Constacyclic codes over finite commutative semi-simple rings

Finite commutative semi-simple rings are direct sum of finite fields. In this study, we investigate the algebraic structure of λ-constacyclic codes over such finite semi-simple rings. Among others, necessary and sufficient conditions for the existence of self-dual, LCD, and Hermitian dual-containing λ-constacyclic codes over finite semi-simple rings are provided. Using the CSS and Hermitian constructions, quantum MDS codes over finite semi-simple rings are constructed.     

Locally recoverable J-affine variety codes

A locally recoverable (LRC) code is a code over a finite field ${\mathbb{F}}_q$ such that any erased coordinate of a codeword can be recovered from a small number of other coordinates in that codeword. We construct LRC codes correcting more than one erasure, which are subfield-subcodes of some J-affine variety codes. For these LRC codes, we compute localities $(r,\delta )$ that determine the minimum size of a set $\bar{R}$ of positions so that any $\delta -1$ erasures in $\bar{R}$ can be recovered from the remaining r coordinates in this set. We also show that some of these LRC codes with lengths $n\gg q$ are $(\delta -1)$-optimal.     

On Hermitian LCD codes and their Gray image

We provide methods and algorithms to construct Hermitian linear complementary dual (LCD) codes over finite fields. We study existence of self-dual basis with respect to Hermitian inner product, and as an application, we construct Euclidean LCD codes by projecting the Hermitian codes over such a basis. Many optimal quaternary Hermitian and ternary Euclidean LCD codes are obtained. Comparisons with classical constructions are made.     

Codes from curves with total inflection points

The concept of pure gaps of a Weierstrass semigroup at several points of an algebraic curve has been used lately to obtain codes that have a lower bound for the minimum distance which is greater than the Goppa bound. In this work, we show that the existence of total inflection points on a smooth plane curve determines the existence of pure gaps in certain Weierstrass semigroups. We then apply our results to the Hermitian curve and construct codes supported on several points that compare better to one-point codes from that same curve.      

A generalization of quasi-twisted codes: Multi-twisted codes

Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, have a special place in algebraic coding theory. Among other things, many of the best-known or optimal codes have been obtained from these classes. In this work we introduce a new generalization of QT codes that we call multi-twisted (MT) codes and study some of their basic properties. Presenting several methods of constructing codes in this class and obtaining bounds on the minimum distances, we show that there exist codes with good parameters in this class that cannot be obtained as QT or constacyclic codes. This suggests that considering this larger class in computer searches is promising for constructing codes with better parameters than currently best-known linear codes. Working with this new class of codes motivated us to consider a problem about binomials over finite fields and to discover a result that is interesting in its own right.     

Characterizations of some two-dimensional cyclic codes correspond to the ideals of F[x,y]/

Two-dimensional cyclic code is one of the natural generalizations of cyclic code. In this paper we study the algebraic structure of some two-dimensional cyclic codes and their dual codes.     

New quantum MDS codes with large minimum distance and short length from generalized Reed–Solomon codes

Quantum maximum-distance-separable (MDS) codes are an important class of quantum codes. In this paper we mainly use classical Hermitian self-orthogonal generalized Reed–Solomon codes to construct three classes of new quantum MDS codes. Further, these quantum MDS codes have large minimum distance and short length.     

Construction of some new quantum MDS codes

Quantum maximum-distance-separable (MDS) codes are an important class of quantum codes. In this paper, we mainly apply a new method of classical Hermitian self-orthogonal codes to construct three classes of new quantum MDS codes, and these quantum MDS codes provide large minimum distance.     

Error-Correcting Codes from Higher-Dimensional Varieties

In this paper we use intersection theory to develop methods for obtaining lower bounds on the parameters of algebraic geometric error-correcting codes constructed from varieties of arbitrary dimension. The methods are sufficiently general to encompass many of the codes previously constructed from higher-dimensional varieties, as well as those coming from curves. And still, the bounds obtained are usually as good as the ones previously known (at least of the same order of magnitude with respect to the size of the ground field). Several examples coming from Deligne–Lusztig varieties, complete intersections of Hermitian hyper-surfaces, and from ruled surfaces (or more generally, projective bundles over a curve) are given.     

