic factors fB„{x) = (—l)"/i(a:)... ft{x). Let Ui = KeT fi{rpa)- It is easy to see that Theorem 2.1.1 and Proposition 2.1.2 are true in this case too. The following statement is clear from the definition.

18 Chapter 2. Some Classes of Codes as Invariant Subspaces

Proposition 2.2.1. A linear code C of length n over F is constacyclic iff C is a ij;a —invariant subspace of F".

The next theorem is analogous to Theorem 2.1.2 and Proposition 2.1.5 and so we omit its proof.

Theorem 2.2.1. Let C be a linear constacyclic code of length n over F. Then the following facts hold:

1) C = Ui^ ® • • • ® C/j, for some minimal xl)a —invariant subspaces Ui^ of F " and k := dimFC = ki^ + • • • + ki^. where ki^ is the dimension ofUi^;

3)ceC iffg{Bn)c =

0-4) the polynomial g{x) has the smallest degree with respect to property 3);

5) rank{g{Bn)) = n — k;

6) The matrix H, the rows of which constitute an arbitrary set of n — k linearly

independent rows of g{A), is a parity check matrix of C.

Proposition 2.2.2. The dual of a linear constacyclic code with respect to a is con-stacyclic with respect to -.

a

Proof. The proof follows from the equality

(^a(c).h) = (B„(a)c,h) = (c.B„,(a)*h) = {c, B„(i)"'h> = a(c, ^ r ' ( h ) ) = 0

for every c e C and h e C ^ . D

Example 2.2.1. As an example of a linear constacyclic code we take n = 8, q = 3 and o = — 1 in (2.7). We then have the following characteristic polynomial

2.2. L i n e a r Constacyclic C o d e s EIS I n v a r i a n t S u b s p a c e s 19

9iBs) = fi{Bs

When splitting this polynomial into irreducible polynomials over GF{3), we find

fix) = fl(x)f2{x) = (X* +X'- 1){X^ -X^- 1),

where the factors fi{x) and ƒ2(x) define minimal ipa —invariant subspaces U\ and U2,

respectively, both of dimension 4 according to Theorem 2.2.1. If we define

C = Uu C' = U2,

then we find, similarly as in Example 2.1.2, that a parity check matrix H for code C is obtained from

/-I 0 0 0 -1 0 -1 0 \

0 - 1 0 0 0 - 1 0 - 1

1 0 - 1 0 0 0 - 1 0

0 1 0 - 1 0 0 0 - 1

1 0 1 0 - 1 0 0 0

0 1 0 1 0 - 1 0 0

0 0 1 0 1 0 - 1 0

by taking 4 independent rows, whereas a parity check matrix H' for C' is obtained in

the same way from

/ - I 0 0 0 - 1 0 1 0 \ 0 - 1 0 0 0 - 1 0 1 - 1 0 - 1 0 0 0 - 1 0 0 - 1 0 - 1 0 0 0 - 1 1 0 - 1 0 - 1 0 0 0 0 1 0 - 1 0 - 1 0 0 0 0 1 0 - 1 0 - 1 0 V 0 0 0 1 0 - 1 0 - 1 / g'{B^) = /2(i?8) =

r

g{Bs) = /i(B8) = ( - 1 0 0 0 - 1 0 - 1 0 ) , ,

and

g'{Bs) = fiiBs)) = ( - 1 0 0 0 - 1 0 1 0)„, ,

respectively. The index ac means that each next row can be obtained from, its

pre-decessor by applying the operator ipa as defined in (2.6). Furthermore, we have the

matrices

h{Bs} = f2{Bs), h'{Bs) = fdBs).

It is an easy task to verify that the following relations hold

giBsMBs) = O, g'iBs)h'iBs) = O.

Actually, both equalities are equivalent to the relation f\{B^)f2{Bs) = O, and the codes C and C' are each other's dual.

2.3 Linear Quasi-Twisted Codes as Invariant

Sub-spaces

Let F = GF((?) and let F " be the n-dimensional vector space over F with the standard basis ei = ( 1 . 0 , . . . , 0), 62 = ( 0 , 1 , . . . , 0 ) , . . . , e„ = ( 0 , 0 , . . . , 1).

