Functional codes arising from quadric intersections with Hermitian varieties

Abstract

We investigate the functional code C h (X) introduced by G. Lachaud [10] in the special case where X is a non-singular Hermitian variety in PG(N, q 2) and h = 2. In [4], F. Edoukou solved the conjecture of Sørensen [11] on the minimum distance of this code for a Hermitian variety X in PG(3, q 2). In this paper, we will answer the question about the minimum distance in general dimension N , with N < O(q 2). We also prove that the small weight codewords correspond to the intersection of X with the union of 2 hyperplanes.

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

“…The number of codewords having these small weights is also calculated. In this way, similar results are obtained to the functional codes C 2 (Q), Q a non-singular quadric [4], and C 2 (X), X a non-singular Hermitian variety [5]. …”

“…In [1,2], Edoukou investigated the functional codes arising from the intersections of quadrics with the non-singular Hermitian variety in PG(3, q 2 ) and PG(4, q 2 ), and the functional codes arising from the intersections of quadrics with the non-singular quadrics and the quadratic cone in PG(3, q). In [5], Hallez and Storme continued this study on the functional codes arising from the intersections of quadrics with the non-singular Hermitian variety in PG(N, q 2 ), N < O(q 2 ). In a first article, the authors extended the results of Edoukou to the functional codes arising from non-singular quadrics in PG(N, q) [4].…”

Structure of functional codes defined on non-degenerate Hermitian varieties

“…143]). Hallez and Storme [10] have proved this conjecture for h = 2 under the condition that n < O (t 2 ).…”

“…The latter conjecture has been solved recently by A. Hallez and L. Storme [10] under the hypothesis n < O (t 2 ). One can also observe in [10, Tables 3.a, 3.b, p. 34] a divisibility condition of the first five weights, though it has not been explicitly mentioned in their paper [10].…”

A study of intersections of quadrics having applications on the small weight codewords of the functional codes C2(Q), Edoukou1, Hallez2, Rodier3 et al. 2010Journal of Pure and Applied AlgebraSelf Cite9000View full textAdd to dashboardCite

“…In [3,4], Edoukou investigated the functional codes arising from the intersections of quadrics with the non-singular Hermitian variety in PG(3, q 2 ) and PG(4, q 2 ), and the functional codes arising from the intersections of quadrics with the non-singular quadrics and the quadratic cone in PG (3, q). In [8], Hallez and Storme continued this study on the functional codes arising from the intersections of quadrics with the non-singular Hermitian variety in PG(N, q 2 ), N < O(q 2 ).…”