A formula on the weight distribution of linear codes with applications to AMDS codes

The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with small Singleton's defect. We propose a new set of linear relations that must be satisfied by the coefficients of the weight distribution. From these relations we are able to derive known identities (in an easier way) for interesting cases, such as extremal codes, Hermitian codes, MDS and NMDS codes. Moreover, we are able to present for the first time the weight distribution of AMDS codes. We also discuss the link between our results and the Pless equations.     

Projective Reed–Muller type codes on rational normal scrolls

In this paper we study an instance of projective Reed–Muller type codes, i.e., codes obtained by the evaluation of homogeneous polynomials of a fixed degree in the points of a projective variety. In our case the variety is an important example of a determinantal variety, namely the projective surface known as rational normal scroll, defined over a finite field, which is the basic underlining algebraic structure of this work. We determine the dimension and a lower bound for the minimum distance of the codes, and in many cases we also find the exact value of the minimum distance. To obtain the results we use some methods from Gröbner bases theory.     

关于循环阵列码

本文用代数观点来研究循环阵列码,证明了一般的阵列码是一些极小循环阵列码的直和,并且对极小循环阵列码给出了明确的刻画．当有限域的特征不整除群的阶时,给出了直接写出相应的多项式环的本原幂等元的方法,从而可以直接写出所有的极小循环码．     

Riemann-Roch spaces of the Hermitian function field with applications to algebraic geometry codes and low-discrepancy sequences

This paper is concerned with two applications of bases of Riemann-Roch spaces. In the first application, we define the floor of a divisor and obtain improved bounds on the parameters of algebraic geometry codes. These bounds apply to a larger class of codes than that of Homma and Kim (J. Pure Appl. Algebra 162 (2001) 273). Then we determine explicit bases for large classes of Riemann-Roch spaces of the Hermitian function field. These bases give better estimates on the parameters of a large class of m-point Hermitian codes. In the second application, these bases are used for fast implementation of Xing and Niederreiter's method (Acta. Arith. 72 (1995) 281) for the construction of low-discrepancy sequences.     

Algebraic geometric codes with applications

The theory of linear error-correcting codes from algebraic geometric curves (algebraic geometric (AG) codes or geometric Goppa codes) has been well-developed since the work of Goppa and Tsfasman, Vladut, and Zink in 1981–1982. In this paper we introduce to readers some recent progress in algebraic geometric codes and their applications in quantum error-correcting codes, secure multi-party computation and the construction of good binary codes.      

A q-polynomial approach to constacyclic codes

As a generalization of cyclic codes, constacyclic codes is an important and interesting class of codes due to their nice algebraic structures and various applications in engineering. This paper is devoted to the study of the q-polynomial approach to constacyclic codes. Fundamental theory of this approach will be developed, and will be employed to construct some families of optimal and almost optimal codes in this paper.     

一类四元环上常循环码是自由码的充要条件

本文研究了环F2 uF2上的奇长度的循环码和(1 u)-循环码.运用代数方法,得到了F2 uF2上的循环码和(1 u)-循环码成为自由码的几个充要条件.推广了Bonnecaze(1999)和Aydin(2002)的关于自由码的结果.     

具有优良渐近参数的代数几何码

本文讨论了一类具有好的渐近参数的代数几何码.通过对除子类数、高次有理除子数以及代数几何码的参数分析,得到一类码其渐近界优于Gilbert-Varshamov界和Xing界.在这两个界的交点处,渐近界有所改进.     

Packing Superballs from Codes and Algebraic Curves

In the present paper, we make use of codes with good parameters and algebraic curves over finite fields with many rational points to construct dense packings of superballs. It turns out that our packing density is quite reasonable. In particular, we improve some values for the best-known lower bounds on packing density.