Lot a be a nonzero element of F and let

V'a

{

pn _ j pn

{xi.X2,...,Xn ) l-> {aXn,Xl,...,Xn-l) ' Then ipa € Hom ( F " , F " ) and it has the following matrix

Bn{a) = B„ /O 0 0 . . . a\ ' 1 0 0 . . . 0 ' 0 1 0 . . . 0

V 0 0 ... 0^

2.3. Linear Quïtói-Twisted Codes as Invariant Subspaces 21

with respect to the basis e = (ei.e2, • •. ,e„). The characteristic polynomial of B„ is - X 0 0

fB„{x)

-X 0

1 -X _{( - l ) " ( a ; " - a ) .} (2.10)

0 0 . . . - X

Let fc be a fixed divisor of n and let n = kl. Let us consider the operator tp = [xpa]''• We define a new basis p = (gi, g2, • • •, gn) of F " as follows:

g l g(+l g(fc-l)/+l = e i . g2 = ei+fc. . = 62, g/+2 = e2+fc. • = e*:, g(fc-l)Z+2 = 62*;. • • • g2/ = e2+(;_l)A:, • , gfc( = efc+(,_i)fc. Then V is represented by the following matrix

(B,

B = B,

\

(2.11)

BtJ

with respect to the basis g. where the k matrices Bi are defined as in (2.9) with n = I. Therefore the characteristic polynomial of B is

/B(x) = (/B,(x))'^ = ( - l ) " ( x ' - a ) ' .

Let us denote by f{x) the polynomial x' ~ a and let f{x) = fi{x)f2{x)... ft{x) be the factorization of ƒ (x) into irreducible factors over F. According to the theorem of Cayley-Hamilton the matrix B of (2.11) satisfies

/(fi) = 0 . (2.12) We assume that {n,q) = 1. In that case f{x) has distinct factors fi{x), i = 1,... ,t,

which are monic. Furthermore, we consider the homogeneous set of equations

fi{B)x = 0. X e F " (2.13) for i = 1 , . . . , t If Ui stands for the solution space of (2.13), then we may write

Ui = Ker/j(V'). We also introduce the following linear subspaces of F " :

Vl =^(gl>g2, • • - , & ) , V2 = ^{ei+i-si+2

g2/)-14 = ^(g(*;-l); + l.g(A.—l)Z+2!- ••'gfc()-Note that Vi,... .Vk are ^—invariant subspaces of F " .

The next proposition is analogous to Theorem 2.1.1 properties 1), 2) and so we omit its proof.

2 2 C h a p t e r 2 . S o m e C l a s s e s o f C o d e s a s I n v a r i a n t S u b s p a c e s

P r o p o s i t i o n 2 . 3 . 1 . The subspaces Ui, U2. • • • ,Ut of F" are tp—invariant. If W is a

tp—invariant subspace of F^ and Wi = WnUi fori = 1,... ,t, then Wi is xp—invariant andW = Wl®•••(BWt•

Corollary 2 . 3 . 1 . F " = C/i e • • • ® {/(.

Proof. This follows from Proposition 2.3.1 with W = F " . D

Let us denote Uij = UiUVj for all i = 1,... ,t a n d j = \,... ,k. T h e n we have t h e following result.

C o r o l l a r y 2 . 3 . 2 . Vj = C/ij 0 • • • ® Utj, j = l,...,k.

Proof. T h i s follows from Proposition 2.3.1 with W = Vj. D

T h e o r e m 2 . 3 . 1 . The subspaces Uij of F" satisfy the following properties:

1) Uij is a ip-invariant subspace of F";

2) ifv is a nonzero vector of Uij, then the vectors v , V'(v), . . . , ip 9 f^-'^{y^ form a basis of Uij and in particular dimpUij = degfi{x);

3) Uij is a minimal tp-invariant subspace of F"^; 4)Uii^Ui2 = ---^Uik;

5)Ui = Ua®---®Uik; 6)F^ = ®Uij.

ij

Proof. 1) T h i s is clear from t h e definition of Uij.

2) Let 0 / V e Uij be an a r b i t r a r y nonzero vector a n d let m > 1 be t h e smallest n a t u r a l n u m b e r with t h e p r o p e r t y t h a t t h e vectors v , ^ ( v ) , . . . , t/)"'(v) are linearly d e p e n d e n t . T h e n t h e r e are elements OQ, . . . . Um-i G F. a t least one of which is nonzero, such t h a t

2 . 3 . L i n e a r Q u a s i - T w i s t e d C o d e s a s I n v a r i a n t S u b s p a c e s 23

Consider t h e polynomial t{x) = x"^ — am-ix"^ ^ — • • • — OQ e F[x]. Since (i(V'))(v) = (/i('/'))(v) = 0, it follows t h a t [{t{x), f,{x)){ip)]{v) = 0. B u t {t{x), fi{x)) is equal t o 1 or t o fi{x). If we assume t h a t {t{x), fi{x)) = 1, t h e n v = 0, which contradicts t h e choice of V. Hence, {t{x), ft{x)) = fi{x) a n d fi{x) divides t{x). T h u s deg/j(a;) < degt(x) = m. O n t h e other hand, t h e vectors v,'i/)(v),...,«/) S/<(v) are linearly dependent, since (/i(«/'))(v) = 0, and from t h e minimality of m wo o b t a i n m = deg fi{x). Therefore dimpUij > d e g / , ( x ) , and so

t t

I = d i m ^ V , = ^ d i m f f / j j > ^ d e g / i ( x ) = d e g / ( x ) = I

i=l i = l and d\m pUij = dog/^(a;).

