1-Generator quasi-cyclic codes over [formula omitted

For R a Galois ring and m 1, . . . , m l positive integers, a generalized quasi-cyclic (GQC) code over R of block lengths (m 1, m 2, . . . , m l ) and length $${\sum_{i=1}^lm_i}$$ is an R[x]-submodule of $${R[x]/(x^{m_1}-1)\times\cdots \times R[x]/(x^{m_l}-1)}$$. Suppose m 1, . . . , m l are all coprime to the characteristic of R and let {g 1, . . . , g t } be the set of all monic basic irreducible polynomials in the factorizations of $${x^{m_i}-1}$$ (1 ≤ i ≤ l). Then the GQC codes over R of block lengths (m 1, m 2, . . . , m l ) and length $${\sum_{i=1}^lm_i}$$ are identified with $${{\mathcal G}_1\times\cdots\times {\mathcal G}_t}$$, where $${{\mathcal G}_j}$$ is an R[x]/(g j )-submodule of $${(R[x]/(g_j))^{n_j}}$$, where n j is the number of i for which g j appears in the factorization of $${x^{m_i}-1}$$ into monic basic irreducible polynomials. This identification then leads to an enumeration of such GQC codes. An analogous result is also obtained for the 1-generator GQC codes. A notion of a parity-check polynomial is given when R is a finite field, and the number of GQC codes with a given parity-check polynomial is determined. Finally, an algorithm is given to compute the number of GQC codes of given block lengths.

Structure and performance of generalized quasi-cyclic codes

Let $$q$$ be a power of a prime integer $$p, m=p^em_0$$ and $$|q|_{m_{0}}$$ the order of $$q$$ modulo $$m_0$$ . By use of finite commutative chain ring theory, an algorithm to construct all distinct 1-generator quasi-cyclic codes with a fixed parity check polynomial over a finite field $$F_q$$ of length $$mn$$ and index $$n$$ , under the condition that $$\mathrm {gcd}(|q|_{m_0},n)=1$$ , are given.

On the ℓ-DLIPs of codes over finite commutative rings

Left dihedral codes over finite chain rings

We introduce (1-γ)-cyclic code and cyclic codes over the finite chain ring R. We prove that the Gray image of a linear (1-γ)-cyclic code over R of length n is a distance invariant quasi-cyclic code over Fpk. We also prove that if (n,p) = 1, then every code over Fpk which is the Gray image of a cyclic code over R of length n is equivalent to a quasi-cyclic code.

In this article, the notion of γ-dual codes over finite chain rings is introduced as an extension of dual codes over finite chain rings. Various characteristics and properties of γ-dual codes over finite chain rings are explored. We provide both necessary and sufficient conditions for the existence of γ-self-dual codes over finite chain rings. Additionally, we investigate the γ-dual of skew ϕ-α-constacyclic codes over finite chain rings.

The ranks of cyclic and negacyclic codes over the finite chain ring R as well as their minimal generating sets are defined, and then the expression forms we presented by studying the structures of cyclic and negacyclic codes over the finite chain ring R. Through the paper, it is assumed that the length of codes n can not be divided by the characteristic of R.

Let m, e be positive integers, p a prime number, $$\mathbb {F}_{p^m}$$ be a finite field of $$p^m$$ elements and $$R=\mathbb {F}_{p^m}[u]/\langle u^e\rangle $$ which is a finite chain ring. For any $$\omega \in R^\times $$ and positive integers k, n satisfying $$\mathrm{gcd}(p,n)=1$$ , we prove that any $$(1+\omega u)$$ -constacyclic code of length $$p^kn$$ over R is monomially equivalent to a matrix-product code of a nested sequence of $$p^k$$ cyclic codes with length n over R and a $$p^k\times p^k$$ matrix $$A_{p^k}$$ over $$\mathbb {F}_p$$ . Using the matrix-product structures, we give an iterative construction of every $$(1+\omega u)$$ -constacyclic code by $$(1+\omega u)$$ -constacyclic codes of shorter lengths over R.

