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formerly Central European Journal of Mathematics

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Volume 15, Issue 1 (Aug 2017)

Issues

Monomial codes seen as invariant subspaces

María Isabel García-Planas / Maria Dolors Magret / Laurence Emilie Um
Published Online: 2017-08-23 | DOI: https://doi.org/10.1515/math-2017-0093

Abstract

It is well known that cyclic codes are very useful because of their applications, since they are not computationally expensive and encoding can be easily implemented. The relationship between cyclic codes and invariant subspaces is also well known. In this paper a generalization of this relationship is presented between monomial codes over a finite field 𝔽 and hyperinvariant subspaces of 𝔽n under an appropriate linear transformation. Using techniques of Linear Algebra it is possible to deduce certain properties for this particular type of codes, generalizing known results on cyclic codes.

Keywords: Monomial codes; Invariant subspaces

MSC 2010: 94B15; 15B33

1 Introduction

It is well known that error correcting codes and cryptographic systems have conflicting objectives, since the first are codes protecting the information of occasional errors due to handling, that is, those searching to solve the difficulties posed by unreliability of the channel, but cryptosystems, also called secret codes, try to ensure its confidentiality, integrity and security. However, they also have complementary objectives. The difficulty for decoding error correcting codes has been used to build cryptographic systems from these codes. Among these systems there is the well known public key McEliece system. In this system the private key of each user is the generator matrix G of a linear code C over a finite field 𝔽q joint with a decoding algorithm. The matrix G is hidden by a permutation matrix thus obtaining the public key, ([1], [2], [3]).

Alongside the use of cryptography to protect communications, there is the technique known as “steganography”whose use is increasing and which consists in the concealment of information. It is used in order to protect information in an anodyne numerical support and it is the support that is sent over a public transmission channel. These techniques of concealment of information are based on cyclic codes over the ring ℤ4, ([4], [5]). Possibly they can improve the efficiency in dissimulation using a generalization of cyclic codes such as the monomial codes.

A first generalization of cyclic codes were constacyclic codes, introduced by E. R. Berlekamp in [6]. Monomial codes are a broader generalization. Linear algebra as a tool to study such codes was introduced in [7]. Monomial codes are widely used because they can be encoded with shift registers.

Let p be a prime number, q = pk for some k ≥ 1. A monomial q-ary code of length n can be defined through a n × n generator-matrix with the property that each row (except the last one) (c1, c2,..., cn), ciGF(q) defines the row as (ancn, a1c1, a2c2,..., an-1cn-1), where a1,..., an are certain fixed elements of GF(q)\{0}. Cyclic codes (a1 =... = an = 1) and constacyclic codes (a1 =... = an-1 = 1) are special subclasses of monomial codes of GF(q)n. Monomial codes can also be described in terms of linear algebra, which constitutes our starting point that will be the characteristic polynomial of the endomorphism of GF(q)n whose matrix in the canonical basis is the one representing the monomial code.

Recall that, given an endomorphism ϕ of a 𝔽-vector space E, a ϕ-invariant subspace VE is hyperinvariant when it is invariant under all linear transformations commuting with ϕ.

2 Invariant subspaces of monomial matrices

Let p be a prime number, q = pk for some k ≥ 1 and 𝔽 = GF(q) and 𝔽n thee n-dimensional 𝔽-vector space.

Let ā = (a1,... an) be a set of n parameters of 𝔽 and consider the following linear map φa¯:FnFn(x1,...,xn)(anxn,a1x1,...,an1xn1)(1) whose associated matrix with respect to the canonical basis {e1 = (1, 0,..., 0), e2 = (0, 1,..., 0), en = (0, 0,..., 1)} is: Aa¯=(000ana10000a20000an10).(2) This matrix is called a monomial matrix. We note that this matrix can be written as the product of a diagonal matrix diag (an, a1,..., an-1) and the permutation matrix (0001100001000010).

Properties

This matrix verifies:

  1. Aa¯n=a1...anIn

  2. ifi=1nai0thenAa¯1=1i=1naiAa¯n1=Aa¯¯t,wherea¯¯=(1a1,...,1an).