3) Let V b e a t / ) - invariant subspacc of F " a n d let { 0 } 5^ V" C Uij. If 0 7^ v e V, then t h e vectors v , ^ ( v ) , . . . , ^ ^^•'^'^'^(v) e V are linearly independent. Therefore dim FV > dim f f/y and V = Uij.

4) T h i s follows from t h e fact t h a t dim pUa = d i m f f / ; 2 = ••• = ^wvipVik = d e g / i ( x ) .

5) Let V € t / j . Since F " = Vi ® • • • © Vfc. we have v = Vi + • • • + v ^ , where v j e V,-, j = 1,...,A;. T h e n /i(V')(v) = / » ( ^ ) ( v i ) + ••• + / K ^ ) ^ ) = 0, so t h a t /t(«/')(Vj) = 0, i.e., Vj € t / j . Hence, Vj € Uij a n d

t/i = t/,1 + • • • + C/ifc.

Assume t h a t v e C/,y n Yl,si.j ^i-i- t h e n v e Vj and v e JDs^j Vs- But V,- n X^^_^j Vg =

{ 0 } , so we o b t a i n t h a t v = 0. T h u s

Ui = Uiie---®Uik.

24 Chapter 2. Some Classes of Codes as Invariant Subspaces

i=l i.j

0

Proposition 2.3.2. Let W be a ip-invariant subspace of Ut. Then there exists a natural number s < k such that W = f//i, where [//j is isomorphic to the direct sum of s copies of

Uii-Proof. Let 0 ^ Wi G H^. Tlien tlie vectors w i . ?/'(wi),..., ?/» S / . - i ( w i ) are linearly

independent. We define Wi := ^(wi.-!/'(wi),..., 0<^eS^*-i(wi)). Let 0 / w j € VF Ije a vector such that W2 ^ Wi. Then the vectors •W2.'i/)(w2): • • • ^ V" ^^•^'"^^2) are linearly independent. Define W2 •= £(w2,i/'(w2),... ,1/" ^/>-i(w2)). Note that dini/rH'^i = dimpW2 = deg/,(x). We vifill prove that the vectors

are also linearly independent. Assume the opposite. Then there exist nonzero poly-nomials hi{x), h2{x) € F[x], deghi{x),degh2{x) < deg/,(x), such that hi{B)wi + /i2(-S)w2 = 0. Since fi(x) is irreducible, we have that {h2{x), fi{x)) = 1. for i = 1 , . . . . i, and therefore by the Euclidean algorithm there are polynomials a{x), b{x) € F[x], such that a{x)h2ix) + b{x)fi{x) = I. Hence. a(B)h2{B)w2 + b{B)fi{B)w2 = W2.

Now W2 G U, and therefore fi(B)'W2 = 0. Thus we obtain that a{B)h2{B)w2 = W2. Fi'om /i2(B)(w2) = —/ii(B)(wi) and the last equality we conclude that W2 € Wi. This contradiction proves the statement. We proceed analogously until we obtain that W = Wi ® • • • ® W,, for some s < k. Since dim FW, = deg ƒ,(x), i = 1 s, it follows that W^Ufi. D

2.3. Linear Quasi-Twisted Codes as Invariant Subspaces 25

Theorem 2.3.2. Let W be a ip—invariant subspace of F". Then

for integers Si < k. 1 < i < t. In particular,

t

dimpW = y^^Sjdegfijx).

i=l

Proof This follows immediately from Proposition 2.3.1 and Proposition 2.3.2. D

Definition 2.3.1. A code C of length n over F is called a k-quasi-twisted code with respect to 0 ^ a € F iff any codeword in C is again a codeword in C after an a-constacyclic shift over k positions.

The following statement is clear from the definition.

Proposition 2.3.3. A linear code C with length n over F is k-quasi-twisted iff C is a ij}—invariant subspace of F".

Theorem 2.3.3. Let C be a linear k—quasi-twisted code of length n over F. Then

for integers Si < k, 1 < i < t. In particular,

t

dimpC = y ^Sjdeg fi{x).

i=l

Proof. This follows from Theorem 2.3.2 and Proposition 2.3.3. G

26 Chapter 2. Some Classes of Codes as Invariant Subspaces

(2.11) gives the representation matrix

B

B. B.

B.

\

BJ

for the operator ip with respect to the basis g, with /O 0 1 B3 = 1 0 Ü

V 1 Ü

For the characteristic polynomial of B we have

fB{x) = {-l)ix'-lf = -{f{x)f,

where f{x) can be factorized into irreducible polynomials over GF{2) as

fix) = fi{x)f2{x) = {X+ 1){X^ +X+1).

