Bounds on quasi-cyclic codes over finite chain rings

Gao, Jian; Shen, Linzhi; Fu, Fang-Wei

doi:10.1007/s12190-015-0885-7

Cited by 8 publications

(6 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 4.3 (BCH-like Bounds). Mattson-Solomon transform can be a great tool for understanding BCH-like bounds that have been established for different types of (chain) rings (see for example [11,16]) or based on invariant spaces for polycyclic codes over fields [26]. Note that the minimum distance of a linear general code over R is the same as the one of its socle [14, Proposition 5].…”

Section: If a Polycyclic Code C Decomposes Asmentioning

confidence: 99%

A transform approach to polycyclic and serial codes over rings

Bajalan¹,

Martı́nez-Moro²,

Szabo³

2021

Preprint

View full text Add to dashboard Cite

In this paper, a transform approach is used for polycyclic and serial codes over finite local rings in the case that the defining polynomials have no multiple roots. This allows us to study them in terms of linear algebra and invariant subspaces as well as understand the duality in terms of the transform domain. We also make a characterization of when two polycyclic ambient spaces are Hamming-isometric.

show abstract

Section: If a Polycyclic Code C Decomposes Asmentioning

confidence: 99%

A transform approach to polycyclic and serial codes over rings

Bajalan¹,

Martı́nez-Moro²,

Szabo³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Quasi-cyclic codes over a finite commutative chain ring are obtained when r = 1, t 1 (X 1 ) = X n1 1 − 1 and l > 1. See for example [3,7] and the references therein.…”

Section: Basic Definitions and Examplesmentioning

confidence: 99%

“…Quasi-cyclic codes over a finite commutative chain ring R can be represented as (R[x]/ x n − 1 )-submodules of (R[x]/ x n − 1 ) l , generalizing the well known construction for finite fields in, for example, [3]. For finite commutative chain ring, the one-generator codes have been extensively studied (see, for example, the classical paper [21] and the references included in [7]), whereas for finite fields the general situation was studied in [11] and recently generalized in [1] to codes over finite quotients of polynomial rings, i.e., to F[x]/ f (x) -submodules of (F[x]/ f (x) ) l where l ∈ N and f (x) is a monic polynomial. Furthermore, Jitman and Ling studied quasi-abelian codes over finite fields using techniques based on the Discrete Fourier Transform.…”

Section: Introductionmentioning

confidence: 99%

“…The proposed approach, which uses the concept of Canonical Generating System introduced in [20], allows the study of quasi-cyclic codes over a finite commutative chain ring and their multivariable generalizations in a broader polynomial way, that is, beyond the one-generator case. Notice that for several generators a trace representation can also be derived from the ideas in [12], see for example Section 4 in [7]. The former approach provides a way for defining codes over some Frobenius local rings which are not chain rings.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Codes over affine algebras with a finite commutative chain coefficient ring

Martı́nez-Moro

Nicolás

Rúa

2018

Finite Fields and Their Applications

View full text Add to dashboard Cite

We consider codes defined over an affine algebra A = R[X1, . . . , Xr]/ t1(X1), . . . , tr(Xr) , where ti(Xi) is a monic univariate polynomial over a finite commutative chain ring R. Namely, we study the A−submodules of A l (l ∈ N). These codes generalize both the codes over finite quotients of polynomial rings and the multivariable codes over finite chain rings. Some codes over Frobenius local rings that are not chain rings are also of this type. A canonical generator matrix for these codes is introduced with the help of the Canonical Generating System. Duality of the codes is also considered.

show abstract

“…Later, circulant matrices have been shown to have applications in many disciplines, e.g., signal processing, image processing, networked systems, communications, and coding theory. Especially, (nonsingular) circulant matrices over finite fields and over commutative finite chain rings are applied in constructions of various families of linear codes (see [1], [5], [7], [9], [10], [11], [14], [17], [18], [19] , and references therein). Circulant matrices have shown to have a closed connection with diagonal matrices (see, for example, [5] and [14]).…”

Section: Introductionmentioning

confidence: 99%

Determinants of some Special Matrices over Commutative Finite Chain Rings

Jitman¹

2020

Preprint

View full text Add to dashboard Cite

Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings R with residue field F q are studied. The number of n × n diagonal matrices over R of determinant a is determined for all elements a in R and for all positive integers n. Subsequently, the enumeration of nonsingular n × n circulant matrices over R of determinant a is given for all units a in R and all positive integers n such that gcd(n, q) = 1. In some cases, the number of singular n × n circulant matrices over R with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinant of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems

show abstract

Bounds on quasi-cyclic codes over finite chain rings

Cited by 8 publications

References 17 publications

A transform approach to polycyclic and serial codes over rings

A transform approach to polycyclic and serial codes over rings

Codes over affine algebras with a finite commutative chain coefficient ring

Determinants of some Special Matrices over Commutative Finite Chain Rings

Contact Info

Product

Resources

About