一类新的广义零差分平衡函数及其应用

2008-2009年,丁存生在构造最佳常组合码与优化及完善差分系统中首次引入了零差分平衡(简称ZDB)函数的概念,据此学者们构造出了最佳组成权重码和最优跳频序列.作者将零差分平衡函数的定义推广到一般的广义零差分平衡函数,并利用2分圆陪集构造了一类广义零差分平衡函数,由此构造出一类新的常组合码和差分系统.     

长为{3,4,5}的完备删位纠错码的组合构造

本文研究了混合长度的删位纠错码的构造问题.利用组合设计的方法构造了长为{3,4,5}的完备删位纠错码T(2,{3,4,5},v),当v为正整数且v≠8时,得到了所有的T(2,{3,4,5},v)-码,并给出码字总数的一个上界,T(2,{3,4,5},v)-码的构造推广了长度为单一值的删位纠错码的构造结果.     

基于RTD(q-2,q)的一类组合批处理码

组合批处理码是为了表示如何把n项数据的子集存储到m个服务器里,使得当我们需要n项数据中的任意k项时.都可以通过从每个服务器里选择至多一项(可一般化为t项)来找到这k项,同时让这些服务器总存储量N尽可能小的一类组合结构.具有参数n,k,m的组合批处理码记作(n,N,k,m)-CBC.本文通过可分解横截设计RTD(q一2,q)构造了一类(q~2+q-2,q~3-q~2-2q,q~2-2q-3,q~2-2q)-CBC.比较具有相同参数n.k,m的CBC的N值,本文的构造优于已有构造.     

两类最优循环局部修复码的构造

局部修复码是一种能修复多个故障节点的纠删码,在分布式存储系统中被广泛使用,构造最优局部修复码是目前分布式存储编码理论研究的热点问题之一.文章利用有限域Fq上循环码构造了以下两类具有局部修复性(r,δ)的最优局部修复码:1)[3(q+1),3(q+1)-3δ+1,δ+2],其中 q ≡ 1(mod 6),r+δ-1=q+...     

两类组合设计及其在认证码中的应用

W．Ogata等定义了两种新的组合设计：外差族（EDF）与外平衡不完全区组设计（E-BIBD）．本文首先用有限域中的分圆类给出EDF的一个构造；接着用EBIBD构造出具有完善保密性的最优分裂A-码，然后证明了由满足一定条件的两个EBIBD通过上述方法构造出的两个认证码是同构的.     

构造小域上的最优局部修复码

局部修复码是近几年分布式存储编码领域一个非常热门的研究方向.满足局部修复性r即要求码字的一位能够被其他至多r位恢复.这种性质对于提高分布式存储系统中失效节点的修复效率非常重要.本文主要考虑了域规模小于n时最优局部修复码的构造.具体地,本文构造了有限域F_q(q=r/(r+1)n_1)满足所有位局部修复性r=2、d=6和d=r+1的最优局部修复码.     

有限域上的广义准多项式码

文章提出了一类新的码——广义准多项式码,它是多项式码的一种推广.文中首先给出了广义准多项式码的概念及其生成矩阵的结构,然后重点研究了1-生成元的广义准多项式码参数的相关性质,并且基于这些性质构造出了一些最优码.     

一类线性流形上矩阵方程X^TAX=B的反问题

设Ω={A∈ASR^nxn｜Ax=C，↓Ax∈RT（S），SS^＋C=0，T2^TC2=-C2^TT2，C2T2^＋72=C2}，考虑问题Ⅰ：给定X∈R^nxm，B∈R^mxm求A∈Ω，使得f（A）=｜｜X^TAX—B｜｜=min；问题Ⅱ：给定A^＋∈R^nxm，求A∈SE，使得｜｜A^＋-A｜｜=minA∈SE｜｜A^＋-A｜｜，SE是问题Ⅰ的解集。本文给出了问题Ⅰ、Ⅱ的解的通式，并给出了问题Ikf（A）=0成立的充分必要条件。     

混合效应模型中的D-最优设计

本研究的是混合效应-Ⅱ模型，首先，给出二阶模型存在D-最优回归设计点的必要条件。其次，组合D-最优回归设计和D-最优区组设计BIBD构造D-最优设计并给出寻找新的D-最优设计点的方法。以两种有代表性的组合误差分布为例，阐明新的D-最优设计点的获得过程，并给出了数值结果。     

环■上基于定义集的几类线性码（英文）

本文研究了有限链环■上基于定义集的线性码的李权重分布,其中p是奇素数,m,k是正整数.通过使用特征和工具,我们计算出上述线性码在Gray映射下的完备权重计数器.此外,还提出了一类满足Griesmer界的常数权重码,它可用于构造满足LVFC界限的最优常维码,本文推广了文献[IEEE Trans.Inf. Theory,2015,61(11):5835-5842]和[Des.Codes Cryptogr.,2019,87(1):15-29]的主要结果.     