Let Ui = Kerfi{tl}) for i = \, 2. We define the following linear code

C = U2.

According to Theorem 2.3.1 we can write

U2 = U21 ^ >U-25;

where U2J = U2 ClVj and U21 = • • • = U25. If we introduce subcodes Ci := t/gi for i = 1....5. then dimCj = degf2{x) = 2, again by Theorem 2.3.1. One can almost

2.3. Linear Quasi-Twisted Codes as Invariant Subspaces 27 ft(S3)=/l(B3) and ^10 1 1 1 0 ^0 1 1

So a parity check matrix for the subcode Ci, i = 1 , . . . , 5 , restricted to its support, is

the row matrix (1,1,1). For C itself we find the parity check matrix

/ I 0 0 0 0\ 0 1 0 0 0 H = 0 0 1 0 0

0 0 0 1 0 Vo 0 0 0 1/

where 1 stands for (1.1, 1) and 0 for (0, 0, 0). Hence, dimC = 15 — 5 = 10, which is

in agreement with Theorem 2.3.3.

Taking two independent columns of h{Bi) yields a generator matrix for Ci

(re-stricted to its support), e.g.

n 1 0 Gr=[

Vo 1 1

This gives rise to the following generator matrix for C itself ^ a 0 0 0 0^y b 0 0 0 0 0 a 0 0 0 0 b 0 0 0 0 0 a 0 0 0 0 b 0 0 0 0 0 a 0 0 0 0 b 0 0 0 0 0 a

Vo 0 0 0 b /

28 Chapter 2. Some Classes of Codes as Invariant Subspaces

with 0 = (0,0,0), a = (1,1,0) and b = (0.1.1). This generator matrix G has been

written with respect to the basis g. When writing the rows of G with respect to the

standard basis e, the matrix takes the following form

G /I 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 '0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 10 0 0 10 0 0 0 10 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0\ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 o /

Example 2.3.2. Now we take n = 18, q = 5, k = '3, 1 = 6 and a = 2, providing us with matrices 0 0 0 0 0 2\ 1 0 0 0 0 0 B (BQ \ Be

V Be/

v

2.3. Linear Q u a s i - T w i s t e d C o d e s £is Invariant Subspaces 29

It turns out that we can write

fix) = fi{x)f2{x)f3{x) = [x^ + 2)(a;2 + ^ + 2){x'' + 4x + 2),

where the fi{x) are irreducible polynomials over GF{5).

Again we define Ut = Ker filij) for i = 1, 2,3, and we introduce the linear code

C = Ui® U2.

The defining polynomial of C is

g{x) = fi{x)f2{x) = x^ + x^ • Ax^ + 2K + 4,

from which we obtain the matrix

9{Bs) = /4 0 2 2 3 4\ 2 4 0 2 2 3 4 2 4 0 2 2 1 4 2 4 0 2 1 1 4 2 4 0 Vo 1 1 4 2 4/

The code of length 6 determined by g{x) is a constacyclic code C with respect to

2 6 GF{5) luith dimension 4 (cf. Theorem 2.2.1). Hence, the matrix g{Be) has rank 6 — 4 = 2, as one can easy verify. By taking two independent rvws, e.g. the first two, one obtains a parity check matrix for C. A generator matrix for C can be constructed

from the polynomial h{x) = f six) = x^ + ix + 2 which determines the matrix

h{Bs /2 0 0 0 2 3\ 4 2 0 0 0 2 1 4 2 0 0 0 0 1 4 2 0 0 0 0 1 4 2 0 Vo 0 0 I 4 2 /

30 Chapter 2. Some Classes of Codes as Invariant Subspaces

taking the first four columns of h{BQ) we obtain a generator matrix for C :

G c =

/ 2 4 1 0 0 0>^ 0 2 4 1 Ü Ü 0 0 2 4 1 0 Vo 0 0 2 4 l /

That this matrix really generates a constacyclic code with respect to 2, can rather

easily be verified. It is sufficient to check that ( 2 0 0 0 2 4 ) -which is the constacyclic

permutation of the last word of the matrix- is a linear combination of the first three.

Just like in Example 2.3.1, it follows that the following matrix generates the

com-plete code C :

(Gc O O^

O G-c O

\0

O

GJ

where O stands for the {i,6) — zeromatrix. The rows in this matrix are codewords of C with respect to the basis g. To obtain a generator with respect to the standard basis

e, one has to carry out the basis transformation.

Example 2.3.3. Like in Example 2.3.2 we take again n = 18, 9 = 5, fc = 3, / = 6 and 0 = 2. Now we consider the codes C\ := U\ and C2 '•= U2.