  3. its characteristic polynomial is pa(s)=det(Aa¯sIn)=(1)n(sni=1nai).

Proposition 2.1

Suppose that a=i=1nai0. Then, the matrix (2) is equivalent under similarity to Aa=(000a100001000010)

Proof

It is easy to prove that Aa¯S=SAa. with =(000i=1naia10000a1a20000i=1n1ai0)

In this section we prove that a ϕā-invariant subspaces are also ϕā-hyperinvariants; that is to say, invariant under all linear maps commuting with ϕā, (see [8] and [9] for more information about these subspaces).

We need to know the centralizer of Aā. To do that, we first calculate the centralizer of the matrix Aa.

Proposition 2.2

([7). ] The centralizer 𝒞(Aa) is the set of the matrices Xa in the form: Xa=(xnax1ax2ax3axn2axn1xn1xnax1abx2axn3axn2x3x4x5x6ax1ax2x2x3x4x5xnax1x1x2x3x4xn1xn)

Proposition 2.3

The centralizer 𝒞(Aā) of Aā is the set of the matrices Yā = SXaS-1, if AāS = SAa.

Proof

Proposition 2.2, we have XaAa = AaXa. Then, SXaS-1Aā = AāSXaS-1. ☐

Note that if v = (v1,..., vn) is an eigenvector of Aā, then: anvn=λv1a1v1=λv2a2v2=λv3an2vn2=λvn1an1vn1=λvn(3) In particular, we have that v=(λn1a1...an1,λn2a2...an1,...,λan1,1)(4) and the following Proposition holds.

Proposition 2.4

Let λGF(q)* be an element such that λn=i=1nai. Then, the one-dimensional subspace [v] spanned by the vector v given in (4) is an hyperinvariant subspace.

Proof

Aa¯v=λv and given any Yā ∈ 𝒞(Aā), then Ya¯v=S(xnI+xn1Aa¯+xn2Aa¯2+...+x1Aa¯n1)S1v=xnv+xn1SAaS1v+xn2SAa2S1v+...++x2SAan2S1v+x1SAa¯n1S1v=xnv+xn1λv+xn2λ2v+...+x1λn1v=αv with α = xn + xn-1λ + x2λ2 + ... + x2λn-2 + x1λn-1 ∈ 𝔽. ☐

Proposition 2.5

Let F be an invariant subspace of Aā. Then, F is hyperinvariant.

Proof

It suffices to observe that, for all Yā ∈ 𝒞(Aā), SXaS1=xnI+xn1Aa¯+xn2Aa¯2+...+x1Aa¯n1.

Therefore, in this case the lattice of invariant subspaces coincides with the lattice of hyperinvariant subspaces: Hinv(Aa¯)=Inv(Aa¯).

Definition 2.6

  1. Let u = (u1,..., un) and v = (v1,..., vn) be two vectors in 𝔽n. We define an inner product over 𝔽 as follows: <u,v>=u1v1+...+unvn.

  2. Two vectors u, v in 𝔽n are said to be orthogonal if < u, v >= 0.

  3. Let F be a subspace of 𝔽n. The dual subspace of F (denoted by F) is F={vFn|uF,<u,v>=0}.

Proposition 2.7

Let F be an invariant subspace of Aā. Then F is an invariant subspace of Aa¯¯1.

Proof

Let vF. For all uF (consequently, AāuF) we have 0=<Aa¯u,v>=<u,Aa¯tv>=<u,i=1naiAa¯¯n1v>=<u,Aa¯¯1v> Thus Aa¯¯1vF.

Let 𝔽[λ1,..., λn] be the algebraic extension of 𝔽 = GF(q) and let λ1,..., λn be the eigenvalues of ϕā with λi=i=1nnλi, i = 1,..., n, where λ is a primitive nth root of unity and i=1nain is a fixed, but otherwise arbitrary zero of the polynomial sni=1nai,where0i=1naiF.

Let vi, i = 1,..., n be the respective eigenvectors. More particularly we have Aa¯vi=λivi,vi=(λin1a1...an1,λin2a2...an1,...,λian1,1), i = 1,..., n,

where Aā is the matrix associated to φa¯:F[λ1,...,λn]nF[λ1,...,λn]n (defined as in 2).

