Related papers: Cyclic Codes and Sequences from Kasami-Welch Funct…
BCH codes with efficient encoding and decoding algorithms have many applications in communications, cryptography and combinatorics design. This paper studies a class of linear codes of length $ \frac{q^m-1}{2}$ over $\mathbb{F}_q$ with…
We are interested in random uniform minimal factorizations of the $n$-cycle which are factorizations of $(1~2\dots n)$ into a product of $n-1$ transpositions. Our main result is an explicit formula for the joint probability that 1 and 2…
In this work, cyclic-skew-cyclic codes and sum-rank BCH codes are introduced. Cyclic-skew-cyclic codes are characterized as left ideals of a suitable non-commutative finite ring, constructed using skew polynomials on top of polynomials (or…
In an interesting paper Professor Cunsheng Ding provided three constructions of cyclic codes of length being a product of two primes. Numerical data shows that many codes from these constructions are best cyclic codes of the same length and…
In this article we study decreasing and increasing factorisations of the cycle, which are decompositions of the cycle $(1~2\dots n)$ into a product of $n-1$ transpositions satisfying monotonicity conditions. We explicit a bijection between…
The paper has a threefold purpose. The first purpose is to present an explicit description of expanded cyclic codes defined in $\GF(q^m)$. The proposed explicit construction of expanded generator matrix and expanded parity check matrix…
Let $p$ be a prime number, $q=p^s$ for a positive integer $s$. For any positive divisor $e$ of $q-1$, we construct an infinite family codes of size $q^{2m}$ with few Lee-weight. These codes are defined as trace codes over the ring…
Detailed information about the weight distribution of a convolutional code is given by the adjacency matrix of the state diagram associated with a controller canonical form of the code. We will show that this matrix is an invariant of the…
Polycyclic codes offer a natural generalization of cyclic codes and provide a broader algebraic framework for constructing linear codes with good parameters. In this paper, we study binary polycyclic codes associated with powers of…
We study the distribution of cycle lengths in models of nonuniform random permutations with cycle weights. We identify several regimes. Depending on the weights, the length of typical cycles grows like the total number $n$ of elements, or a…
We develop an algorithm for computing the weight distribution of a linear $[n,k]$ code over a finite field $\mathbb{F}_q$. We represent the codes by their characteristic vector with respect to a given generator matrix and a special type of…
Quasi-cyclic codes form an important class of algebraic codes that includes cyclic codes as a special subclass. This chapter focuses on the algebraic structure of quasi-cyclic codes, first. Based on these structural properties, some…
By exploiting the connection between scattered $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^3$ and minimal non degenerate $3$-dimensional rank metric codes of $\mathbb{F}_{q^m}^{n}$, $n \geq m+2$, described in [2], we will exhibit a new…
Two classes of turbo codes over high-order finite fields are introduced. The codes are derived from a particular protograph sub-ensemble of the (dv=2,dc=3) low-density parity-check code ensemble. A first construction is derived as a…
Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Perfect nonlinear monomials were employed to…
Usually, it is difficult to determine the weight distribution of an irreducible cyclic code. In this paper, we discuss the case when an irreducible cyclic code has the maximal number of distinct nonzero weights and give a necessary and…
Linear codes with a few weights can be applied to secrete sharing, authentication codes, association schemes and strongly regular graphs. For an odd prime power $q$, we construct a class of three-weight $\F_q$-linear codes from quadratic…
Given a finite extension $K/F$ of degree $r$ of a finite field $F$, we enumerate all selfdual skew cyclic codes in the Ore quotient ring $K[X;\text{Frob}]/(X^{rk}-1)$ for any positive integer $k$ coprime to the characteristic $p$ (separable…
Cyclic codes have efficient encoding and decoding algorithms over finite fields, so that they have practical applications in communication systems, consumer electronics and data storage systems. The objective of this paper is to give eight…
Let $\mathbf{F}_q$ be a finite field of $q$ elements. For multiplicative characters $\chi_1,\dots, \chi_m$ of $\mathbf{F}_q^\times$, we let $J(\chi_1,\dots, \chi_m)$ denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for…