The code Ci is defined by gi{x) = fi{x) = x^ -(- 2. Similarly as in all previous

2.3. Linear Quasi-Twisted Codes as Invariant Subspaces 31 Pl(S6) = /2 0 0 0 2 0\ 0 2 0 0 0 2 1 0 2 0 0 0 0 1 0 2 Ü 0 0 0 1 0 2 0 \0 0 0 1 0 2/ and hiiBe /4 0 2 0 1 0\ 0 4 0 2 0 1 3 0 4 0 2 0 0 3 0 4 0 2 1 0 3 0 4 0 Vo 1 0 3 0 4^

Since dimCi = 2, a generator matrix G^- for Ci (the restriction of Ci with respect

to its support) is obtained by taking 2 independent columns o//ii(i?g).

The code Cg is defined by g2ix) = f2{x) = x'^ + x + 2. For this code we find the

matrices g2{Be) /2 0 0 0 2 2\ 1 2 0 0 0 2 1 1 2 0 0 0 0 1 1 2 0 0 0 0 1 1 2 0 Vo 0 0 1 1 2^

32

_{Chapter 2. Some Classes of Codes as Invariant Subspaces}

A generator matrix G-Q- for C^ can be obtained by taking 2 independent columns of

h2{Be).

Finally, the code C3 :— f/3 is defined by gsix) = fsix) = x^ + 4x + 2. This code is

the dual of C = Ci ® C2. So, the matrix gsiB^) is equal to the matrix h{BQ) presented in Example 2.3.2. Indeed, we find

/2 0 0 0 2 3\ 4 2 0 0 0 2 1 4 2 0 0 0 0 1 4 2 0 0 0 0 1 4 2 0 Vo 0 0 1 4 2^ while /4 0 2 2 3 4\ 2 4 0 2 2 3 4 2 4 0 2 2 1 4 2 4 0 2 1 1 4 2 4 0 Vo 1 1 4 2 4^^

A generator matrix GQ- for C3 is obtained by taking 2 independent columns ofh-siBa)

It will be obvious that the matrix 9i{Be)

2.3. Linear Quasi-Twisted Codes as Invariant Subspaces

Gi={ o Gc; o

\o

O

GJ

is a generator matrix for the complete code Ci, for i = 1,2.3.

One can easily check that the six rows of the matrices G,, i = 1, 2,3, are

indepen-dent. So. it follows that

F" = (7i ® f/2 e Us

(cf. Corollary 2.3.1). Furthermore, the minimal ^—invariant subspace Ui, is spanned

by the rows of the submatrix {G^j- O O). We shall denote this fact by

Ua = £{G-^^0 0), i = 1,2,3.

Similarly, we can write

Ui2=^{0G-^0), i= 1,2,3,

and

Ua = i{OOG-c;), i = 1.2,3.

It follows immediately that

Ur = Ua ® Ui2 ® Ua

and

Vj =Uij®U2j®U3j,

Chapter 3

The Case W h e n t h e Field

Characteristic Divides the

Length

3.1 Linear Cyclic Codes of Length Divisible by t h e

Field Characteristic

Let F = GF{q) a n d let F " be t h e n-dimensional vector space over F with t h e s t a n d a r d basis e i = ( 1 , 0 , . . . . 0), 62 = ( 0 , 1 , . . . , 0 ) , . . . , e„, = (0, 0 . . . . , 1).

Let

(F" —» F "

X2,--- ,Xn) I-» {x„.xi,.. . , a ; n - i ) (3.1)

T h e n ip € Horn ( F " . F " ) a n d it has t h e following m a t r i x

A = 0 0 0 0 0 0 1 0

n

Vo

l\

oJ

Chapter 3. The Case W h e n the Field Characteristic Divides the Length

with respect to the basis e = (ei,e2, • • • ,e„). The characteristic polynomial of A is

fA{x)= U i - X . . . U = (_l)"(a;" _ 1). (3.3) -X 1 0 0 0 —X 1 0 0 .. 0 .. — X . . 0 ..

• ^ 1

• °

Let us write briefly f{x) for //i(a;). We assume that (n.q) = p" = d and n = dni, where (p, ni) = 1 and p = c h a r F . Let a:"' — 1 = fi{x)... ft{x) be the factorization of x"' — 1 into irreducible monic factors over F. Then the factorization of ƒ (x) is

f(x) = (-1)"(X" - 1) = (-1)"(X"' - 1)' = {-ir[f,{x))\f2{x)Y . . . [ft{x)Y.

Furthermore, we consider the homogeneous set of equations

ƒ/(A)x = 0. X e F " (3.4)

for z = 1 f. If Ui stands for the solution space of (3.4). then we may write Ui = Kerft{^).