Generalized quasi-cyclic (GQC) codes form a wide and useful class of linear codes that includes thoroughly quasi-cyclic codes, finite geometry (FG) low density parity check (LDPC) codes, and Hermitian codes. Although it is known that the systematic encoding of GQC codes is equivalent to the division algorithm in the theory of Grobner basis of modules, there has been no algorithm that computes Grobner basis for all types of GQC codes. In this paper, we propose two algorithms to compute Grobner basis for GQC codes from their parity check matrices: echelon canonical form algorithm and transpose algorithm. Both algorithms require sufficiently small number of finite-field operations with the order of the third power of code-length. Each algorithm has its own characteristic; the first algorithm is composed of elementary methods, and the second algorithm is based on a novel formula and is faster than the first one for high-rate codes. Moreover, we show that a serial-in serial-out encoder architecture for FG LDPC codes is composed of linear feedback shift registers with the size of the linear order of code-length; to encode a binary codeword of length n, it takes less than 2n adder and 2n memory elements. Keywords: automorphism group, Buchberger's algorithm, division algorithm, circulant matrix, finite geometry low density parity check (LDPC) codes.

The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(p e , l) (including $${\mathbb{Z}_{p^e}}$$ ). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual codes over Galois rings GF(2 e , l) of length n = 2 l for any a ? 1 and l ? 2. Torsion codes over residue fields of finite chain rings are introduced, and some of their properties are derived. Finally, we describe MDS codes and self-dual codes over finite principal ideal rings by examining codes over their component chain rings, via a generalized Chinese remainder theorem.

Circulant matrices form an important class of matrices that have been continuously studied due to their nice algebraic structures and wide applications. In this study, we focus specifically on negacirculant matrices, which are known as extensions of the classical circulant matrices. The algebraic structures of the rings of n × n n\times n negacirculant matrices over finite fields and over finite commutative chain rings are presented. Subsequently, the algebraic structures and enumeration of the unit groups of such matrix rings are established. Additionally, the number of non-singular n × n n\times n negacirculant matrices with prescribed determinant is given in some cases. Conjectures and open problems are proposed as well as a brief discussion in the case where the underlying ring is a finite commutative ring with identity is also presented.

Construction and enumeration for self-dual cyclic codes of even length over [formula omitted

Let R be an arbitrary commutative finite chain ring and \(\gamma \) a fixed generator of the maximal ideal of R. Suppose s is the nilpotency index of \(\gamma \) and F is the residue field of R modulo its ideal \(\gamma R\), i.e. \(F = R / \gamma R\), \(\vert F \vert = q\) with \(q=p^{\alpha }\) for some prime number p and let \(R^{\times }\) denote the multiplicative group of units of R. For any \(\omega \in R^{\times }\) and \(t \ge \lceil \frac{s}{2}\rceil \), the structural properties and dual codes of \((1+ \omega \gamma ^{t})\)-quasi-twisted (QT) codes of length \(n=\ell m\), with \((m, p)=1\), over R are given. The key idea is to view a \((1 + \omega \gamma ^{t})\)-QT code over R as a linear code over \(R_{m} = R[x] / \langle x^{m} - (1 + \omega \gamma ^{t}) \rangle \). Furthermore, given the decomposition of a \((1+ \omega \gamma ^{t})\)-QT code, we provide the decomposition of its dual code. As a result, a characterization of self-dual \((1+ \omega \gamma ^{t})\)-QT codes over a finite chain ring R, with \((1+ \omega \gamma ^{t})=(1+ \omega \gamma ^{t})^{-1}\), is provided. By using the Chinese remainder theorem or the discrete Fourier transform, the ring \(R[x] / \langle x^{m} - (1 + \omega \gamma ^{t}) \rangle \) can be decomposed into a direct sum of finite chain rings. The inverse transform of the discrete Fourier transform produces a method for deriving \((1+ \omega \gamma ^{t})\)-QT codes from codes of lower lengths.

<abstract><p>An associative Artinian ring with an identity is a chain ring if its lattice of left (right) ideals forms a unique chain. In this article, we first prove that for every chain ring, there exists a certain finite commutative chain subring which characterizes it. Using this fact, we classify chain rings with invariants $ p, n, r, k, k', m $ up to isomorphism by finite commutative chain rings ($ k' = 1 $). Thus the classification of chain rings is reduced to that of finite commutative chain rings.</p></abstract>