Structure of functional codes defined on non-degenerate Hermitian varieties

Abstract

We study the functional codes of order h defined by G. Lachaud on a non-degenerate Hermitian variety, by exhibiting a result on divisibility for all the weights of such codes. In the case where the functional code is defined by evaluating quadratic functions on the non-degenerate Hermitian surface, we list the first five weights, describe the geometrical structure of the corresponding quadrics and give a positive answer to a conjecture formulated on this question by Edoukou (2009) [8]. The paper ends with two conjectures. The first is about the divisibility of the weights in the functional codes. The second is about the minimum distance and the distribution of the codewords of the first 2h + 1 weights.

The small weight codewords of the functional codes associated to non-singular Hermitian varieties

“…This result is a particular case of a more general result on the divisibility of the functional codes C h (X), defined on the non-singular Hermitian variety X of PG(N, q 2 ) by the hypersurfaces of degree h [3]. To achieve this goal, the known result is first of all mentioned that a Hermitian variety X in PG(N, q 2 ) can be made to correspond to a quadric in PG(2N + 1, q).…”

A note on diagonal and Hermitian hypersurfaces

“…where the function B(d, s), given in Equation (2), will be useful in much of the sequel. The Tsfasman-Serre-Sorensen bound [10,21] for the same quantity is given by:…”

Linear codes from Denniston maximal arcs

“…The length of C V (X ) is n, the dimension is dim Fq η(V), and the minimum distance coincides with n − max f ∈V, η(f ) =0 #(V f ∩ X ), where V f denotes the set of zeros of f . The case where X consists of the set of F q -rational points of a quadric or a hermitian variety and V is a vector space of polynomials of a given degree has been thoroughly investigated; see for instance [3,4,[11][12][13][14]. In particular, functional codes from the Hermitian curve, the so-called Hermitain codes, have performances which are sometimes comparable with those of BCH codes; see e.g.…”