Theorem 3.1.1. The subspaces Ui of F" satisfy the following conditions: 1) Ui is a (f-invariant subspace of F";

2) if W is a ip—invariant subspace of F " and Wi = W (1 Ui for i = i,... ,t, then

Wi is tp—invariant and W = W\ ® • • • 0 Wf-,

3) F'' = Ui®---®Ut;

4) ff{x) is the monic polynomial of minimal degree in F[x] such that ff{A)u = 0

for all u e Ui;

5; ^ | „ (x) = i-lfdegftfd(^j.y [n particular, dim pU; = deg f^^„^{x) =ddegf,{x)

6) there exists a vector u, 6 Ui such that the vectors

Ui,ip{\li),.

form a basis of Ui;

,<pdimu,-i^^,)

7) if the vector u, is as in 6), then for each vector u G Ui there exists a polynomial

3.1. Linear Cyclic Codes of Length Divisible by t h e Field Characteristic 37

Proof. Tlie proofs of 1) and 2) are completely similar to the proofs of corresponding

proporties of Theorem 2.1.1 in Chapter 2.

3) It follows from 2) with W = F " .

4) Let mi{x) G F[x] be the monic polynomial of smallest degree such that m , ( ^ ) u = 0 for all u € Ui- By the division algorithm in F[x] there are polynomials qi{x),ri{x), such that ff{x) = mi{x)qi{x) + ri{x), where degri(a;) < Aagmi{x). Then for each vector u € ?7i we have ff{A)vi = qi{A)mi{A)u + ri{A)u and hence ri{A)u = 0. But this contradicts the choice of Tni{x) unless ri(x) is identically zero. Thus, mi{x) di-vides ff{x) for all i = 1 , . . . , i. Therefore, there exist integers 0 < Sj < d such that mi{x) = fi'{x). Set m{x) = Y\.i=i fni{x). Since m{A)u = 0 for all u 6 F". and m{x)

divides the minimal polynomial x" — 1 of A, we conclude that x " — 1 = m{x). Hence,

ft{x)...ft{x) = x''-l=fl^ix)...ft{x).

Now the statement follows from the uniqueness of the factorization of polynomials into irreducible factors.

5) Let ki = dim pUi and let / , ( i ) = /^|,,_ for i = 1 . . . . , t. Let g('^ = (gf \ . . . , g^^') be a basis of Ui over F and let At be the matrix of ip\u^ with respect to that basis.

By property 3) we obtain that g = {g\ , . . . . g [ . . . , g p , . . . , g^ ') is a basis of F" and ip is represented by the following matrix

M l \ A^

A' =

\ A J

38 Chapter 3. The Case W h e n the Field Characteristic Divides the Length

matrix from the standard basis of F " to the basis g. Then

(UM)

MA')

fiiM

= MT-'AT) = T-'MA)T.

MAt) J

Ji) _ \(i) J")

Note that fi{Ai) = O. Let g^' = A)Vei + • • • + A^^e^, j = l,...,k^. Since g}" e U, we obtain that MA)

(^\

\^l/

:

TMA')T-^'n

\^U

TMA')

I 1

where 1 is on the {kl^ 1- ki-i + j)—th position. Therefore, ffix) divides fi{x) for all i = 1 . . . . , t Let f^{x) = ff{x)gi{x). Then

fix) = Mx)... Mx) = ftix)... ft{x)gr{x). ..gtix).

It follows from the last identity that gi{x) = (-l)'^deg/j(a:)

6) Let ei = Ui + U2 + • • • + Ut for Uj € t/j, i = 1 , . . . , t Then

62= f{ei) = <p{ui) + ip{u2) H h V'(ut), 63= <^(e2) = v'^(ui) + ¥32(u2) + • • • + V'^(Ut),

3.1. Linear Cyclic Codes of Length Divisible by the Field Characteristic 39

Let V be an arbitrary vector in F " . Then

V = A i e i + A2e2 H h A „ e „ =

= A l ( U i + U2 + h Uf) + A2(¥j(Ul) + </5(U2) -i 1- IfiiUt)) + + ••• + A „ ( ^ " - i ( u i ) + 'fi"-\u2) + ••• + ^"-^(uO) =

= (AlUi + A2v3(ui) + • • • + A„(^"-i(ui)) + + • • • + (AlUj + \2<fi{ut) + ••• + A„(^"-i(ut)).

Hence, every vector v; € Ui has the form Vj = AiU, + A2i/?(ui) H h A„<^"~'(u,), i =

I,... ,t, and so Ui = l{\ii, ip{u.i),... ,</?"'^(ui)}. Since dimp-t/, = ki, the vectors

U i , v ? ( u i ) , . . . , v p ' = ' " ^ ( u i )

form a basis of Ui.

7) This follows immediately from 6). D

Theorem 3.1.2. Let U be a (p—invariant subspace of Ui for some 1 < i < t. Then there exists an integer k, 0 < k < d, such that U = Imfj'{(p^^j) = Kerff~''(f\^, ) = Kerff-\p).

