Two-weight and three-weight linear codes based on Weil sums

Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. In this paper, several classes of two-weight and three-weight linear codes are presented and their weight distributions are determined using Weil sums. Some of the linear codes obtained are optimal or almost optimal with respect to the Griesmer bound.     

Three-weight ternary linear codes from a family of cyclic difference sets

Linear codes with a few weights have applications in data storage systems, secret sharing schemes, and authentication codes. Recently, Ding (IEEE Trans. Inf. Theory 61(6):3265–3275, 2015) proposed a class of ternary linear codes with three weights from a family of cyclic difference sets in \(({\mathbb {F}}_{3^m}^*/{\mathbb {F}}_{3}^*,\times )\), where \(m=3k\) and k is odd. One objective of this paper is to construct ternary linear codes with three weights from cyclic difference sets in \(({\mathbb {F}}_{3^m}^*/{\mathbb {F}}_{3}^*,\times )\) derived from the Helleseth–Gong functions. This construction works for any positive integer \(m=sk\) with an odd factor \(s\ge 3\), and thus leads to three-weight ternary linear codes with more flexible parameters than earlier ones mentioned above. Another objective of this paper is to determine the weight distribution of the proposed linear codes.     

Quantum codes from nearly self-orthogonal quaternary linear codes

Construction X and its variants are known from the theory of classical error control codes. We present instances of these constructions that produce binary stabilizer quantum error control codes from arbitrary quaternary linear codes. Our construction does not require the classical linear code \(C\) that is used as the ingredient to satisfy the dual containment condition, or, equivalently, \(C^{\perp _h}\) is not required to satisfy the self-orthogonality condition. We prove lower bounds on the minimum distance of quantum codes obtained from our construction. We give examples of record breaking quantum codes produced from our construction. In these examples, the ingredient code \(C\) is nearly dual containing, or, equivalently, \(C^{\perp _h}\) is nearly self-orthogonal, by which we mean that \(\dim (C^{\perp _h})-\dim (C^{\perp _h}\cap C)\) is positive but small.     

Minimal linear codes from Maiorana-McFarland functions

Minimal linear codes have important applications in secret sharing schemes and secure multi-party computation, etc. In this paper, we study the minimality of a kind of linear codes over $\mathrm{GF}(p)$ from Maiorana-McFarland functions. We first obtain a new sufficient condition for this kind of linear codes to be minimal without analyzing the weights of its codewords, which is a generalization of some works given by Ding et al. in 2015. Using this condition, it is easy to verify that such minimal linear codes satisfy $\frac{w_{\min }}{w_{\max }}\le \frac{p-1}{p}$ for any prime p, where $w_{\min }$ and $w_{\max }$ denote the minimum and maximum nonzero weights in a code, respectively. Then, by selecting the subsets of $\mathrm{GF}{(p)}^s$, we present two new infinite families of minimal linear codes with $\frac{w_{\min }}{w_{\max }}\le \frac{p-1}{p}$ for any prime p. In addition, the weight distributions of the presented linear codes are determined in terms of Krawtchouk polynomials or partial spreads.     

A family of linear codes from constant dimension subspace codes

Designs, Codes and Cryptography - Linear codes with good parameters have wide applications in secret sharing schemes, authentication codes, association schemes, consumer electronics and...     

New quinary linear codes from quasi-twisted codes and their duals

One of the central problems in algebraic coding theory is construction of linear codes with best possible parameters. Quasi-twisted (QT) codes have been promising to solve this problem. Despite extensive search in this class and discovery of a large number of new codes, we have been able to find still more new codes that are QT over the alphabet ${\mathbb{F}}_5$ using a more comprehensive search strategy.     

Evaluation of the Hamming weights of a class of linear codes based on Gauss sums

Linear codes with a few weights have been widely investigated in recent years. In this paper, we mainly use Gauss sums to represent the Hamming weights of a class of q-ary linear codes under some certain conditions, where q is a power of a prime. The lower bound of its minimum Hamming distance is obtained. In some special cases, we evaluate the weight distributions of the linear codes by semi-primitive Gauss sums and obtain some one-weight, two-weight linear codes. It is quite interesting that we find new optimal codes achieving some bounds on linear codes. The linear codes in this paper can be used in secret sharing schemes, authentication codes and data storage systems.     

