Abelian Codes Research Articles

Complementary dual abelian codes in group algebras of some finite abelian groups

Complementary dual abelian codes in group algebras of some finite abelian groups

Duadic codes over Fp + uFp + vFp + uvFp

Duadic codes constitute a well-known class of cyclic codes. In this paper, we study the structure of duadic codes of length n over the ring R = Fp + uFp + vFp + uvFp, u2 = v2 = 0, uv = vu, where p is prime and (n, p) = 1. These codes have been studied here in the setting of abelian codes over R, and we have used Fourier transform and idempotents to study them. We have characterized abelian codes over R by studying their torsion and residue codes. It is shown that the Gray image of an abelian code of length n over R is a binary abelian code of length 4n. Conditions for self-duality and self-orthogonality of duadic codes over R are derived. Some conditions on the existence of self-dual augmented and extended codes over R are presented. We have also studied Type II self-dual augmented and extended codes over R. Some results related to theminimum Lee distances of duadic codes over R are presented. We have also presented a sufficient condition for abelian codes of the same length over R to have the same minimum Hamming distance. Some optimal binary linear codes of length 36 and ternary linear codes of length 16 have been obtained as Gray images of duadic codes of length 9 and 4, respectively, over R using the computational algebra system Magma.

Duadic codes over <inline-formula><tex-math id="M1">$ \mathbb{Z}_4+u\mathbb{Z}_4 $</tex-math></inline-formula>

In this paper, we study the structure of duadic codes of an odd length $ n $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $, $ u^2 = 0 $, (more generally over $ \mathbb{Z}_{q}+u\mathbb{Z}_{q} $, $ u^2 = 0 $, where $ q = p^r $, $ p $ a prime and $ (n, p) = 1 $) using the discrete Fourier transform approach. We study these codes by considering them as a class of abelian codes. Some results related to self-duality and self-orthogonality of duadic codes are presented. Some conditions on the existence of self-dual augmented and extended duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $ are determined. We present a sufficient condition for abelian codes of the same length over $ \mathbb{Z}_4+u\mathbb{Z}_4 $ to have the same minimum Hamming distance. A new Gray map over $ \mathbb{Z}_4+u\mathbb{Z}_4 $ is defined, and it is shown that the Gray image of an abelian code over $ \mathbb{Z}_4+u\mathbb{Z}_4 $ is an abelian code over $ \mathbb{Z}_4 $. We have obtained five new linear codes of length $ 18 $ over $ \mathbb{Z}_4 $ from duadic codes of length $ 9 $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $ through the Gray map and a new map from $ \mathbb{Z}_4+u\mathbb{Z}_4 $ to $ \mathbb{Z}_4^2 $.

Self-Dual 2-Quasi Abelian Codes

A class of self-dual quasi-abelian codes of index 2 over any finite field <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> is introduced. By counting the number of such codes and the number of the codes in this class whose relative minimum weights are small, such codes are proved to be asymptotically good provided −1 is a square in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> . Moreover, a class of self-orthogonal quasi-abelian codes of index 2 is defined; and such codes always exist. In a way similar to that for self-dual quasi-abelian codes of index 2, it is proved that these self-orthogonal quasi-abelian codes of index 2 are asymptotically good.

On equivalent classes of minimal Abelian codes

Let Fq be a finite field and G a finite abelian group. An abelian code is an ideal of Fq[G]. Two abelian codes c1 and c2 of Fq[G] are equivalent if there exists an automorphism of G whose linear extension to Fq[G] maps c1 onto c2. MacWilliams determined the number of equivalent classes of minimal abelian codes (minimal ideals) in F2[G] for cyclic group G of odd cardinality. Miller claimed that MacWilliams' result remains true in general, which is however not correct as pointed out by Ferraz, Guerreiro and Polcino Milies. In this paper we completely determine the number of equivalent classes of minimal abelian codes for any Fq[G].

On ⁎-clean group rings over finite fields

A ring R is called clean if every element of R is the sum of a unit and an idempotent. Motivated by a question proposed by Lam on the cleanness of von Neumann Algebras, Vaš introduced a more natural concept of cleanness for ⁎-rings, called the ⁎-cleanness. More precisely, a ⁎-ring R is called a ⁎-clean ring if every element of R is the sum of a unit and a projection (⁎-invariant idempotent). Let F be a finite field and G a finite abelian group. In this paper, we introduce two classes of involutions on group rings of the form FG and characterize the ⁎-cleanness of these group rings in each case. When ⁎ is taken as the classical involution, we also characterize the ⁎-cleanness of FqG in terms of LCD abelian codes and self-orthogonal abelian codes in FqG.

On the number of non-G-equivalent minimal abelian codes

Let $G$ be a finite abelian group. Ferraz, Guerreiro and Polcino Milies prove that the number of $G$-equivalence classes of minimal abelian codes is equal to the number of $G$-isomorphism classes of subgroups for which corresponding quotients are cyclic. In this article, we prove that the notion of $G$-isomorphism is equivalent to the notion of isomorphism on the set of all subgroups $H$ of $G$ with the property that $G/H$ is cyclic. As an application, we calculate the number of non-$G$-equivalent minimal abelian codes for some specific family of abelian groups. We also prove that the number of non-$G$-equivalent minimal abelian codes is equal to number of divisors of the exponent of $G$ if and only if for each prime $p$ dividing the order of $G$, the Sylow $p$-subgroups of $G$ are homocyclic.