Proof. Let the vector Uj € Ui be as in Theorem 3.1.1 and let us consider the set

J = {g{x)eF[x]\{g{A)){vL,)eU}.

One may easily verify that J is a principal ideal in F[x\. So. there exists a monic polynomial h{x) e F[x\ such that J =< h{x) >. We shall show that U = hn h{(p\^, ). Let u e [/. Then according to Theorem 3.1.1 u = g{A)ui for a suitable polynomial g{x) e F[x]. Since g{x) e J, then g{x) = h{x)gi{x). Hence u = {hgi){A)\ii = h{A)gi(A)ni = h{A)'Vi, where v, e Ui. Thus u e lm h{i.p^^, ) . Conversely, suppose that

40 Chapter 3. T h e Case W h e n the Field Characteristic Divides the Length

u € Im/i((/3|,^ ), i.e., u = h{A)v for some v € f/j. Hence, v = g{A)ui for a suitable polynomial g{x) e F[x] and so u = h{A)g{A)Ui = {hg){A)ui. Since h{x)g{x) G J, we conclude that u G f/.

We shall prove that h{x) = f^{x) for some G < k < d. Since / f (A)u,: = 0, it follows that ff{x) 6 J. So, h{x) divides ff{x). Since fi{x) is an irreducible polynomial, h{x) = / f (x) for some 0 < k < d. Hence. U = lmf^{ipi^^ ). We shall complete the proof by showing that U = Ker/^ ~ (</3|j^ ). We have

/ f - ^ A O / f (A,) = ff{A,) = O.

where A, is the matrix of (^|,, . Since each column of //^(A;) is a solution of the ho-mogeneous set of equations /f"*(Ai)x = 0, then U = I r a / f (<(j|^, ) C Ker ff~''{ip^^, ). One may easily verify that Ker/^^'^((^i^, ) = Kerff~''{ip). Now suppose that u e Kerff^'^iif), i.e., /f"*(A)u = 0. Hence, u e Ker ƒ/((/?) = f/, and u = ff(A)u, for a

suitable polynomial g{x) e -F[a:]. So, ff~''(A)g{A)Ui = 0. Since / / ( x ) is the minimal polynomial with the property ff(A)ui = 0, we conclude that f^{x) divides g{x). Thus g{x) e J and u e U, which proves the statement, n

Proposition 3.1.1. Let U be a (p-invariant subspace of F". Then U is a direct sum of subspaces of F " of the form. Kerf^' (ip), where 0 < Sj < d.

Proof. It follows immediately from property 2) of Theorem 3.1.1 and from

Theo-rem 3.1.2. D

The following statement is clear from the definitions.

Proposition 3.1.2. A linear code C of length n over F is cyclic iffC is a p—invariant subspace of F".

Theorem 3.1.3. Let C be a linear cyclic code of length n over F. Then the following facts hold:

3.1. Linear Cyclic Codes of Length Divisible by the Field Characteristic

1) C = Uij ® • • • (B Ui^ for some ip—invariant subspaces Ut^ = Ker f^''{(p) of F " ,

Q < Sr < d, and k := dinifC = Ylr=i Srdegfi^.{x);

2) f^ici^) = (-i)v;;'w.--C(^) = 5W;

3) c e C i£f g{A)c = 0:

4) the polynomial g(x) has the smallest degree with respect to property 3);

5) rank{g{A)) = n — k.

Proof. 1) The first part of the statement follows from Proposition 3.1.1. Now we shall show that dimf-Ker/*"" = Srdeg/j^(x). Let us consider the following chain of linear subspaces of F "

KevfiM C Kerfli^p) C • • • C Ker/^.(v') =

t^v-Since the characteristic polynomial of the restriction of ip on Kerfl{ip) is a divisor of /^|„ (x) = {-lr'^^-^J''ff^(x) for a l l / = 1 . . . . rf, we obtain for the dimensions of the respective subspaces the following inequalities for natural rmmbers

hdegfi,.{x) < hdegfi^Xx) < ••• < /ddeg/j,,(i) = rfdeg/i,.(z).

Thus /j = z for i = 1 , . . . , d, which proves the statement. In particular, it follows from the proof that f^^_ (x) = (-I)'''*»-''-/;';(a-).

2) Let us denote ki^ = dim pUi^ = s,.dog ƒ;,., r = 1 . . . m. Let (uj'' . . . u^'' ) be a basis of U,^ over F and let Bi,_ be the matrix of (yj|, with respect to that basis. Then (Uj , . . . . u[' , u / " , . . . . u^'" ) is a basis of C over F and ip\c is represented by the following matrix

42 Chapter 3. T h e Case W h e n the Field Characteristic Divides the Length

with respect to that basis. Hence,

/^icW = ^i£, w---^,- {x) = {-irf!^^{x)...f^-{x).

