Related papers: A lower bound on permutation codes of distance $n-…
Let $A_2(n,d)$ be the maximum size of a binary code of length $n$ and minimum distance $d$. In this paper we present the following new lower bounds: $A_2(18,4) \ge 5632$, $A_2(21,4) \ge 40960$, $A_2(22,4) \ge 81920$, $A_2(23,4) \ge 163840$,…
In this paper, two classes of quantum MDS codes are constructed. The main tools are multiplicative structures on finite fields. Carefully choosing different cosets can make the corresponding generalized Reed-Solomon codes Hermitian…
Quantum maximum-distance-separable (MDS for short) codes are an important class of quantum codes. In this paper, by using Hermitian self-orthogonal generalized Reed-Solomon (GRS for short) codes, we construct five new classes of $q$-ary…
The Ulam distance of two permutations on $[n]$ is $n$ minus the length of their longest common subsequence. In this paper, we show that for every $\varepsilon>0$, there exists some $\alpha>0$, and an infinite set $\Gamma\subseteq…
It is an important task to construct quantum maximum-distance-separable (MDS) codes with good parameters. In the present paper, we provide six new classes of q-ary quantum MDS codes by using generalized Reed-Solomon (GRS) codes and…
A set of linearly constrained permutation matrices are proposed for constructing a class of permutation codes. Making use of linear constraints imposed on the permutation matrices, we can formulate a minimum Euclidian distance decoding…
A code of length $n$ is said to be (combinatorially) $(\rho,L)$-list decodable if the Hamming ball of radius $\rho n$ around any vector in the ambient space does not contain more than $L$ codewords. We study a recently introduced class of…
We design a heuristic method, a genetic algorithm, for the computation of an upper bound of the minimum distance of a linear code over a finite field. By the use of the row reduced echelon form, we obtain a permutation encoding of the…
In this paper we study collections of mutually nearly orthogonal Latin squares ($\text{MNOLS}$), which come from a modification of the orthogonal condition for mutually orthogonal Latin squares. In particular, we find the maximum $\mu$ such…
A Latin square of order $n$ is an $n\times n$ array which contains $n$ distinct symbols exactly once in each row and column. We define the adjacent distance between two adjacent cells (containing integers) to be their difference modulo $n$,…
Let $B_p$ be the Latin square given by the addition table for the integers modulo an odd prime $p$. Here we consider the properties of Latin trades in $B_p$ which preserve orthogonality with one of the $p-1$ MOLS given by the finite field…
In this work we investigate unions of lifted MRD codes of a fixed dimension and minimum distance and derive an explicit formula for the cardinality of such codes. This will then imply a lower bound on the cardinality of constant dimension…
We prove that the combinatorial distance between any two reduced expressions of a given permutation of {1, ..., n} in terms of transpositions lies in O(n^4), a sharp bound. Using a connection with the intersection numbers of certain curves…
In this paper, we introduce a new parameter of a code, referred to as the remoteness, which can be viewed as a dual to the covering radius. Indeed, the remoteness is the minimum radius needed for a single ball to cover all codewords. After…
A linear code with parameters $[n, k, n-k+1]$ is called a maximum distance separable (MDS for short) code. A linear code with parameters $[n, k, n-k]$ is said to be almost maximum distance separable (AMDS for short). A linear code is said…
The main result here is a characterisation of binary $2$-neighbour-transitive codes with minimum distance at least $5$ via their minimal subcodes, which are found to be generated by certain designs. The motivation for studying this class of…
A pair of orthogonal latin cubes of order $q$ is equivalent to an MDS code with distance $3$ or to an ${\rm OA}_1(3,5,q)$ orthogonal array. We construct pairs of orthogonal latin cubes for a sequence of previously unknown orders…
This paper examines the construction of low-density parity-check (LDPC) codes from transversal designs based on sets of mutually orthogonal Latin squares (MOLS). By transferring the concept of configurations in combinatorial designs to the…
A permutation array(or code) of length $n$ and distance $d$, denoted by $(n,d)$ PA, is a set of permutations $C$ from some fixed set of $n$ elements such that the Hamming distance between distinct members $\mathbf{x},\mathbf{y}\in C$ is at…
Permutation codes of length $n$ and distance $d$ is a set of permutations on $n$ symbols, where the distance between any two elements in the set is at least $d$. Subgroup permutation codes are permutation codes with the property that the…