On Construction of Optimal A2-Codes

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　Ｔｈｅａｕｔｈｅｎｔｉｃａｔｉｏｎｃｏｄｅｓｗｉｔｈａｒｂｉｔｒａｔｉｏｎ (Ａ2 ｃｏｄｅｓ)ａｒｅｉｎｔｒｏｄｕｃｅｄｂｙＳｉｍｍｏｎｓ[1]ａｎｄｓｔｕｄｉｅｄｉｎｍａｎｙｐａｐｅｒｓ (ｆｏｒｅｘａｍｐｌｅ ,[1— 8] ) .Ｏｎｅｏｆｔｈｅｍｏｓｔｉｍｐｏｒｔａｎｔｐｒｏｂｌｅｍｓｉｎｔｈｅｓｔｕｄｙｏｆａｕｔｈｅｎｔｉｃａｔｉｏｎｃｏｄｅｓｉｓｔｏｆｉｎｄｌｏｗｅｒｂｏｕｎｄｓｏｎｃｈｅａｔｉｎｇｐｒｏｂａｂｉｌｉｔｉｅｓａｎｄｏｎｔｈｅｎｕｍｂｅｒｓｏｆｅｎｃｏｄｉ…     

On constructions for optimal two-dimensional optical orthogonal codes

Two-dimensional optical orthogonal codes (2-D OOCs) are of current practical interest in fiber-optic code-division multiple-access networks as they enable optical communication at lower chip rate to overcome the drawbacks of nonlinear effects in large spreading sequences of one-dimensional codes. A 2-D OOC is said to be optimal if its cardinality is the largest possible. In this paper, we develop some constructions for optimal 2-D OOCs using combinatorial design theory. As an application, these constructions are used to construct an infinite family of new optimal 2-D OOCs with auto-correlation 1 and cross-correlation 1.     

Combinatorial Classification of Optimal Authentication Codes with Arbitration

Unconditionallysecure authentication codes with arbitration ( A ²-codes)protect against deceptions from the transmitter and the receiveras well as that from the opponent. We first show that an optimalA ²-code implies an orthogonal array and an affine-resolvable design. Next we define a new design,an affine -resolvable + BIBD,and prove that optimal A ²-codes are equivalentto this new design. From this equivalence, we derive a conditionon the parameters for the existence of optimal A ²-codes.Further, we show tighter lower bounds on the size of keys thanbefore for large sizes of source states which can be consideredas an extension of the bounds on the related designs.     

Perfect codes in the graphs Ok

In this paper we consider the existence of perfect codes in the infinite class of distance-transitive graphs O_k. Perfect 1-codes correspond to certain Steiner systems and necessary conditions for the existence of such a code are satisfied if k + 1 is prime. We give some nonexistence results for perfect 2-, 3-, and 4-codes and for perfect e-codes in general, including a lower bound for k in terms of e.     

Bounds and Combinatorial Structure of (k,n) Multi-Receiver A-Codes

In the model of(k,n) multi-receiver authentication codes ( A-codes),a transmitter broadcasts a message m to nreceivers in such a way that not only an outside opponent butalso any k-1 receivers cannot cheat any other receiver.In this paper, we derive lower bounds on the cheating probabilitiesand the sizes of keys of (k,n) multi-receiver A-codes.The scheme proposed by Desmedt, Frankel and Yung meets all ourbounds with equalities. This means that our bounds are tightand their scheme is optimum. We further show a combinatorialstructure of optimum (k,n) multi-receiver A-codes.A notion of TWOOAs is introduced. A TWOOA is a pair of orthogonalarrays which satisfy a certain condition. We then prove thatan optimum (k,n) multi-receiver A-codeis equivalent to a TWOOA.     

Combinatorial Bounds on Authentication Codes with Arbitration

Unconditionallysecure authentication codes with arbitration ( A²-codes)protect against deceptions from the transmitter and the receiveras well as that from the opponent. In this paper, we presentcombinatorial lower bounds on the cheating probabilities andthe sizes of keys of A²-codes. These bounds areall tight. Our main technique is a reduction of an A²-codeto a splitting A-code.     

Generic constructions for partitioned difference families with applications: a unified combinatorial approach

Partitioned difference families are an interesting class of discrete structures which can be used to derive optimal constant composition codes. There have been intensive researches on the construction of partitioned difference families. In this paper, we consider the combinatorial approach. We introduce a new combinatorial configuration named partitioned relative difference family, which proves to be very powerful in the construction of partitioned difference families. In particular, we present two general recursive constructions, which not only include some existing constructions as special cases, but also generate many new series of partitioned difference families. As an application, we use these partitioned difference families to construct several new classes of optimal constant composition codes.     

Some combinatorial constructions for optimal perfect deletion-correcting codes

There are two kinds of perfect t-deletion-correcting codes of length k over an alphabet of size v, those where the coordinates may be equal and those where all coordinates must be different. We call these two kinds of codes T*(k − t, k, v)-codes and T(k − t, k, v)-codes respectively. The cardinality of a T(k − t, k, v)-code is determined by its parameters, while T*(k − t, k, v)-codes do not necessarily have a fixed size. Let N(k − t, k, v) denote the maximum number of codewords in any T*(k − t, k, v)-code. A T*(k − t, k, v)-code with N(k − t, k, v) codewords is said to be optimal. In this paper, some combinatorial constructions for optimal T*(2, k, v)-codes are developed. Using these constructions, we are able to determine the values of N(2, 4, v) for all positive integers v. The values of N(2, 5, v) are also determined for almost all positive integers v, except for v = 13, 15, 19, 27 and 34.      

Strongly regular graphs associated with ternary bent functions

We prove a new characterization of weakly regular ternary bent functions via partial difference sets. Partial difference sets are combinatorial objects corresponding to strongly regular graphs. Using known families of bent functions, we obtain in this way new families of strongly regular graphs, some of which were previously unknown. One of the families includes an example in [N. Hamada, T. Helleseth, A characterization of some {3v₂+v₃,3v₁+v₂,3,3}-minihypers and some [15,4,9;3]-codes with B₂=0, J. Statist. Plann. Inference 56 (1996) 129-146], which was considered to be sporadic; using our results, this strongly regular graph is now a member of an infinite family. Moreover, this paper contains a new proof that the Coulter-Matthews and ternary quadratic bent functions are weakly regular.     

q-Coverings,codes, and line graphs

In this paper we consider the relationship between q-coverings of a regular graph and perfect 1-codes in line graphs. An infinite class of perfect 1-codes in the line graphs L(I_k) is constructed.