3) Let c € C. Then c = u^^ + • • • + u^„, for some u^;^ € f/^^, r = 1 , . . . , ÏTI, and

g{A)c = (-!)''-[(ƒ- ... f!:){A)n,, + . . • + ( ƒ - . . . f::KA)u,J = 0.

Conversely, suppose that g{A)c = 0 for some c e F " . According to Theorem 3.1.1 we have c = ui 4- ••• + u „ u,- € U,. Then g(^)c = {-lf[{ftl---rC)(^)^i + • • • + (//'i' • • • /*j:)(^)"tl = 0, so that (/(A)[uj, + • • • + Uj,] = 0, where { j i , . . . j j = { 1 . . . . , i } \ { 2 i , . . . ,im}. Let VjY = g{A)\ij^ for r = 1 , . . . , / . Hence, Vj^ € t/^,. and Vji + ••• + Vj, = 0 . Therefore, v^v = 0 for r = 1 , . . . / . Since (3(2;),//,.(a^)) = 1, there are polynomials a{x),h{x) e F[x\ so that a(x)p(j:) + b{x)ff^{x) = 1. Hence Uj^ = a{A)g{A)uj^ + b(A)ff{A)uj^ = 0 and c = Uj, H 1- u^^ e C

4) Suppose that h{x) e F[x\ is a nonzero polynomial of smallest degree such that b{A)c = 0 for all c € C By the division algorithm in F[x] there are polynomials q{x),r{x) such that g{x) = b{x)q{x) + r(x), where degr(a;) < deg6(x). Then for each

vector c € C we have g(A)c = q{A)b{A)c + r{A)c and hence, r{A)c = 0. But this contradicts the choice of b{x) unless r{x) is identically zero. Thus, b(x) divides g{x). If deg b{x) < deg g{x), then b(x) is a product of some of the irreducible factors of g(x) and without loss of generality we can assume that b{x) = ( — 1)*^*! +'•+'=<t f^^ . • • fi^ and t < m. Let us consider the code C" = V,, © • • • ® Vi, C C. Then b{x) = f^\^, and

by the equation g{A)c = 0 for all c € C we obtain that C C C'. This contradiction proves the statement.

3.1. Linear Cyclic Codes of Length Divisible by the Field Characteristic 43

5) By property 3) C is the solutions space of the homogeneous set of equations g{A)'x. = 0. Then dinif C = k = n — Teink{g{A)). which proves the statement. D

Proposition 3.1.3. The matrix H, the rows of which constitute an arbitrary set of n — k linearly independent rows of g(A). is a parity check matrix of C.

Proof. The proof follows from the equation g{A)c = 0 for every vector c e C and

from the fact that rank (ff(A)) = n — A:. D

h{x)

{'\T-^ft'\x)...ft'^{x\

Let

ƒ(£)

where 0 < s,. < rf for r = 1 , . . . , i.

Let ( g ( , , . . . , g(„_t) be a basis of C^, where g;^ is a Z^^th vector row of g{A). By the equation giyA)h{A) = O we obtain that (g(,.,hi) = 0 for each i = 1 , . . . ,n, r = \.... .n — k. The last equation gives us that the columns hj of h{A) are codewords in C.

We shall show that rank {h{A)) = k. By the inequality of Sylvester we obtain that rank(O) = 0 > rank(p(A)) + rank(/i(A)) - n. Thus rank{h{A)) <n- rank{g{A)) = n — (n — k) = k. On the other hand the inequality of Sylvester, applied to the product h{A) = {-ir-''ft-'^iA)...f^-''(A), gives us that iank{h{A)) > r{ff (A)) + ••• + riff-"'(A)) - n{t - 1) = ni - d ^ L i ^egfi + E ' = I s,dcgfi - nt + n = k. Therefore rank(/i(y4)) = k. Thus we have proved the following proposition.

Proposition 3.1.4. The matrix G, the rows of which constitute an arbitrary set of k linearly independent rows of {h{A)) , is a generator matrix of the code C.

Let fip\ ^ [x] = h{x). By Theorem 3.1.3 it follows that h{x) is the polynomial of the smallest degree such that h{A)\i = 0 for every u 6 C""". Let h*{x) = h{x)q{x) + r(x), where degr(a:) < degh{x). Then h*{A) = A"-''{h{A*)) = h{A)q{A) + r{A), hence for every vector u € C ^ the assertion A"^''{h{A)) u = q{A)h{A)u + r{A)u holds, so that r{x) = 0. Thus h{x) divides h*{x). Since both are polynomials of the same degree. h*{x) = ah{x). where o G F is the leading coefficient of the product

(/r(a:))''-^'...(/;(x)r-^'.Thus

hix) = ^h*ix) = {-ir-''-{f^{x))

a a d—si

_{• • ( / ; w )}