Xiang-dong Hou
Let $Ψ_{k,n}$ denote the number of inequivalent binary self-orthogonal $[n,k]$ codes. We present a method which allows us to compute $Ψ_{k,n}$ explicitly for a moderate $k$ and an arbitrary $n$. Included in this paper are explicit formulas for $Ψ_{k,n}$ with $k\le 5$.
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The purpose of this paper is to complete the classification of binary self-dual [48,24,10] codes with an automorphism of odd prime order. We prove that if there is a self-dual [48, 24, 10] code with an automorphism of type p-(c,f) with p being an odd prime, then p=3, c=16, f=0. By considering only an automorphism of type 3-(16,0), we prove that there are exactly 264 inequivalent self-dual [48, 24, 10] codes with an automorphism of odd prime order, equivalently, there are exactly 264 inequivalent cubic self-dual [48, 24, 10] codes.
Generalized perfect binary arrays (GPBAs) were used by Jedwab to construct perfect binary arrays. A non-trivial GPBA can exist only if its energy is $2$ or a multiple of $4$. This paper introduces generalized optimal binary arrays (GOBAs) with even energy not divisible by $4$, as analogs of GPBAs. We give a procedure to construct GOBAs based on a characterization of the arrays in terms of $2$-cocycles. As a further application, we determine negaperiodic Golay pairs arising from generalized optimal binary sequences of small length.
In recent times, the construction of deterministic matrices has gained popularity as an alternative of random matrices as they provide guarantees for recovery of sparse signals. In particular, the construction of binary matrices has attained significance due to their potential for hardware-friendly implementation and appealing applications. Our present work aims at constructing incoherent binary matrices consisting of orthogonal blocks with small block coherence. We show that the binary matrices constructed from Euler squares exhibit block orthogonality and possess low block coherence. With a …
It is known that correlation-immune (CI) Boolean functions used in the framework of side-channel attacks need to have low Hamming weights. The supports of CI functions are (equivalently) simple orthogonal arrays when their elements are written as rows of an array. The minimum Hamming weight of a CI function is then the same as the minimum number of rows in a simple orthogonal array. In this paper, we use Rao's Bound to give a sufficient condition on the number of rows, for a binary orthogonal array (OA) to be simple. We apply this result for determining the minimum number of rows in all simple…
In this note, we demonstrate that every binary doubly even self-dual code of length $40$ can be realized as the residue code of some extremal Type II $\mathbb{Z}_4$-code. As a consequence, it is shown that there are at least $94356$ inequivalent extremal Type II $\mathbb{Z}_4$-codes of length $40$.
In this paper we generalize the construction of binary self-orthogonal codes obtained from weakly self-orthogonal designs described by Tonchev in [12] in order to obtain self-orthogonal codes over an arbitrary field. We extend construction self-orthogonal codes from orbit matrices of self-orthogonal designs and weakly self-orthogonal 1-designs such that block size is odd and block intersection numbers are even described in [5]. Also, we generalize mentioned construction in order to obtain self-orthogonal codes over an arbitrary field. We construct weakly self-orthogonal designs invariant under…
Binary self-orthogonal codes and balanced incomplete block designs are two combinatorial configurations that have been much studied because of their wide areas of application. In this paper, we have shown the distribution of (16; 6; 3)-designs in binary (25,12) self-orthogonal codes. The paper also presents the relationships among the codes with embedded designs.
Maximal sets of mutually orthogonal frequency squares and Doehlert-Klee designs
本文证明了二元相互正交频率方阵（MOFS）集合等价于特定的Doehlert-Klee设计，并利用该联系及循环生成方法给出了类型极大二元MOFS集合的新构造。
First Subconstituents of Orthogonal Graphs in Even Characteristic:Automorphisms, Cliques, Cores, and Binary Triple Transitivity
本文研究偶特征正交图第一子构成的结构，证明其为仿射极图补图，确定其自同构群、独立数、色数、最大团及核，并给出二元情形三重传递性的简短证明。
A \emph{frequency square} is a matrix in which each row and column is a permutation of the same multiset of symbols. We consider only {\em binary} frequency squares of order $n$ with $n/2$ zeroes and $n/2$ ones in each row and column. Two such frequency squares are \emph{orthogonal} if, when superimposed, each of the 4 possible ordered pairs of entries occurs equally often. In this context we say that a $k$-MOFS$(n)$ is a set of $k$ binary frequency squares of order $n$ in which each pair of squares is orthogonal. A $k$-MOFS$(n)$ must satisfy $k\le(n-1)^2$, and any MOFS achieving this bound ar…
Mutually orthogonal binary frequency squares of mixed type
本文研究混合型二元频率方集合的互正交性，推广了 MOFS 基数的经典上界，并利用大于 2 的模的论证给出新的类型极大性充分条件。
With a computer-aided approach based on the connection with equitable partitions, we establish the uniqueness of the orthogonal array OA$(1536,13,2,7)$, constructed in [D.G.Fon-Der-Flaass. Perfect $2$-Colorings of a Hypercube, Sib. Math. J. 48 (2007), 740-745] as an equitable partition of the $13$-cube with quotient matrix $[[0,13],[3,10]]$. By shortening the OA$(1536,13,2,7)$, we obtain $3$ inequivalent orthogonal arrays OA$(768,12,2,6)$, which is a complete classification for these parameters too. After our computing, the first parameters of unclassified binary orthogonal arrays OA$(N,n,2,t)…
The minimum distance graph of a code has the codewords as vertices and edges exactly when the Hamming distance between two codewords equals the minimum distance of the code. A constructive proof for reconstructibility of an extended perfect binary one-error-correcting code from its minimum distance graph is presented. Consequently, inequivalent such codes have nonisomorphic minimum distance graphs. Moreover, it is shown that the automorphism group of a minimum distance graph is isomorphic to that of the corresponding code.