Let us consider the basis v = (v1,..., vn) of eigenvectors of ϕā. Applying basis change to Aā, we obtain the following diagonal matrix Da¯=(λ1000λ2000λn)=S1AS and taking into account (4) we have S=(λ1n1a1...an1λ2n1a1...an1λnn1a1...an1λ1n2a2...an1λ2n2a2...an1λnn2a2...an1λ1an1λ2an1λnan1111)

We define now the following vectors: ui=(λia1...an1,λi2a2...an1,...,λn1an1,λin),1in(5)

Proposition 2.8

The set of vectors defined in (5) verify the following relationship. <ui,vj>={a1...annifi=j0ifij

Proof

<ui,vj>==1nλiλjn==1nλiλjnλjλj==1n(λiλj)λjn=a1...an=1n(λiλj)={a1...an=1n1=a1...annifi=ja1...an=1n(λ)=0(withλarootofunit)ifij

From this Proposition the inverse matrix of the matrix S can easily obtained.

3 Monomial codes as invariant subspaces

Definition 3.1

A code C of length n over the field 𝔽 is called monomial with respect to a1, ..., an, if whenever c = (c1,..., cn) belongs to C, then sc = (ancn, a1c1,..., an-1cn-1) is also in C.

The shift (the map csc) can be represented in a matrix form (000ana10000a20000an10)(c1c2c3cn)=(ancna1c1a2c2an1cn) Note that this matrix is the matrix (2).

In the particular case where ai = 1, for all i, the code is a cyclic code and if a1 = ... = an-1 = 1 is a constacyclic code (see [10]).

Applying Proposition 2.1 the study can be reduced to the case of constacyclic codes. Nevertheless, we will not make use of this result, but directly consider monomial codes.

We are interested in the case where anai, for some i = 2,..., an-1 and i=1nai0. In particular, we need to consider q > 2. As an immediate consequence of Definition 3.1 we have the following Proposition.

Proposition 3.2

A linear code C with length n over the field 𝔽 is monomial if, and only if, C is an Aā-invariant subspace of 𝔽n.

And after Proposition 2.5 we have the following result.

Proposition 3.3

A linear code C with length n over the field 𝔽 is monomial if, and only if, C is an Aā-hyperinvariant subspace of 𝔽n.

Suppose now that (n, q) = 1 and pa¯(t)=(1)n(tni=1nai) has no multiple roots and splits into distinct irreducible factors.

General Linear Algebra theory over finite fields yields the following statement.

Proposition 3.4

Let C be a monomial code, and pa¯(s)=(1)npa¯1(s)...pa¯r(s) the decomposition of pā(s) into irreducible factors. Then C = Ker pāi1 (Aā) ⊕ ... ⊕ Ker pāis (Aā) = Ker h(Aā), h(s) = pāi1 (s) : . : pāit (s) for some minimal Aā-invariant subspaces Ker pāij (Aā) de 𝔽n.

Example 3.5

Consider the matrix Aā with an = 2, a1 = 4, a2 = ... = an-1 = 1, n = 8 and q = 5. Then p(s) = pā(s) = s8 - 1. Factorizing p(s) into irreducible factors over 𝔽 = GF(5) we have pā(s) = p1(s)p2(s)p3(s)p4(s)p5(s)p6(s) = (s + 1)(s + 2)(s + 3)(s + 4 )(s2 + 2 )(s2 + 3). The factors pi(s) define minimal Aā-invariant subspaces, Fi = Ker pi(Aā), for i = 1, 2, 3, 4, 5, 6.

Let us consider C=F1F5=Ker(p1(Aa¯))Ker(p5(Aa¯))Ker(Aa¯3+Aa¯2+2Aa¯+2I)=Ker(2000013322000034242000034442000003442000003442000003442000003442) This is a monomial code, C = Kerh(Aā) with h(s) = p1(s)p5(s) = s3 + s2 + 2s + 2.

