Hermitian varieties as codewords

We show that the incidence vector of any hermitian variety in the projective geometry PG _m–1(F _q ²), where q = p ^t , and p is a prime, is in the code over F _p of the symmetric design of points and hyperplanes of the geometry by using the theorem of Delsarte [8] that identifies this code with a nonprimitive generalized Reed-Muller code.     

Structure of functional codes defined on non-degenerate Hermitian varieties

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.     

On the functional codes defined by quadrics and Hermitian varieties

In recent years, functional codes have received much attention. In his PhD thesis, F.A.B. Edoukou investigated various functional codes linked to quadrics and Hermitian varieties defined in finite projective spaces (Edoukou, PhD Thesis, 2007). This work was continued in (Edoukou et al., Des Codes Cryptogr 56:219–233, 2010; Edoukou et al., J Pure Appl Algebr 214:1729–1739, 2010; Hallez and Storme, Finite Fields Appl 16:27–35, 2010), where the results of the thesis were improved and extended. In particular, Hallez and Storme investigated the functional codes ${C_2(\mathcal{H})}$ , with ${\mathcal{H}}$ a non-singular Hermitian variety in PG(N, q ²). The codewords of this code are defined by evaluating the points of ${\mathcal{H}}$ in the quadratic polynomials defined over ${\mathbb{F}_{q^2}}$ . We now present the similar results for the functional code ${C_{Herm}(\mathcal{Q})}$ . The codewords of this code are defined by evaluating the points of a non-singular quadric ${\mathcal{Q}}$ in PG(N, q ²) in the polynomials defining the Hermitian varieties of PG(N, q ²).     

Small weight codewords in the codes arising from Desarguesian projective planes

We study codewords of small weight in the codes arising from Desarguesian projective planes. We first of all improve the results of K. Chouinard on codewords of small weight in the codes arising from PG(2, p), p prime. Chouinard characterized all the codewords up to weight 2p in these codes. Using a particular basis for this code, described by Moorhouse, we characterize all the codewords of weight up to 2p + (p−1)/2 if p ≥ 11. We then study the codes arising from . In particular, for q ₀ = p prime, p ≥ 7, we prove that the codes have no codewords with weight in the interval [q + 2, 2q − 1]. Finally, for the codes of PG(2, q), q = p ^h , p prime, h ≥ 4, we present a discrete spectrum for the weights of codewords with weights in the interval [q + 2, 2q − 1]. In particular, we exclude all weights in the interval [3q/2, 2q − 1]. Geertrui Van de Voorde research is supported by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen) Joost Winne was supported by the Fund for Scientific Research - Flanders (Belgium).     

Small weight codewords in the LDPC codes arising from linear representations of geometries

In this article, we investigate the minimum distance and small weight codewords of the LDPC codes of linear representations, using only geometrical methods. First, we present a new lower bound on the minimum distance and we present a number of cases in which this lower bound is sharp. Then we take a closer look at the cases and with a hyperoval, hence q even, and characterize codewords of small weight. When investigating the small weight codewords of , we deal with the case of a regular hyperoval, that is, a conic and its nucleus, separately, since in this case, we have a larger upper bound on the weight for which the results are valid. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 1–24, 2009     

Linear codes associated to skew-symmetric determinantal varieties

In this article we consider linear codes coming from skew-symmetric determinantal varieties, which are defined by the vanishing of minors of a certain fixed size in the space of skew-symmetric matrices. In odd characteristic, the minimum distances of these codes are determined and a recursive formula for the weight of a general codeword in these codes is given.     

The second generalized Hamming weight for two-point codes on a Hermitian curve

The aim of this article is the determination of the second generalized Hamming weight of any two-point code on a Hermitian curve of degree q + 1. The determination involves results of Coppens on base-point-free pencils on a plane curve. To avoid non- essential trouble, we assume that q > 4.      

Small weight codewords in LDPC codes defined by (dual) classical generalized quadrangles

We find lower bounds on the minimum distance and characterize codewords of small weight in low-density parity check (LDPC) codes defined by (dual) classical generalized quadrangles. We analyze the geometry of the non-singular parabolic quadric in PG(4,q) to find information about the LDPC codes defined by Q (4,q), and . For , and , we are able to describe small weight codewords geometrically. For , q odd, and for , we improve the best known lower bounds on the minimum distance, again only using geometric arguments. Similar results are also presented for the LDPC codes LU(3,q) given in [Kim, (2004) IEEE Trans. Inform. Theory, Vol. 50: 2378–2388]     

The dual geometry of Hermitian two-point codes

In this paper, we study the algebraic geometry of any two-point code on the Hermitian curve and reveal the purely geometric nature of their dual minimum distance. We describe the minimum-weight codewords of many of their dual codes through an explicit geometric characterization of their supports. In particular, we show that they appear as sets of collinear points in many cases.     

LDPC codes from the Hermitian curve

In this paper, we study the code which has as parity check matrix the incidence matrix of the design of the Hermitian curve and its (q + 1)-secants. This code is known to have good performance with an iterative decoding algorithm, as shown by Johnson and Weller in (Proceedings at the ICEE Globe com conference, Sanfrancisco, CA, 2003). We shall prove that has a double cyclic structure and that by shortening in a suitable way it is possible to obtain new codes which have higher code-rate. We shall also present a simple way to constructing the matrix via a geometric approach.      

The lengths of Hermitian self-dual extended duadic codes

Duadic codes are a class of cyclic codes that generalize quadratic residue codes from prime to composite lengths. For every prime power q, we characterize integers n such that there is a duadic code of length n over F_q² with a Hermitian self-dual parity-check extension. We derive asymptotic estimates for the number of such n as well as for the number of lengths for which duadic codes exist.     

On the subfield subcodes of Hermitian codes

We present a fast algorithm using Gröbner basis to compute the dimensions of subfield subcodes of Hermitian codes. With these algorithms we are able to compute the exact values of the dimension of all subfield subcodes up to q ≤  32 and length up to 2¹⁵. We show that some of the subfield subcodes of Hermitian codes are at least as good as the previously known codes, and we show the existence of good long codes.     

Minimum distance of Hermitian two-point codes

We prove a formula for the minimum distance of two-point codes on a Hermitian curve.