Cyclic [formula omitted]-additive 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.

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.

Motivated by the recent studies on Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> -additive cyclic codes, Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> [u]-cyclic codes and Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2(s)</sub> -additive codes have been introduced by Aydogdu et al.. In this paper, we study Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> R-linear cyclic codes where R = Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> + uZ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , q is a prime number and u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = 0. There are two major ingredients in this paper. The first is to investigate the algebraic structure of cyclic codes and their duals over ring Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> R, the spanning sets, the types and sizes of Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> R-cyclic codes and their duals as well. Based on this, the second ingredient is to present an infinite family of MDSS codes and obtain some illustrative examples of q-ary cyclic codes with optimal parameters derived from the Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> R-cyclic codes.

Coded Distributed Matrix Multiplication (CDMM) is a distributed matrix multiplication (DMM) for large-scale matrices through a coding scheme such that any <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> worker nodes among all <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> worker nodes can recover the final product, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> corresponds to the length of the code and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> is called the recovery threshold. The state-of-art CDMM schemes, such as EP codes for Single DMM and GCSA codes for batch DMM, are defined over a Galois field GF(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i>) of size <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> ≥ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i>. These are inefficient for small Galois fields such as GF(2) and the integer residue ring Z<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>p</i></sub><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup> due to the lack of invertible elements for interpolation. DMM over Z<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">_p</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup> (such as Z<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">64</sup> ) is well-motivated in practice due to their direct compatibility with hardware. In this work, we construct efficient CDMM over the Galois ring GR(p<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup>, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i>) which is an extension ring over Z<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>p</i></sub><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup> of degree <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i>, particularly, GR(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p, d</i>) = GF(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup>) is the Galois field and GR(<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>p</i></sub><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup>, 1) = Z<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">_p</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup>. We first give a general CDMM framework for the batch of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> matrix multiplications via the famous RMFE (Cascudo et al. Crypto’18). Compared with GCSA, our construction has a smaller recovery threshold by a factor of 1/<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>. Next, we optimize EP codes via batch preprocessing of the input matrices. We give two types of Single CDMM, which can achieve almost the same performance as EP codes over a Galois field with size <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> ≥ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i>. Finally, we present the experimental analysis of our CDMM on Galois rings.

Counting points on diagonal equations over Galois rings GR(p2,2n)

From a linear block code B over the Galois ring GR(4, to) with a k x n generator matrix and minimum Hamming distance d, a rate-k/n convolutional code over the ring Z4 with squared Euclidean free distance at least 2d and a non-recursive encoder with memory at most to m-1 is constructed. When the generator matrix of B is systematic, the convolutional encoder is systematic, basic, non-catastrophic and minimal. Long codes constructed in this manner are shown to satisfy a Gilbert-Varshamov bound.

Self-dual codes over the Galois ring GR(4,2) are investigated. Of special interest are quadratic double circulant codes. Euclidean self-dual (Type II) codes yield self-dual (Type II) ℤ4-codes by projection on a trace orthogonal basis. Hermitian self-dual codes also give self-dual ℤ4-codes by the cubic construction, as well as Eisenstein lattices by Construction A. Applying a suitable Gray map to self-dual codes over the ring gives formally self-dual 𝔽4-codes, most notably in length 12 and 24. Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension 44 are constructed.

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.

On the ℓ-DLIPs of codes over finite commutative rings

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

The main objective of coding theory is to find codes with efficient encoding and decoding techniques and the greatest possible value of distance for a given code length, code size, and code alphabet cardinality. Various distances like Hamming distances, symbol-pair distances, etc., have been established and investigated in coding theory to analyse a code’s error-detecting and error-correcting capabilities with respect to various communication channels. A maximum distance separable code (MDS code) is a code that meets the Singleton bound. The MDS codes have the greatest error correcting capability. When the length and size of the codes are fixed, an MDS code has the largest distance. As a result, nowadays, researching and discovering MDS codes with respect to various distances is a hot topic in coding theory. In the theory of error-correcting codes, constacyclic codes over finite fields play a crucial role. Constacyclic codes, moreover, have many real-world applications. Shift registers can effectively encode and decode these codes because they have rich algebraic structures. They also offer excellent error-correction capabilities. All of this clarifies their chosen engineering role. In this monograph, we determine the multi-symbol distances of various classes of repeated-root constacyclic codes over some classes of finite commutative chain rings and investigate various classes of non-trivial MDS codes using these multi-symbol distance distributions.

Repeated root Cyclic and Negacyclic codes over Galois rings havebeen studied much less than their simple root counterparts. This situation isbeginning to change. For example, repeated root codes of length ps, where $p$ isthe characteristic of the alphabet ring, have been studied under some additionalhypotheses. In each one of those cases, the ambient space for the codes hasturned out to be a chain ring. In this paper, all remaining cases of cyclicand negacyclic codes of length ps over a Galois ring alphabet are considered.In these cases the ambient space is a local ring with simple socle but not achain ring. Nonetheless, by reducing the problem to one dealing with uniserialsubambients, a method for computing the Hamming distance of these codes isprovided.

In this paper, we study constacyclic codes over finite principal ideal rings. An isomorphism between constacyclic codes and cyclic codes over finite principal ideal rings is given. Further, an open question is partially answered by giving necessary and sufficient conditions for the existence of non-trivial cyclic self-dual codes over finite principal ideal rings. As an example of codes over a finite principal ideal ring, we study constacyclic codes over R+vR where v2=v and R is a finite chain ring.

We give the structure of constacyclic codes over some chain rings.We also provide conditions on the equivalence between constacycliccodes and cyclic codes over finite chain rings.As a special case,weconsider the structure of $(\alpha + \beta p)$-constacyclic codes oflength $p^s$ over $GR(p^e,r)$.

Left dihedral codes over finite chain rings