Codes Closed under Arbitrary Abelian Group of Permutations

Algebraic structure of codes closed under arbitrary abelian group G of permutations is investigated resulting in insight into dual of G-invariant codes and self-dual G-invariant codes. For special types of the groups, these codes give cyclic, abelian, quasi-cyclic and quasi-abelian codes. Karlin's decoding algorithm for systematic one-generator quasicyclic codes is extended for systematic quasi-abelian codes with any number of generators.

The finite field $\mathbb {F}_{q^\ell }$ of $q^\ell $ elements contains $\mathbb {F}_{q}$ as a subfield. If $\theta \in \mathbb {F}_{q^\ell }$ is of degree $\ell $ over $\mathbb {F}_{q}$ , it can be used to unfold elements of $\mathbb {F}_{q^\ell }$ to vectors in $\mathbb {F}_{q}^\ell $ . We apply the unfolding to the coordinates of all codewords of a cyclic code $\mathcal {C}$ over $\mathbb {F}_{q^\ell }$ of length $n$ . This generates a quasi-cyclic code $\mathcal {Q}$ over $\mathbb {F}_{q}$ of length $n\ell $ and index $\ell $ . We focus on the class of quasi-cyclic codes resulting from the unfolding of cyclic codes. Given a generator polynomial $g(x)$ of a cyclic code $\mathcal {C}$ , we present a formula for a generator polynomial matrix for the unfolded code $\mathcal {Q}$ . On the other hand, for any quasi-cyclic code $\mathcal {Q}$ with a reduced generator polynomial matrix $G$ , we provide a necessary and sufficient condition on $G$ that determines whether or not the code $\mathcal {Q}$ can be represented as the unfolding of a cyclic code. Furthermore, as an application, we discuss the reversibility of the class of quasi-cyclic codes resulting from unfolding of cyclic codes. Specifically, we provide a necessary and sufficient condition on the defining set $\mathcal {T}$ of the cyclic code $\mathcal {C}$ that ensures the reversibility of the unfolded code. Numerical examples are used to illustrate theoretical results. Some of these examples show that quasi-cyclic codes reversibility does not necessarily require a self-reciprocal generator polynomial for the cyclic code. Since reversibility is essential in constructing DNA codes, some DNA codes are designed as examples.

Let p be a prime number and m be a positive integer. In this paper, we investigate cyclic codes of length n over the local non-Frobenius ring R=GR(p2,m)[u], where u2=0 and pu=0. We first determine the algebraic structure of cyclic codes of arbitrary length n. For the case gcd(n,p)=1, we explicitly describe the generators of cyclic codes over R. Moreover, we establish necessary and sufficient conditions for the existence of self-dual and LCD codes, together with their enumeration. Several illustrative examples and tables are presented, highlighting the mass formula for cyclic self-orthogonal codes, cyclic LCD codes, and families of new cyclic codes that arise from our results.

We study the problem of constructing balanced generator matrices for Reed-Solomon and Tamo-Barg codes. More specifically, we are interested in realizing generator matrices, for the full-length cyclic versions of these codes, where all rows have the same weight and the difference in weight between any columns is at most one. The results presented in this paper translate to computationally balanced encoding schemes, which can be appealing in distributed storage applications. Indeed, the balancedness of these generator matrices guarantees that the computation effort exerted by any storage node is essentially the same. In general, the framework presented can accommodate various values for the required row weight. We emphasize the possibility of constructing sparsest and balanced generator matrices for Reed-Solomon codes, i.e., each row is a minimum distance codeword. The number of storage nodes contacted once a message symbol is updated decreases with the row weight, so sparse constructions are appealing in that context. Results of similar flavor are presented for cyclic Tamo-Barg codes. In particular, we show that for a code with minimum distance d and locality r, a construction in which every row is of weight d+r-1 is possible. The constructions presented are deterministic and operate over the codes' original underlying finite field. As a result, efficient decoding from both errors and erasures is possible thanks to the plethora of efficient decoders available for the codes considered.

In this paper, we investigate the structure and properties of cyclic codes over the ring <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vF</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> . We first study the relationship between cyclic codes over <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vF</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> and binary cyclic codes. Then we prove that cyclic codes over the ring are principally generated, and give the generator polynomial of cyclic codes over the ring. Finally, we obtain the unique idempotent generators for cyclic codes of odd length and determine the number of cyclic codes for a given length <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> over <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vF</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> .

In this paper we study the family of cyclic codes such that its minimum distance reaches the maximum of its BCH bounds. We also show a way to construct cyclic codes with that property by means of computations of some divisors of a polynomial of the form $x^n-1$. We apply our results to the study of those BCH codes $C$, with designed distance $\delta$, that have minimum distance $d(C)=\delta$. Finally, we present some examples of new binary BCH codes satisfying that condition. To do this, we make use of two related tools: the discrete Fourier transform and the notion of apparent distance of a code, originally defined for multivariate abelian codes.