Constructing quasi-cyclic codes from linear algebra theory

Let F _q be a finite field of cardinality q, l and m be positive integers and M _l (F _q ) the F _q -algebra of all l × l matrices over F _q . We investigate the relationship between monic factors of X ^m ? 1 in the polynomial ring M _l (F _q )[X] and quasi-cyclic (QC) codes of length lm and index l over F _q . Then we consider the idea of constructing QC codes from monic factors of X ^m ? 1 in polynomial rings over F _q -subalgebras of M _l (F _q ). This idea includes ideas of constructing QC codes of length lm and index l over F _q from cyclic codes of length m over a finite field F _q l, the finite chain ring F _q  + uF _q  + · · · + u ^{l ? 1} F _q (u ^l  = 0) and other type of finite chain rings.     

Designs from subcode supports of linear codes

We present new constructions of t-designs by considering subcode supports of linear codes over finite fields. In particular, we prove an Assmus-Mattson type theorem for such subcodes, as well as an automorphism characterization. We derive new t-designs (t ≤ 5) from our constructions.      

Incomplete sums,DC-constrained codes,and codes that maintain synchronization

The codes formed by vectors of values of characters of polynomials with argument running over (a subset of) points of a finite field are a frequent object of investigation since one can estimate their parametes via the classical inequalities for the values of exponential sums. In [2] it was observed that applying estimates of incomplete sums, one can prove that some codes of this type have small running digital sum compared to the block length. In the present paper, we derive an extension of the results of [2] to nonprime fields and to the nonbinary case, which leads to new families of error-correcting dc-constrained codes. We show also that constructed codes are comma-free and detect synchronization errors even for a high rate of additive errors.The results of this paper were presented in part at the French-Soviet Workshop on Algebraic Coding, Paris, July 22–24, 1991, and published in Springer Lect. Notes Comput. Sci., 573, 1992, pp. 16–22.     

Four classes of linear codes from cyclotomic cosets

This paper presents four classes of linear codes from coset representatives of subgroups and cyclotomic coset families of certain finite field, and determines their weight enumerators. These linear codes may have applications in consumer electronics, communications and secret sharing schemes.     

Difference sets and three-weight linear codes from trinomials

We confirm a conjecture of Cunsheng Ding claiming that the punctured value-sets of a list of eleven trinomials over odd-degree extensions of the binary field give rise to difference sets with Singer parameters. In the course of confirming the conjecture, we show that these trinomials share the remarkable property that every element of the value-set of each trinomial has either one or four preiamges. We also give the partial resolution of another conjecture of Cunsheng Ding claiming that linear codes constructed from those eleven trinomials are three-weight.     

Perfect powers from the sums of terms of linear recurrences

Let Gi= {Gi n1 n=0 (i = 1;2 ;m; m 2) be linear recursive se- quences of order ki 2 andGn 2 Z. Further let K >1 be a fixed real number and S = s 2 Z : s = p e 11p ev v ; ei 2 N , wherepi"s are fixed primes. In this paper, under some restrictions, we prove that, if m P i=1 G (i) n i = sw q holds for positive in- tegers w 2; q; n1; n2; nm with n1 > K max (n2; nm), then q tegers w 2; q; n1; n2; nm with n1 > K max (n2; nm), then q...     

Three-weight generalization of Hardy's inequality

Translated from Matematicheskie Zametki, Vol. 54, No. 2, pp. 56–62, August, 1993.     

Optimal ternary linear codes

Let n _q (k, d) denote the smallest value of n for which there exists a linear [n, k, d]-code over GF(q). An [n, k, d]-code whose length is equal to n _q (k, d) is called optimal. The problem of finding n _q (k, d)has received much attention for the case q = 2. We generalize several results to the case of an arbitrary prime power q as well as introducing new results and a detailed methodology to enable the problem to be tackled over any finite field.In particular, we study the problem with q = 3 and determine n ₃(k, d) for all d when k 4, and n ₃(5, d) for all but 30 values of d.