The Two-Point Codes with the Designed Distance on a Hermitian Curve in Even Characteristic

In Homma M and Kim SJ [2], the authors considered two-point codes on a Hermitian curve defined over fields of odd characteristic. In this paper, we study the geometry of a Hermitian curve over fields of even characteristic and classify the two-point codes whose minimum distances agree with the designed ones.     

Self-Orthogonal 3-(56,12,65) Designs and Extremal Doubly-Even Self-Dual Codes of Length 56

In this paper, we show that the code generated by the rows of a block-point incidence matrix of a self-orthogonal 3-(56,12,65) design is a doubly-even self-dual code of length 56. As a consequence, it is shown that an extremal doubly-even self-dual code of length 56 is generated by the codewords of minimum weight. We also demonstrate that there are more than one thousand inequivalent extremal doubly-even self-dual [56,28,12] codes. This result shows that there are more than one thousand non-isomorphic self-orthogonal 3-(56,12,65) designs. AMS Classification: 94B05, 05B05     

环F2+uF2+vF2上线性码广义Gray像

本文定义了环F2+uF2+vF2到域F2的广义Gray映射φ像,研究了环F2+uF2+vF2上线性码的广义Gray像.利用广义Gray映射φ的线性性,证明了环F2+uF2+vF2上线性码C的广义Gray像φ(C)满足dH(C)=dH(φ(C))且φ(C⊥)φ(C)⊥.同时,给出了F2+uF2+vF2上循环码C的广义Gray像φ(C)为F2上的4-拟循环码.     

前缀码的嵌入定理

设X*是字母表X上的自由幺半群,以X*为顶点集构造一个语言图Γ(X*),引入语言图Γ(X*)的横截集的概念,给出了前缀码嵌入到极大前缀码的一个构造.     

Types of superregular matrices and the number of n‐arcs and complete n‐arcs in PG (r,q)

Based on the classification of superregular matrices, the numbers of non‐equivalent n‐arcs and complete n‐arcs in PG(r, q) are determined (i) for 4 ≤ q ≤ 19, 2 ≤ r ≤ q ? 2 and arbitrary n, (ii) for 23 ≤ q ≤ 32, r = 2 and n ≥ q ? 8<$>. The equivalence classes over both PGL (k, q) and PΓL(k, q) are considered throughout the examinations and computations. For the classification, an n‐arc is represented by the systematic generator matrix of the corresponding MDS code, without the identity matrix part of it. A rectangular matrix like this is superregular, i.e., it has only non‐singular square submatrices. Four types of superregular matrices are studied and the non‐equivalent superregular matrices of different types are stored in databases. Some particular results on t(r, q) and m′(r, q)—the smallest and the second largest size for complete arcs in PG(r, q)—are also reported, stating that m′(2, 31) = 22, m′(2, 32) = 24, t(3, 23) = 10, and m′(3, 23) = 16. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 363–390, 2006     

The largest strongly consecutive repeat-free code

Answering a question of Körner and Simonyi, this paper gives a strongly consecutive repeat-free code of maximal size in [b]ⁿ.     

Moments of distributions related to digital expansions

The paper studies certain (probability) measures of binomial type defined in a recursive way on the unit interval. These measures are related to the sum-of-digit function and similar quantities. In particular, we undertake an asymptotic analysis of the moments of the corresponding distributions. This is done by a combination of a method based on the Mellin transform and the depoissonisation technique.     

On Balanced Optimal Optical Orthogonal Codes

Variable‐weight optical orthogonal codes (OOCs) were introduced by G.‐C. Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirement. In this paper, by using incomplete difference matrices and perfect relative difference families, a balanced ‐OOC is obtained for every positive integer .     

Spreading Code Design Using a Global Optimization Method

The performance of a code division multiple access system depends on the correlation properties of the employed spreading code. Low cross-correlation values between spreading sequences are desired to suppress multiple access interference and to improve bit error performance. An auto-correlation function with a distinct peak enables proper synchronization and suppresses intersymbol interference. However, these requirements contradict each other and a trade-off needs to be established. In this paper, a global two dimensional optimization method is proposed to minimize the out-of-phase average mean-square aperiodic auto-correlation with average mean-square aperiodic cross-correlation being allowed to lie within a fixed region. This approach is applied to design sets of complex spreading sequences. A design example is presented to illustrate the relation between various correlation characteristics. The correlations of the obtained sets are compared with correlations of other known sequences.     

Skew codes of prescribed distance or rank

In this paper we generalize the notion of cyclic code and construct codes as ideals in finite quotients of non-commutative polynomial rings, so called skew polynomial rings of automorphism type. We propose a method to construct block codes of prescribed rank and a method to construct block codes of prescribed distance. Since there is no unique factorization in skew polynomial rings, there are much more ideals and therefore much more codes than in the commutative case. In particular we obtain a [40, 23, 10]₄ code by imposing a distance and a [42,14,21]₈ code by imposing a rank, which both improve by one the minimum distance of the previously best known linear codes of equal length and dimension over those fields. There is a strong connection with linear difference operators and with linearized polynomials (or q-polynomials) reviewed in the first section.