The failure of spectral synthesis on some types of discrete Abelian groups

We consider the Kac–Bernstein functional equation $$\begin{aligned} f(x+y)g(x-y)=f(x)f(y)g(x)g(-y), \quad x, y\in X, \end{aligned}$$ on an arbitrary Abelian group X. We give a complete description of the solutions of this equation in the class of positive functions. This result is a generalization of a theorem by Pl. Kannappan. We also study the solutions of this equation in the class of complex-valued Hermitian functions.

В наших предыдущих работах мы продемонстрировали, что теория конечных подмножеств различных ассоциативных алгебр позволяет интерпретировать элементарную арифметику, в частности, она неразрешима. Например, это было показано для любых бесконечных абелевых групп. Возникает естественный вопрос: можно ли обобщить этот результат на более широкий класс алгебр, скажем, все коммутативные моноиды. В некоторых случаях нами ответ тоже получен ранее: для коммутативных моноидов с сокращением, имеющим элемент бесконечного порядка или произвольных абелевых групп. Сейчас же мы продемонстрируем, что не для всяких коммутативных моноидов это верно. Более того, мы дадим описание конструкции, которая позволяет строить такого рода системы из разного рода исходных алгебр. Вместе с тем, будут указаны и некоторые границы её применимости. In our previous works, we have proved for various associative algebras that the finite subsets theory allows to interpret elementary arithmetic, in particular, such theory is undecidable. For example, this is proved for all infinite Abelian groups. A natural question arises: can we generalize this result to a wider class of algebras, for example, all commutative monoids. In some cases, we also have proved analogous result: for commutative cancellative monoids with an element of infinite order, or arbitrary Abelian groups. In this paper we prove that this is not true for arbitrary commutative monoids. Moreover, we propose a method that allows to construct such algebras by various original algebras. Also, we have found a limitation of this method.

Cohomological dimension and acyclic resolutions

The transform domain characterization of linear cyclic codes over finite fields using Discrete Fourier Transform (DFT) over an appropriate extension field is well known. In this paper, we extend this transform domain characterization for linear quasi-cyclic codes over finite fields. We show how one can derive a lower bound on the minimum Hamming distance of a quasi-cyclic code and decode the code upto that minimum Hamming distance using this characterization.

Cyclic [formula omitted]-additive codes

In this paper, rate 1/p q-ary systematic quasi-cyclic codes are constructed based on matroid theory. The relationship between the generator matrix and minimum distance d is derived. Examples and algorithm are presented.

For an odd prime p, we obtain algebraic structure of cyclic codes of length n over a finite commutative non-chain semi-local ring \(\mathfrak {R}=\mathbb {F}_{p}[u,v,w]/\langle u^{2}-u,v^{2}-1,w^2-1,uv-vu,vw-wv,wu-uw\rangle \). These codes of length n can be viewed as principal ideals of the quotient ring \(\mathfrak {R}[x]/\langle x^n-1\rangle \). Here, a Gray map is defined to obtain p-ary quasi-cyclic codes of length 8n with index 8 as \(\mathbb {F}_p\)-images of cyclic codes of length n over \(\mathfrak {R}\). Also, we present necessary and sufficient conditions for a cyclic code to be an LCD (linear complementary dual) code over \(\mathfrak {R}\). Moreover, it is shown that the Gray image of an LCD code of length n over \(\mathfrak {R}\) is an LCD code of length 8n over \(\mathbb {F}_{p}\). Finally, a few non-trivial examples are given in support of our derived results.KeywordsCyclic codeNon-chain ringSemi-local ringGray mapLCD code

An [$$n, k, d$$n,k,d] code is a binary linear code of block length $$n$$n, dimension $$k$$k and minimum Hamming distance $$d$$d. Since the minimum distance determines the error detection or correction capability, it is desired that $$d$$d is as large as possible for the given block length $$n$$n and dimension $$k$$k. One of the most fundamental problems in coding theory is to construct codes with best possible minimum distances. This problem is very difficult in both theory and practice. During the last decades, it has proved that the class of quasi-cyclic (QC) codes contain many such codes. In this paper, augmentation of binary QC codes is studied. A new augmentation algorithm is presented, and 10 new $$h$$h-generator QC codes that are better than previously known code have been constructed. Furthermore, Construction X has been applied to obtain another 18 new improved binary linear codes. With the standard construction techniques, a total of 124 new binary linear codes that improve the lower bound on the minimum distance have been obtained.