Binary LCD Codes and Their Graph Representations
本文研究了生成二进制线性互补双码的简单图的特征，统一了完整图、海明图、约翰逊图和格拉索姆图等案例，并证明了非同构会议图在特定条件下生成不同码。
Shortest self-orthogonal embeddings of binary linear codes
本文利用线性码的 hull 性质确定二元线性码的最短自正交嵌入长度，提出从汉明码构造自对偶码的算法，构造出多个最优自正交码及具新参数的 8 维码。
Orthogonal designs are fundamental mathematical notions used in the construction of space time block codes for wireless transmissions. Designs have two important parameters, the rate and the decoding delay; the main problem of the theory is to construct designs maximizing the rate and minimizing the decoding delay. All known constructions of CODs are inductive or algorithmic. In this paper, we present an explicit construction of optimal CODs. We do not apply recurrent procedures and do calculate the matrix elements directly. Our formula is based on a cubic function in two binary n-vectors. In …
Correlation immune Boolean functions play an important role in the implementation of efficient masking countermeasures for side-channel attacks in cryptography. In this paper, we investigate a method to construct correlation immune functions through families of mutually orthogonal cellular automata (MOCA). First, we show that the orthogonal array (OA) associated to a family of MOCA can be expanded to a binary OA of strength at least 2. To prove this result, we exploit the characterization of MOCA in terms of orthogonal labelings on de Bruijn graphs. Then, we use the resulting binary OA to defi…
Self-Orthogonal Cellular Automata
本文研究双置换元胞自动机生成自正交拉丁方的条件，枚举小直径规则并给出线性CA自正交性的完整理论刻画，证明二元情形下不可约性为充分条件，并将判定简化为局部规则中心系数奇偶性计算。
Linear codes with large minimal distances are important error correcting codes in information theory.Orthogonal codes have more applications in the other fields of mathematics. In this paper, we study the binary and ternary orthogonal codes generated by the weight matrices on finite-dimensional modules of simple Lie algebras. The Weyl groups of the Lie algebras act on these codes isometrically. It turns out that certain weight matrices of $sl(n,\mbb{C})$ and $o(2n,\mbb{C})$ generate doubly-even binary orthogonal codes and ternary orthogonal codes with large minimal distances. Moreover, we prov…
The classification of orthogonal arrays OA(2048,14,2,7) and some completely regular codes
本文对 OA(2048,14,2,7) 及其等价的 14 维立方体完全正则码进行完全分类，并确定派生正交表和相关完全正则码的等价类数量。
The design of binary error-correcting codes is a challenging optimization problem with several applications in telecommunications and storage, which has also been addressed with metaheuristic techniques and evolutionary algorithms. Still, all these efforts focused on optimizing the minimum distance of unrestricted binary codes, i.e., with no constraints on their linearity, which is a desirable property for efficient implementations. In this paper, we present an Evolutionary Strategy (ES) algorithm that explores only the subset of linear codes of a fixed length and dimension. To that end, we re…
From a given $[n, k]$ code $C$, we give a method for constructing many $[n, k]$ codes $C'$ such that the hull dimensions of $C$ and $C'$ are identical. This method can be applied to constructions of both self-dual codes and linear complementary dual codes (LCD codes for short). Using the method, we construct 661 new inequivalent extremal doubly even $[56, 28, 12]$ codes. Furthermore, constructing LCD codes by the method, we improve some of the previously known lower bounds on the largest minimum weights of binary LCD codes of length $n=26,28 \le n \le 40$.
A classic result of Delsarte connects the strength (as orthogonal array) of a linear code with the minimum weight of its dual: the former is one less than the latter. We show that Delsarte's observation extends to codes over arbitrary finite rings. Since the paper of Hammons \emph{et al.}, there is a lot of interest in codes over rings, especially in codes over $Z_4$ and their (usually non-linear) binary Gray map images. We show that Delsarte's observation extends to codes over arbitrary finite commutative rings with identity. Also, we show that the strength of the Gray map image of a $Z_4$ co…
On the weight distribution of random binary linear codes
该研究聚焦随机二元线性码的重量分布问题，通过构造特定正交补码空间，推导了计数特定汉明重量码字的随机变量的各阶矩的渐近性质，为编码理论相关研究提供了理论支撑。
We give new proofs of asymptotic upper bounds of coding theory obtained within the frame of Delsarte's linear programming method. The proofs rely on the analysis of eigenvectors of some finite-dimensional operators related to orthogonal polynomials. The examples of the method considered in the paper include binary codes, binary constant-weight codes, spherical codes, and codes in the projective spaces.
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