Hermitian codes as generalized Reed-Solomon codes

Hermitian codes obtained from Hermitian curves are shown to be concatenated generalized Reed-Solomon codes. This interpretation of Hermitian codes is used to investigate their structure. An efficient encoding algorithm is given for Hermitian codes. A new general decoding algorithm is given and applied to Hermitian codes to give a decoding algorithm capable of decoding up to the full error correcting capability of the code.This work is supported by a Natural Science and Engineering Research Council Grant A7382.     

Hermitian LCD codes from cyclic codes

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. It was proved that asymptotically good Hermitian LCD codes exist. The objective of this paper is to construct some cyclic Hermitian LCD codes over finite fields and analyse their parameters. The dimensions of these codes are settled and the lower bounds on their minimum distances are presented. Most Hermitian LCD codes presented in this paper are not BCH codes.     

Hermitian rank distance codes

Let \(X=X(n,q)\) be the set of \(n\times n\) Hermitian matrices over \(\mathbb {F}_{q^2}\). It is well known that X gives rise to a metric translation association scheme whose classes are induced by the rank metric. We study d-codes in this scheme, namely subsets Y of X with the property that, for all distinct \(A,B\in Y\), the rank of \(A-B\) is at least d. We prove bounds on the size of a d-code and show that, under certain conditions, the inner distribution of a d-code is determined by its parameters. Except if n and d are both even and \(4\le d\le n-2\), constructions of d-codes are given, which are optimal among the d-codes that are subgroups of \((X,+)\). This work complements results previously obtained for several other types of matrices over finite fields.     

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.      

Euclidean and Hermitian LCD MDS codes

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. On the other hand, being optimal codes, maximum distance separable codes (abbreviated MDS) are of much interest from many viewpoints due to their theoretical and practical properties. However, little work has been done on LCD MDS codes. In particular, determining the existence of q-ary [n, k] LCD MDS codes for various lengths n and dimensions k is a basic and interesting problem. In this paper, we firstly study the problem of the existence of q-ary [n, k] LCD MDS codes and solve it for the Euclidean case. More specifically, we show that for \(q>3\) there exists a q-ary [n, k] Euclidean LCD MDS code, where \(0\le k \le n\le q+1\), or, \(q=2^{m}\), \(n=q+2\) and \(k= 3 \text { or } q-1\). Secondly, we investigate several constructions of new Euclidean and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and Hermitian LCD MDS codes use some linear codes with small dimension or codimension, self-orthogonal codes and generalized Reed-Solomon codes.     

On maximum additive Hermitian rank-metric codes

Journal of Algebraic Combinatorics - Inspired by the work of Zhou (Des Codes Cryptogr 88:841–850, 2020) based on the paper of Schmidt (J Algebraic Combin 42(2):635–670, 2015), we...     

Minimum distance of Hermitian two-point codes

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

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.     

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.     

Construction of binary LCD codes,ternary LCD codes and quaternary Hermitian LCD codes

We give two methods for constructing many linear complementary dual (LCD for short) codes from a given LCD code, by modifying some known methods for constructing self-dual codes. Using the methods, we construct binary LCD codes and quaternary Hermitian LCD codes, which improve the previously known lower bounds on the largest minimum weights.

     

Quantum MDS codes with relatively large minimum distance from Hermitian self-orthogonal codes

It has become common knowledge that constructing q-ary quantum MDS codes with minimum distance bigger than \(q/2+1\) is significantly more difficult than constructing those with minimum distance less than or equal to \(q/2+1\). Despite of various constructions of q-ary quantum MDS codes, all known q-ary quantum MDS codes have minimum distance bounded by \(q/2+1\) except for some lengths. The purpose of the current paper is to provide some new q-ary quantum MDS codes with minimum distance bigger than \(q/2+1\). In this paper, we provide several classes of quantum MDS codes with minimum distance bigger than \(q/2+1\). For instance, some examples in these classes include q-ary \([n,n-2k, k+1]\)-quantum MDS codes for cases: (i) \(q\equiv -1\bmod {5}, n=(q^2+4)/5\) and \(1\le k\le (3q-2)/5\); (ii) \(q\equiv -1\bmod {7}, n=(q^2+6)/7\) and \(1\le k\le (4q-3)/7\); (iii) \(2|q, q\equiv -1\bmod {3}, n=2(q^2-1)/3\) and \(1\le k\le (2q-1)/3\); and (iv) \(2|q, q\equiv -1\bmod {5}, n=2(q^2-1)/5\) and \(1\le k\le (3q-2)/5\).     

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.     

Generalized Almost Hermitian Spaces and Holomorphically Projective Mappings

Mediterranean Journal of Mathematics - We derive some interesting properties of generalized almost Hermitian spaces. Additionally, we study basic equations of holomorphically projective mappings...     

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).     

Generalized weighing matrices and self-orthogonal codes

It is shown that the row spaces of certain generalized weighing matrices, Bhaskar Rao designs, and generalized Hadamard matrices over a finite field of order q are hermitian self-orthogonal codes. Certain matrices of minimum rank yield optimal codes. In the special case when q=4, the codes are linked to quantum error-correcting codes, including some codes with optimal parameters.     

Bounds on collaborative decoding of interleaved Hermitian codes and virtual extension

We derive the maximum decoding radius for interleaved Hermitian (IH) codes if a collaborative decoding scheme is used. A decoding algorithm that achieves this bound, which is based on a division decoding algorithm, is given. Based on the decoding radius for the interleaved codes, we derive a bound on the code rate below which virtual extension of non-interleaved Hermitian codes can improve the decoding capabilities.