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Minimal codes are linear codes where all non-zero codewords are minimal, i.e., whose support is not properly contained in the support of another codeword. The minimum possible length of such a $k$-dimensional linear code over $\mathbb{F}_q$…
The Gilbert type bound for codes in the title is reviewed, both for small and large alphabets. Constructive lower bounds better than these existential bounds are derived from geometric codes, either over Fp or Fp2 ; or over even degree…
Neural codes, represented as collections of binary strings called codewords, are used to encode neural activity. A code is called convex if its codewords are represented as an arrangement of convex open sets in Euclidean space. Previous…
We consider linear error correcting codes associated to higher dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult…
A binary linear error correcting codes represented by two code families Kronecker products sum are considered. The dimension and distance of new code is investigated. Upper and lower bounds of distance are obtained. Some examples are given.…
We prove that a quasiisometric map between rank one symmetric spaces is within bounded distance from a unique harmonic map. In particular, this completes the proof of the Schoen-Li-Wang conjecture.
Grassmannian codes are known to be useful in error-correction for random network coding. Recently, they were used to prove that vector network codes outperform scalar linear network codes, on multicast networks, with respect to the alphabet…
We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show…
The paper deals with the problem of deciding if two finite-dimensional linear subspaces over an arbitrary field are identical up to a permutation of the coordinates. This problem is referred to as the permutation code equivalence. We show…
The present paper mainly studies limits and constructions of insertion and deletion (insdel for short) codes. The paper can be divided into two parts. The first part focuses on various bounds, while the second part concentrates on…
We derive new estimates of the size of codes and orthogonal arrays in the ordered Hamming space (the Niederreiter-Rosenbloom-Tsfasman space). We also show that the eigenvalues of the ordered Hamming scheme, the association scheme that…
The hull of a linear code (i.e., a finite field vector space)~\({\mathcal C}\) is defined to be the vector space formed by the intersection of~\({\mathcal C}\) with its dual~\({\mathcal C}^{\perp}.\) Constructing vector spaces with a…
We present a one-to-one correspondence between equivalence classes of embeddings of a manifold (into a larger manifold of the same dimension) and equivalence classes of certain distances on the manifold. This correspondence allows us to use…
In this paper we consider the weighted $k$-Hamming and $k$-Edit distances, that are natural generalizations of the classical Hamming and Edit distances. As main results of this paper we prove that for any $k\geq 2$ the DECIS-$k$-Hamming…
Let $A(n,d)$ be the maximum number of $0,1$ words of length $n$, any two having Hamming distance at least $d$. We prove $A(20,8)=256$, which implies that the quadruply shortened Golay code is optimal. Moreover, we show $A(18,6)\leq 673$,…
We study the upper bounds for $A(n,d)$, the maximum size of codewords with length $n$ and Hamming distance at least $d$. Schrijver studied the Terwilliger algebra of the Hamming scheme and proposed a semidefinite program to bound $A(n, d)$.…
In this paper we present an extension of known semidefinite and linear programming upper bounds for spherical codes and consider a version of this bound for distance graphs. We apply the main result for the distance distribution of a…
Maximum rank-distance (MRD) codes are extremal codes in the space of $m\times n$ matrices over a finite field, equipped with the rank metric. Up to generalizations, the classical examples of such codes were constructed in the 1970s and are…
In this paper, we produce some new classes of entanglement-assisted quantum MDS codes(EAQMDS codes for short) via generalized Reed-Solomon codes over finite fields of odd characteristic. Among our constructions, there are many EAQMDS codes…
We study bounded deviation of non-deterministic finite transducers under the Hamming distance: the bounded comparison problem asks, given two transducers and $k \in \mathbb{N}$, whether for every input the two transducers produce words at…