Example 3.6

With the notations as in the Example above, g(s)=Pa¯(s)h(s)=p2(s)p3(s)p4(s)p6(s)=s5+9s4+29s3+53s2+78s+72 It is straightforward to check that (Aa¯5+9Aa¯4+29Aa¯3+53Aa¯2+78Aa¯+72I)(v1)=0(Aa¯5+9Aa¯4+29Aa¯3+53Aa¯2+78Aa¯+72I)(v2)=0(Aa¯5+9Aa¯4+29Aa¯3+53Aa¯2+78Aa¯+72I)(v3)=0 with v1=(1,4,2,1,3,4,2,1),v2=(1,0,4,0,2,0,1,0),v3=(0,3,0,4,0,2,0,1).

Corollary 3.7

Let C be a monomial code. There exists g(s) verifying pā(s) = g(s) · h(s) with gcd(g(s), h(s)) = 1 such that g(Aā)c = 0, ∀cC.

Considering the inner product introduced in definition 2.6.

Proposition 3.8

Let C be a monomial code with respect a1, ..., an. Then, its dual code Cis a monomial code with respect 1a1,...,1an.

Proof

The statement follows from Proposition 2.7. ☐

In the case a1 = ... = an = 1 we obtain the well known result about cyclic codes.

Corollary 3.9

The dual of a cyclic code is a cyclic code.

4 Parity matrices of monomial codes

Let 𝔽[λ1,..., λn] be the algebraic extension considered in Section 2.

Let C be a monomial code and g(s) as in corollary 3.7. Let us consider a basis v = (v1,..., vn) of eigenvectors of ϕā.

In this basis, the matrix of ϕā is a diagonal matrix, which will be denoted by Dā.

Since Dā is a diagonal matrix, the matrix g(Dā) is also diagonal and g(Aa¯)=g(SDa¯S1)=Sg(Da¯)S1 Condition g(Aā)c = 0 is equivalent to g(Dā)c′ = 0 where c′ = S-1c.

Without loss of generality we can assume that λ1,..., λn are ordered in such a way that g(λi) = 0, for all 1 ≤ ik and g(λi) = αi ≠ 0, for all k + 1 ≤ in. With h(s) as in corollary 3.7, h(λi) ≠ 0, for all 1 ≤ ik and h(λi) = 0, for all k + 1 ≤ in. Given c = (c1,..., cn) ∈ C and c=(c1,...,cn)=s1cwehavethatg(Da¯)=(0,...,0,αk+1ck+1,...,αncn). Equivalently: 1a1...ann(00αk+1αn)(λ1a1an1λ12a2an1λ1n1an1λ1nλ2a1an1λ22a2an1λ2n1an1λ2nλna1an1λn2a2an1λnn1an1λnn)(c1c2cn)=1a1...an1n(00αk+1λk+1a1...an1αk+1λk+1nαnλna1...an1αnλnn)(c1c2cn)=0 Then we can deduce the following proposition.

Proposition 4.1

Let uij, 1 ≤ jr = nk be a family of vectors as in (5) corresponding to λij, with g(λij) = αij ≠ 0. Then c is a codeword of the monomial code C if and only if uijc=0,forall1jr. As a consequence the matrix A = (uij) ∈ M(n-kn(𝔽[λ1,..., λn]) is a parity matrix of the monomial code over the field 𝔽[λ1,..., λn].

Example 4.2

Over 𝔽 = GF(5) we consider a monomial code C with an = a1 = 2, a2 = ... = a-1 = 1 and n = 4 defined by g(s) = s2 − 2. Over F[2,42,3,43],thepolynomialh(s)=s2+2=(s3)(s43).

Then (323334333234) is a parity matrix of the code C over F[2,42,3,43].

5 Hamming distance of monomial codes

Remember that the Hamming weight (for short, weight) of a vector v is the number of its nonzero entries and is denoted wH(v). We have wH(x) = dH(x, 0). The minimum weight of a code C is the minimum nonzero weight among all codewords of C, wmin(C)=min0xC(wH(x)) Taking into account that dH(x, y) = dH(xz, yz) for all z and that in particular dH(x, y) = dH(xy, yy) = dH(xy, 0) we have that over a field, the Hamming distance is translation invariant and, in particular, for linear codes, the minimum weight is equal to the minimum distance.

