Related papers: Some $3$-designs and shortened codes from binary c…
In this paper we develop a technique to extend any bound for cyclic codes constructed from its defining sets (ds-bounds) to abelian (or multivariate) codes. We use this technique to improve the searching of new bounds for abelian codes.
A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find…
Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by $\mathcal{C}(f)=\left\{ {\rm Tr}(af(x)+bx)_{x \in \mathbb{F}_{q^m}^*}: a,b \in…
We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings F_2 + uF_2 and F_4 + uF_4; using groups of order 3, 7, 9, 13, and 15.…
Mathematically, ternary coding is more efficient than binary coding. It is little used in computation because technology for binary processing is already established and the implementation of ternary coding is more complicated, but remains…
We present a new application of multi-orbit cyclic subspace codes to construct large optical orthogonal codes, with the aid of the multiplicative structure of finite fields extensions. This approach is different from earlier approaches…
This paper deals with the problem of increasing the minimum distance of a linear code by adding one or more columns to the generator matrix. Several methods to compute extensions of linear codes are presented. Many codes improving the…
LCD codes are linear codes with important cryptographic applications. Recently, a method has been presented to transform any linear code into an LCD code with the same parameters when it is supported on a finite field with cardinality…
Quaternary code (QC) designs form an attractive class of nonregular factorial fractions. We develop a complementary set theory for characterizing optimal QC designs that are highly fractionated in the sense of accommodating a large number…
The hull of a linear code over finite fields is the intersection of the code and its dual, and linear codes with small hulls have applications in computational complexity and information protection. Linear codes with the smallest hull are…
In this paper, a class of two-weight and three-weight linear codes over $\gf(p)$ is constructed, and their application in secret sharing is investigated. Some of the linear codes obtained are optimal in the sense that they meet certain…
Best and Brouwer [Discrete Math. 17 (1977), 235-245] proved that triply-shortened and doubly-shortened binary Hamming codes (which have length $2^m-4$ and $2^m-3$, respectively) are optimal. Properties of such codes are here studied,…
Cyclic codes are an interesting class of linear codes due to their efficient encoding and decoding algorithms as well as their theoretical importance. BCH codes form a subclass of cyclic codes and are very important in both theory and…
In this paper, we show infinite series of $3$-designs in the extended quadratic residue codes over $\mathbb {F}_{r^2}$ for a prime $r$.
Linear codes are the most important family of codes in cryptography and coding theory. Some codes have only a few weights and are widely used in many areas, such as authentication codes, secret sharing schemes and strongly regular graphs.…
Let $R=\mathbb{Z}_4$ be the integer ring mod $4$. A double cyclic code of length $(r,s)$ over $R$ is a set that can be partitioned into two parts that any cyclic shift of the coordinates of both parts leaves invariant the code. These codes…
We prove that a binary linear code of block length $n$ that is locally correctable with $3$ queries against a fraction $\delta > 0$ of adversarial errors must have dimension at most $O_{\delta}(\log^2 n \cdot \log \log n)$. This is almost…
For the past decades, linear codes with few weights have been widely studied, since they have applications in space communications, data storage and cryptography. In this paper, a class of binary linear codes is constructed and their weight…
We prove that a certain binary linear code associated with the incidence matrix of a quasi-symmetric 2-(37,9,8) design with intersection numbers 1 and 3 must be contained in an extremal doubly even self-dual code of length 40. Using the…
There are many results on the minimum distance of a cyclic code of the form that if a certain set T is a subset of the defining set of the code, then the minimum distance of the code is greater than some integer t. This includes the BCH,…