On the size of identifying codes in binary hypercubes

In this paper, we consider identifying codes in binary Hamming spaces Fn, i.e., in binary hypercubes. The concept of (r,??)-identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. Currently, the subject forms a topic of its own with several possible applications, for example, to sensor networks.Let us denote by the smallest possible cardinality of an (r,??)-identifying code in Fn. In 2002, Honkala and Lobstein showed for ?=1 that

     

New upper bounds on Lee codes

The paper considers exact values of and upper bounds on the maximal cardinality of a q-ary Lee code of length n with a minimum distance ?d. Special attention is paid to small parameters. Some new results are presented and tables with the presently known best upper bounds are given for q∈{5,6,7} and n?7.     

常重复合码的构造和界

摘要给出了一种Chebyshev距离下的常重复合码的构造,并在其基础上讨论了它的译码算法和优化处理.考虑了Chebyshev距离下的界及其改进.研究了具有Chebyshev距离和Hamming距离的常重复合码的构造,给出了Hamming距离为4的常重复合码的一个结论.     

Constructions and bounds on linear error-block codes

We obtain new bounds on the parameters and we give new constructions of linear error-block codes. We obtain a Gilbert–Varshamov type construction. Using our bounds and constructions we obtain some infinite families of optimal linear error-block codes over . We also study the asymptotic of linear error-block codes. We define the real valued function α _q,m,a (δ), which is an analog of the important real valued function α _q (δ) in the asymptotic theory of classical linear error-correcting codes. We obtain both Gilbert–Varshamov and algebraic geometry type lower bounds on α _q,m,a (δ). We compare these lower bounds in graphs.      

On the asymptotic number of inequivalent binary self-dual codes

Let Ψn be the number of inequivalent self-dual codes in . We prove that , where . Let Δn be the number of inequivalent doubly even self-dual codes in . We also prove that .     

Weight distributions and weight hierarchies of two classes of binary linear codes

Linear codes with a few weights can be applied to communication, consumer electronics and data storage system. In addition, the weight hierarchy of a linear code has many applications such as on the type II wire-tap channel, dealing with t-resilient functions and trellis or branch complexity of linear codes and so on. In this paper, we present a formula for computing the weight hierarchies of linear codes constructed by the generalized method of defining sets. Then, we construct two classes of binary linear codes with a few weights and determine their weight distributions and weight hierarchies completely. Some codes of them can be used in secret sharing schemes.     

Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

An identifying code is a subset of vertices of a graph with the property that each vertex is uniquely determined (identified) by its nonempty neighbourhood within the identifying code. When only vertices out of the code are asked to be identified, we get the related concept of a locating-dominating set. These notions are closely related to a number of similar and well-studied concepts such as the one of a test cover. In this paper, we study the decision problems Identifying Code and Locating-Dominating Set (which consist in deciding whether a given graph admits an identifying code or a locating-dominating set, respectively, with a given size) and their minimization variants Minimum Identifying Code and Minimum Locating-Dominating Set. These problems are known to be NP-hard, even when the input graph belongs to a number of specific graph classes such as planar bipartite graphs. Moreover, it is known that they are approximable within a logarithmic factor, but hard to approximate within any sub-logarithmic factor. We extend the latter result to the case where the input graph is bipartite, split or co-bipartite: both problems remain hard in these cases. Among other results, we also show that for bipartite graphs of bounded maximum degree (at least 3), the two problems are hard to approximate within some constant factor, a question which was open. We summarize all known results in the area, and we compare them to the ones for the related problem Dominating Set. In particular, our work exhibits important graph classes for which Dominating Set is efficiently solvable, but Identifying Code and Locating-Dominating Set are hard (whereas in all previous works, their complexity was the same). We also introduce graph classes for which the converse holds, and for which the complexities of Identifying Code and Locating-Dominating Set differ.     

A class of binary cyclic codes with generalized Niho exponents

In this paper, a class of binary cyclic codes with three generalized Niho-type nonzeros is introduced. Based on some techniques in solving certain equations over finite fields, the proposed cyclic codes are shown to have six nonzero weights and the weight distribution is also completely determined.     

On Codes Identifying Sets of Vertices in Hamming Spaces

A code is called (t, 2)-identifying if for all the words x, y(x y) and the sets (B _t (x) B _t (y)) C and are nonempty and different. Constructions of such codes and a lower bound on the cardinality of these codes are given. The lower bound is shown to be sharp in some cases. We also discuss a more general notion of -identifying codes and introduce weakly identifying codes.     

