Quantum codes from nearly self-orthogonal quaternary linear codes

Construction X and its variants are known from the theory of classical error control codes. We present instances of these constructions that produce binary stabilizer quantum error control codes from arbitrary quaternary linear codes. Our construction does not require the classical linear code \(C\) that is used as the ingredient to satisfy the dual containment condition, or, equivalently, \(C^{\perp _h}\) is not required to satisfy the self-orthogonality condition. We prove lower bounds on the minimum distance of quantum codes obtained from our construction. We give examples of record breaking quantum codes produced from our construction. In these examples, the ingredient code \(C\) is nearly dual containing, or, equivalently, \(C^{\perp _h}\) is nearly self-orthogonal, by which we mean that \(\dim (C^{\perp _h})-\dim (C^{\perp _h}\cap C)\) is positive but small.     

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.     

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.     

Lower bounds on the minimum distance in Hermitian one-point differential codes

Korchmáros and Nagy [Hermitian codes from higher degree places. J Pure Appl Algebra, doi: 10. 1016/j.jpaa.2013.04.002, 2013] computed the Weierstrass gap sequence G(P) of the Hermitian function field Fq2( H ) at any place P of degree 3, and obtained an explicit formula of the Matthews-Michel lower bound on the minimum distance in the associated differential Hermitian code CΩ(D, mP ) where the divisor D is, as usual, the sum of all but one 1-degree Fq2-rational places of Fq2( H ) and m is a positive integer. For plenty of values of m depending on q, this provided improvements on the designed minimum distance of CΩ(D, mP). Further improvements from G(P) were obtained by Korchmáros and Nagy relying on algebraic geometry. Here slightly weaker improvements are obtained from G(P) with the usual function-field method depending on linear series, Riemann-Roch theorem and Weierstrass semigroups. We also survey the known results on this subject.     

Ternary self-orthogonal codes of dual distance three and ternary quantum codes of distance three

Ternary self-orthogonal codes with dual distance three and ternary quantum codes of distance three constructed from ternary self-orthogonal codes are discussed in this paper. Firstly, for given code length n ≥ 8, a ternary [n, k]₃ self-orthogonal code with minimal dimension k and dual distance three is constructed. Secondly, for each n ≥ 8, two nested ternary self-orthogonal codes with dual distance two and three are constructed, and consequently ternary quantum code of length n and distance three is constructed via Steane construction. Almost all of these quantum codes constructed via Steane construction are optimal or near optimal, and some of these quantum codes are better than those known before.     

Minimum distance of Hermitian two-point codes

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

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.     

MDS symbol-pair codes from repeated-root cyclic codes

Designs, Codes and Cryptography - Symbol-pair codes are proposed to protect against pair-errors in symbol-pair read channels. The research of symbol-pair codes with the largest possible minimum...     

On codes with covering radius 1 and minimum distance 2

We show that a code C of length n over an alphabet Q of size q with minimum distance 2 and covering radius 1 satisfies |C| ≥ qⁿ⁻¹/(n − 1). For the special case n = q = 4 the smallest known example has |C| = 31. We give a construction for such a code C with |C| = 28.     

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.      

Lower bounds on the minimum average distance of binary codes

Let β(n,M) denote the minimum average Hamming distance of a binary code of length n and cardinality M. In this paper we consider lower bounds on β(n,M). All the known lower bounds on β(n,M) are useful when M is at least of size about 2ⁿ⁻¹/n. We derive new lower bounds which give good estimations when size of M is about n. These bounds are obtained using a linear programming approach. In particular, it is proved that lim_n→∞β(n,2n)=5/2. We also give a new recursive inequality for β(n,M).     

On the minimum average distance of binary constant weight codes

Let β(n,M,w) denote the minimum average Hamming distance of a binary constant weight code with length n, size M and weight w. In this paper, we study the problem of determining β(n,M,w). Using the methods from coding theory and linear programming, we derive several lower bounds on the average Hamming distance of a binary constant weight code. These lower bounds enable us to determine the exact value for β(n,M,w) in several cases.     

Quantum codes from caps

Caps in a finite projective geometry over GF(4) are used for the construction of some quantum error-correcting codes, including an optimal ?27,13,5? code.     

Some new quantum MDS codes from generalized Reed–Solomon codes

Quantum maximum distance separable (MDS) codes form a significant class of quantum codes. In this paper, by using Hermitian self-orthogonal generalized Reed–Solomon codes, we construct two new classes of $q$-ary quantum MDS codes, which have minimum distance greater than $\frac{q}{2}$. Most of these quantum MDS codes are new in the sense that their parameters are not covered by the codes available in the literature.     

关于Goppa 几何码最小距离的一个新方法

引进一个关于Goppa几何码(代数几何码)最小距离界的一个新方法.应用Maharaj的思想(即用显示基来近似表达Riemann-Roch空间)到Goppa几何码的最小距离的界上去.通过厄米特曲线上的代数几何码的一类例子,来证明标准的几何码的下界在某些情形下可以被显著地改进.进一步地,我们给出了这些码的最小距离上界,并说明了我们的下界非常接近这个上界.     

Further improvements on the designed minimum distance of algebraic geometry codes

In the literature about algebraic geometry codes one finds a lot of results improving Goppa’s minimum distance bound. These improvements often use the idea of “shrinking” or “growing” the defining divisors of the codes under certain technical conditions. The main contribution of this article is to show that most of these improvements can be obtained in a unified way from one (rather simple) theorem. Our result does not only simplify previous results but it also improves them further.