Further improvements on asymptotic bounds for codes using distinguished divisors

For a prime power q, let ${\alpha }_q$ be the standard function in the asymptotic theory of codes, that is, ${\alpha }_q(\delta )$ is the largest asymptotic information rate that can be achieved for a given asymptotic relative minimum distance δ of q-ary codes. In recent years the Tsfasman–Vlăduţ–Zink lower bound on ${\alpha }_q(\delta )$ was improved by Elkies, Xing, and Niederreiter and Özbudak. In this paper we show further improvements on these bounds by using distinguished divisors of global function fields.     

Invariant sets under permutation,extremal graphs,and covering codes

Let $c_q(n,R)$ denote the minimum cardinality of a subset H in ${\mathbb{F}}_q^n$ such that every word in this space differs in at most R coordinates from a multiple of a vector in H, where q is a prime or a prime power. In order to explore the symmetries of such coverings, we investigate a few algebraic properties of invariant sets under permutation. Extremal problems arising from invariant sets are also studied on a graph theoretical viewpoint. As an application, a new class of upper bounds on $c_q(n,R)$ is constructed.     

The minimum size of a finite subspace partition

A subspace partition of P=PG(n,q) is a collection of subspaces of P whose pairwise intersection is empty. Let σ_q(n,t) denote the minimum size (i.e., minimum number of subspaces) in a subspace partition of P in which the largest subspace has dimension t. In this paper, we determine the value of σ_q(n,t) for . Moreover, we use the value of σ_q(2t+2,t) to find the minimum size of a maximal partial t-spread in PG(3t+2,q).     

On the distance between linear codes

Let V be an n-dimensional vector space over the finite field consisting of q elements and let ${\mathrm{\Gamma }}_k(V)$ be the Grassmann graph formed by k-dimensional subspaces of V, $1<k<n-1$. Denote by $\mathrm{\Gamma }{(n,k)}_q$ the restriction of ${\mathrm{\Gamma }}_k(V)$ to the set of all non-degenerate linear ${[n,k]}_q$ codes. We show that for any two codes the distance in $\mathrm{\Gamma }{(n,k)}_q$ coincides with the distance in ${\mathrm{\Gamma }}_k(V)$ only in the case when $n<{(q+1)}^2+k-2$, i.e. if n is sufficiently large then for some pairs of codes the distances in the graphs ${\mathrm{\Gamma }}_k(V)$ and $\mathrm{\Gamma }{(n,k)}_q$ are distinct. We describe one class of such pairs.     

The Galois variance of constacyclic codes

This paper studies the Galois images of constacyclic codes over ${\mathbb{F}}_{q^m}$ of length relatively prime to q, and determines when those images are equal and when they intersect only at the zero codeword. The subfield subcodes and trace codes of constacyclic codes are also determined.     

Codes defined by forms of degree 2 on Hermitian surfaces and Sørensen's conjecture

We study the functional codes $C_h(X)$ defined by Lachaud in [G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties, in: Arithmetic, Geometry, and Coding Theory, Luminy, France, 1993, de Gruyter, Berlin, 1996, pp. 77–104] where $X\subset {\mathbb{P}}^N$ is an algebraic projective variety of degree d and dimension m. When X is a Hermitian surface in $\mathit{PG}(3,q)$, Sørensen in [A.B. Sørensen, Rational points on hypersurfaces, Reed–Muller codes and algebraic-geometric codes, PhD thesis, Aarhus, Denmark, 1991], has conjectured for $h⩽t$ (where $q=t^2$) the following result:$\mathrm{\#}{X}_{Z(f)}({\mathbb{F}}_q)⩽h(t^3+t^2-t)+t+1$ which should give the exact value of the minimum distance of the functional code $C_h(X)$. In this paper we resolve the conjecture of Sørensen in the case of quadrics (i.e. $h=2$), we show the geometrical structure of the minimum weight codewords and their number; we also estimate the second weight and the geometrical structure of the codewords reaching this second weight.