New lower bounds for the three-dimensional finite bin packing problem

The three-dimensional finite bin packing problem (3BP) consists of determining the minimum number of large identical three-dimensional rectangular boxes, bins, that are required for allocating without overlapping a given set of three-dimensional rectangular items. The items are allocated into a bin with their edges always parallel or orthogonal to the bin edges. The problem is strongly NP-hard and finds many practical applications. We propose new lower bounds for the problem where the items have a fixed orientation and then we extend these bounds to the more general problem where for each item the subset of rotations by 90° allowed is specified. The proposed lower bounds have been evaluated on different test problems derived from the literature. Computational results show the effectiveness of the new lower bounds.     

Two new bounds on the size of binary codes with a minimum distance of three

We prove two new upper bounds on the size of binary codes with a minimum distance of three, namelyA(10, 3)76 andA(11, 3)152.     

On the general excess bound for binary codes with covering radius one

Let K(n,1)$K(n,1)$ denote the minimal cardinality of a binary code of length n$n$ and covering radius one. Fundamental for the theory of lower bounds for K(n,1)$K(n,1)$ is the covering excess method introduced by Johnson and van Wee. Let _δi${\delta }_i$ denote the covering excess on a sphere of radius i$i$, 0≤i≤n$0\le i\le n$. Generalizing an earlier result of van Wee, Habsieger and Honkala showed _δp−1≥p−1${\delta }_{p-1}\ge p-1$ whenever n≡−1$n\equiv -1$ (mod p$p$) for an odd prime p$p$ and _δ0=_δ1=?=_δp−2=0${\delta }_0={\delta }_1=?={\delta }_{p-2}=0$ holds. In the present paper we give the new estimation _δp−1≥(p−2)p−1${\delta }_{p-1}\ge (p-2)p-1$ instead. This answers a question of Habsieger and yields a “general improvement of the general excess bound” for binary codes with covering radius one. The proof uses a classification theorem for certain subset systems as well as new congruence properties for the δ$\delta$-function, which were conjectured by Habsieger.     

Improving bounds on the minimum Euclidean distance for block codes by inner distance measure optimization

The minimum Euclidean distance is a fundamental quantity for block coded phase shift keying (PSK). In this paper we improve the bounds for this quantity that are explicit functions of the alphabet size q, block length n and code size |C|. For q=8, we improve previous results by introducing a general inner distance measure allowing different shapes of a neighborhood for a codeword. By optimizing the parameters of this inner distance measure, we find sharper bounds for the outer distance measure, which is Euclidean.The proof is built upon the Elias critical sphere argument, which localizes the optimization problem to one neighborhood. We remark that any code with q=8 that fulfills the bound with equality is best possible in terms of the minimum Euclidean distance, for given parameters n and |C|. This is true for many multilevel codes.     

Lower bounds of Ramsey numbers based on cubic residues

A method to improve the lower bounds for Ramsey numbers R(k,l) is provided: one may construct cyclic graphs by using cubic residues modulo the primes in the form p=6m+1 to produce desired examples. In particular, we obtain 16 new lower bounds, which are

R(6,12)230, R(5,15)242, R(6,14)284, R(6,15)374,R(6,16)434, R(6,17)548, R(6,18)614, R(6,19)710,R(6,20)878, R(6,21)884, R(7,19)908, R(6,22)1070,R(8,20)1094, R(7,21)1214, R(9,20)1304, R(8,21)1328.

     

Lower bounds on covering codes via partition matrices

Let Kq(n,R) denote the minimal cardinality of a q-ary code of length n and covering radius R. Let σq(n,s;r) denote the minimal cardinality of a q-ary code of length n, which is s-surjective with radius r. In order to lower-bound Kq(n,n−2) and σq(n,s;s−2) we introduce partition matrices and their transversals. Our approach leads to a short new proof of a classical bound of Rodemich on Kq(n,n−2) and to the new bound Kq(n,n−2)?3q−2n+2, improving the first iff 5?n<q?2n−4. We determine Kq(q,q−2)=q−2+σ₂(q,2;0) if q?10. Moreover, we obtain the new powerful recursive bound Kq+1(n+1,R+1)?min{2(q+1),Kq(n,R)+1}.