A characterization of Hermitian varieties as codewords

It is known that the Hermitian varieties are codewords in the code defined by the points and hyperplanes of the projective spaces $PG(r,q^2)$. In finite geometry, also quasi-Hermitian varieties are defined. These are sets of points of $PG(r,q^2)$ of the same size as a non-singular Hermitian variety of $PG(r,q^2)$, having the same intersection sizes with the hyperplanes of $PG(r,q^2)$. In the planar case, this reduces to the definition of a unital. A famous result of Blokhuis, Brouwer, and Wilbrink states that every unital in the code of the points and lines of $PG(2,q^2)$ is a Hermitian curve. We prove a similar result for the quasi-Hermitian varieties in $PG(3,q^2)$, $q=p^{h}$, as well as in $PG(r,q^2)$, $q=p$ prime, or $q=p^2$, $p$ prime, and $r\geq 4$

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

This article studies the small weight codewords of the functional code C (Herm) (X), with X a non-singular Hermitian variety of PG(N, q (2)). The main result of this article is that the small weight codewords correspond to the intersections of X with the singular Hermitian varieties of PG(N, q (2)) consisting of q + 1 hyperplanes through a common (N - 2)-dimensional space I , forming a Baer subline in the quotient space of I . 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 (Edoukou et al., J. Pure Appl. Algebra 214:1729-1739, 2010), and C (2)(X), X a non-singular Hermitian variety (Hallez and Storme, Finite Fields Appl. 16:27-35, 2010)

Functional codes arising from quadric intersections with Hermitian varieties

AbstractWe investigate the functional code Ch(X) introduced by G. Lachaud (1996) [10] in the special case where X is a non-singular Hermitian variety in PG(N,q2) and h=2. In [4], F.A.B. Edoukou (2007) solved the conjecture of Sørensen (1991) [11] on the minimum distance of this code for a Hermitian variety X in PG(3,q2). In this paper, we will answer the question about the minimum distance in general dimension N, with N<O(q2). We also prove that the small weight codewords correspond to the intersection of X with the union of 2 hyperplanes

Intersection sets, three-character multisets and associated codes

In this article we construct new minimal intersection sets in ${\mathrm{AG}}(r,q^2)$ sporting three intersection numbers with hyperplanes; we then use these sets to obtain linear error correcting codes with few weights, whose weight enumerator we also determine. Furthermore, we provide a new family of three-character multisets in ${\mathrm{PG}}(r,q^2)$ with $r$ even and we also compute their weight distribution.Comment: 17 Pages; revised and corrected result

Line Hermitian Grassmann codes and their parameters

In this paper we introduce and study line Hermitian Grassmann codes as those subcodes of the Grassmann codes associated to the 2-Grassmannian of a Hermitian polar space defined over a finite field. In particular, we determine the parameters and characterize the words of minimum weight