We are going to obtain a bound for the minimum distance of two parametric monomial codes in a similar way to that presented by Roos in [11] for cyclic codes.

Let 𝔽 be a finite field and A=(a1...an)=(a11an1a1an).

Let C be a linear code over 𝔽 having A as a parity matrix and dH(A) the minimum distance of C.

Remember that dH(A) = d if and only if every set of d − 1 columns is linearly independent and some set of d columns of A is linearly dependent.

For any matrix X=(x11x1nxm1xmn) with nonzero columns xi ∈ 𝔽m for 1 ≤ in we consider A(X)=(x1a1xnan) The following result is well known due to Roos ([12]).

Lemma 5.1

If dH(A) ≥ 2 and every m × (m + dH(A) − 2) submatrix of X has full rank, then dH(A(X)) ≥ dH(A) + m − 1.

Definition 5.2

Let M = [λi1,..., λil] be a set ofroots of Sni=1nai in 𝔽[λ1,..., λn]. We will say that M is a consecutive set of length ℓ, if there exists a primitive n-root of the unit λ and an exponent i such that M=[i=1nainλi,...,i=1nainλi+1].

Definition 5.3

  1. Let Λ = [λj1,..., λj] be a set of zeros of the polynomial sni=1nai. We define the matrix AΛ=(λj1a1an1λj12a2an1λj1nλja1an1λj2a2an1λjn)M×n(F[λ1,...,λn]).

  2. Let U = [x1,..., xm] be a set of consecutive zeros of the polynomial sn − 1. We define the matrix XU=(x1x12x1nxmxm2xmn)Mm×n(F[λ1,...,λn]).

Let C be the monomial code defined by the polynomial pā(s) = g(s) · h(s) over the splitting field 𝔽[λ1,..., λn] of pā(s) and consider now as Λ the set of all zeros of h(s). Following 4.1 the matrix AΛ is a parity matrix of the code C, if the minimum distance of C over 𝔽[λ1,..., λn] is dH(AΛ). Then, the minimum distance of C over 𝔽 is at least dH(AΛ), since C over 𝔽 is a subfield subcode of C over 𝔽[λ1,..., λn].

Remark 5.4

Notice that the minors of AΛ a are of Vandermonde type.

Theorem 5.5

Let Λ be the set defined in 5.3 and U be a consecutive set of roots of sn − 1 such that dH(AΛ)−2 ≥ 0. Then, dH(AΛ(XU)) ≥ dH(Aλ) + card U − 1.

Proof

It suffices to observe that in this particular setup dH(AΛ) ≥ 2, then we can apply Lemma 5.1. ☐

As a Corollary we obtain the following result.

Theorem 5.6

Let C be a monomial code of length n over 𝔽, and pā = g(s)h(s). For some integers ℓ, m ≥ 1, suppose that h(s) has a string of m consecutive zeros: h(λ) = h(λℓ+1) = ... = h(λℓ+m-1) = 0. Then, the minimum distance of C is at least d.

Example 5.7

Let n = 9, q = 7, an = 2, a1 = 3, a2 = ... = an-1 = 1 and let α be a 18th-primitive root of unity. Taking into account that (x18 − 1) = (x9 − 1)(x9 + 1), α is a root of x9 + 1 and α2 = β is a primitive root of x9 − 1. We want to classify the zeros with respect to the various irreducible polynomial divisors of x9 + 1. We will determine the cyclotomic cosets of 7 modulo 18 containing the odd integers: C1 = [ 1, 7, 13], C3 = [3], C5 = [5, 17, 11], C9 = [9], C15 = [15].

Let the zeros of h(s) be αi with iC1C5, so h(s) = (sα)(sα7)(sα13)(sα5)(sα17)(sα11). Given that βi = αβi = α2i+1 the zeros of h(s) can be written as β2, β3; β5,β6; β8, β9. Since h(s) has a string of two consecutive zeros. Then, the two parametric monomial code has a minimum distance d ≥ 3.

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About the article

Received: 2016-05-02

Accepted: 2017-05-17

Published Online: 2017-08-23


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0093.

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© 2017 García-Planas et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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