On the dimension of ideals in group algebras, and group codes

Several relations and bounds for the dimension of principal ideals in group algebras are determined by analyzing minimal polynomials of regular representations. These results are used in the two last sections. First, in the context of semisimple group algebras, to compute, for any abelian code, an element with Hamming weight equal to its dimension. Finally, to get bounds on the minimum distance of certain MDS group codes. A relation between a class of group codes and MDS codes is presented. Examples illustrating the main results are provided.

A New Approach to the Berlekamp-Massey-Sakata Algorithm: Improving Locator Decoding

We study the problem of the computation of Groebner basis for the ideal of linear recurring relations of a doubly periodic array. We find a set of indexes such that, along with some conditions, guarantees that the set of polynomials obtained at the last iteration in the Berlekamp-Massey-Sakata algorithm is exactly a Groebner basis for the mentioned ideal. Then, we apply these results to improve locator decoding in abelian codes.

Some arithmetical properties of cyclotomic cosets and their applications

The q-orbits of the permutation μq of 1+rZnr, denoted by (1+rZnr)∕μq, are characterized by using Möbius inversion formula and the Generalized Chinese Remainder Theorem. Then the (−ph)-orbits of the permutation μ−ph of (1+rZnr)∕μq are characterized and enumerated. As applications, the explicit enumeration formulas for constacyclic codes concerned with ph-dualities are obtained, and several classes of ph-LCD MDS constacyclic codes are constructed. These characterizations are generalized to the q-cyclotomic cosets modulo (n1,…,nr) which find applications in studying abelian codes.

Constructions of Abelian Codes Multiplying Dimension of Cyclic Codes

In this note, we apply some techniques developed in Bernal et al. (IEEE Trans Inf Theory 62(2):655–668, 2016; Adv Math Commun 10:459–474, 2016; IEEE Trans Inf Theory 2018. https://doi.org/10.1109/TIT.2018.2868446) to give a particular construction of bivariate Abelian Codes from cyclic codes, multiplying their dimension and preserving their apparent distance. We show that, in the case of cyclic codes whose maximum BCH bound equals its minimum distance the obtained abelian code verifies the same property; that is, the strong apparent distance and the minimum distance coincide. We finally use this construction to multiply Reed–Solomon codes to abelian codes.

The Shift Bound for Abelian Codes and Generalizations of the Donoho-Stark Uncertainty Principle

Let $G$ be a finite abelian group. If $f: G\rightarrow {\mathbf{C}} $ is a nonzero function with Fourier transform $\hat {f}$ , the Donoho-Stark uncertainty principle states that $| {\mathrm{ supp}}(f)|| {\mathrm{ supp}}(\hat {f})|\geq |G|$ . The purpose of this paper is twofold. First, we present the shift bound for abelian codes with a streamlined proof. Second, we use the shifting technique to prove a generalization and a sharpening of the Donoho-Stark uncertainty principle. In particular, if $f: G\rightarrow {\mathbf{F}} $ is a non-zero function from $G$ to a field ${\mathbf{F}}$ , and if $f$ has a Fourier transform $\hat {f}$ , the sharpened uncertainty principle states that $| {\mathrm{ supp}}(f)|| {\mathrm{ supp}}(\hat {f})|\geq |G|+| {\mathrm{ supp}}(f)|-|H({\mathrm{ supp}}(f))|$ , where $H({\mathrm{ supp}}(f))$ is the stabilizer of ${\mathrm{ supp}}(f)$ in $G$ .

LCD and Self-Orthogonal Group Codes in a Finite Abelian $p$ -Group Algebra

Let $\Bbb F_{q}$ be a finite field with $q$ elements and $p$ be a prime with $\gcd (p,q)=1$ . Let $G$ be a finite abelian $p$ -group and $\Bbb F_{q}(G)$ be a group algebra. In this paper, we find all primitive idempotents and minimal abelian group codes in the group algebra $\Bbb F_{q}(G)$ . Furthermore, we give all LCD abelian codes (linear code with complementary dual) and self-orthogonal abelian codes of $\Bbb F_{q}(G)$ .

Self-dual and complementary dual abelian codes over Galois rings

Self-dual and complementary dual cyclic/abelian codes over finite fields form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, abelian codes over Galois rings are studied in terms of the ideals in the group ring ${ GR}(p^r,s)[G]$, where $G$ is a finite abelian group and ${ GR}(p^r,s)$ is a Galois ring. Characterizations of self-dual abelian codes have been given together with necessary and sufficient conditions for the existence of a self-dual abelian code in ${ GR}(p^r,s)[G]$. A general formula for the number of such self-dual codes is established. In the case where $\gcd(|G|,p)=1$, the number of self-dual abelian codes in ${ GR}(p^r,s)[G]$ is completely and explicitly determined. Applying known results on cyclic codes of length $p^a$ over ${ GR}(p^2,s)$, an explicit formula for the number of self-dual abelian codes in ${ GR}(p^2,s)[G]$ are given, where the Sylow $p$-subgroup of $G$ is cyclic. Subsequently, the characterization and enumeration of complementary dual abelian codes in ${ GR}(p^r,s)[G]$ are established. The analogous results for self-dual and complementary dual cyclic codes over Galois rings are therefore obtained as corollaries.

On $\sigma$ -LCD Codes

Linear complementary pairs (LCPs) of codes play an important role in armoring implementations against sidechannel attacks and fault injection attacks. One of the most common ways to construct LCP of codes is to use Euclidean linear complementary dual (LCD) codes. In this paper, we first introduce the concept of linear codes with o complementary dual (σ-LCD), which includes known Euclidean LCD codes, Hermitian LCD codes, and Galois LCD codes. Like Euclidean LCD codes, σ-LCD codes can also be used to construct LCP of codes. We show that for q 2, all q-ary linear codes are σ-LCD, and for every binary linear code C, the code {0} × C is σ-LCD. Furthermore, we study deeply σ-LCD generalized quasi-cyclic (GQC) codes. In particular, we provide the characterizations of σ-LCD GQC codes, self-orthogonal GQC codes, and self-dual GQC codes, respectively. Moreover, we provide the constructions of asymptotically good σ-LCD GQC codes. Finally, we focus on σ-LCD abelian codes and prove that all abelian codes in a semisimple group algebra are σ-LCD. The results derived in this paper extend those on the classical LCD codes and show that σ-LCD codes allow the construction of LCP of codes more easily and with more flexibility.

From ds-Bounds for Cyclic Codes to True Minimum Distance for Abelian Codes

In this paper, we develop a technique to extend any bound for the minimum distance of cyclic codes constructed from its defining sets (ds-bounds) to Abelian (or multivariate) codes through the notion of ${\mathbb B}$ -apparent distance. We also study conditions for an Abelian code to verify that its ${\mathbb B}$ -apparent distance reaches its (true) minimum distance. Then, we construct some codes as an application.

Implementation of an algorithm to compute the strong apparent distance of bivariate codes

The BCH bound is the oldest lower bound for the minimum distance of a cyclic code. The study of this bound and its generalizations are classical topics, which includes the study of the very well-known family of BCH codes. In 1970, P. Camion extended the notion of BCH bound to the family of abelian codes by introducing the apparent distance of polynomials. Camion showed that the minimum value of the apparent distance of certain polynomials associated to codewords is less than or equal to the minimum distance of the code. The mentioned minimum value is known as the apparent distance of an abelian code. In 2016, Bernal-Bueno-Simón introduced the notion of strong apparent distance of polynomials and hypermatrices and developed an algorithm to compute the minimum strong apparent distance of a hypermatrix based on g-orbits manipulations. In this work, we will present the implementation of an algorithm to compute the strong apparent distance of bivariate codes.

Information Sets From Defining Sets for Reed–Muller Codes of First and Second Order

Reed–Muller codes belong to the family of affine-invariant codes. As such codes, they have a defining set that determines them uniquely, and they are extensions of cyclic group codes. In this paper, we identify those cyclic codes with multidimensional abelian codes and we use the techniques introduced by Bernal and Simon to construct information sets for them from their defining set. For first- and second-order Reed–Muller codes, we describe a direct method to construct information sets in terms of their basic parameters.

Hermitian self-dual quasi-abelian codes

Quasi-abelian codes constitute an important class of linear codes containing theoretically and practically interesting codes such as quasi-cyclic codes, abelian codes, and cyclic codes. In particular, the sub-class consisting of 1-generator quasi-abelian codes contains large families of good codes. Based on the well-known decomposition of quasi-abelian codes, the characterization and enumeration of Hermitian self-dual quasi-abelian codes are given. In the case of 1-generator quasi-abelian codes, we offer necessary and sufficient conditions for such codes to be Hermitian self-dual and give a formula for the number of these codes. In the case where the underlying groups are some $p$-groups, the actual number of resulting Hermitian self-dual quasi-abelian codes are determined.

Characterization and enumeration of complementary dual abelian codes

Abelian codes and complementary dual codes form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, a family of abelian codes with complementary dual in a group algebra $${\mathbb {F}}_{p^\nu }[G]$$ has been studied under both the Euclidean and Hermitian inner products, where p is a prime, $$\nu $$ is a positive integer and G is an arbitrary finite abelian group. Based on the discrete Fourier transform decomposition for semi-simple group algebras and properties of ideas in local group algebras, the characterization of such codes have been given. Subsequently, the number of complementary dual abelian codes in $${\mathbb {F}}_{p^\nu }[G]$$ has been shown to be independent of the Sylow p-subgroup of G and it has been completely determined for every finite abelian group G. In some cases, a simplified formula for the enumeration has been provided as well. The known results for cyclic complementary dual codes can be viewed as corollaries.