Search

Diameter perfect codes form a natural generalization for perfect codes. They are based on the code-anticode bound which generalizes the sphere-packing bound. The code-anticode bound was proved by Delsarte for distance-regular graphs and it holds for some other metrics too. In this paper we prove the bound for non-binary constant-weight codes with the Hamming metric and characterize the diameter perfect codes and the maximum size anticodes for these codes. We distinguish between six families of non-binary diameter constant-weight codes and four families of maximum size non-binary constant-weight anticodes. Each one of these families of diameter perfect codes raises some different questions. We consider some of these questions and leave lot of ground for further research. Finally, as a consequence, some $t$ -intersecting families related to the well-known Erdös-Ko-Rado theorem, are constructed. [ABSTRACT FROM AUTHOR]

An $[{n}, {k}, {n}-{k}+1]$ linear code is called an MDS code. An $[{n}, {k}, {n}-{k}]$ linear code is said to be almost maximum distance separable (almost MDS or AMDS for short). A code is said to be near maximum distance separable (near MDS or NMDS for short) if the code and its dual code both are almost maximum distance separable. The first near MDS code was the [11, 6, 5] ternary Golay code discovered in 1949 by Golay. This ternary code holds 4-designs, and its extended code holds a Steiner system ${S}(5, 6, 12)$ with the largest strength known. In the past 70 years, sporadic near MDS codes holding t-designs were discovered and a lot of infinite families of near MDS codes over finite fields were constructed. However, the question as to whether there is an infinite family of near MDS codes holding an infinite family of t-designs for $ {t}\geq 2$ remains open for 70 years. This paper settles this long-standing problem by presenting an infinite family of near MDS codes over $ {GF}(3^{ {s}})$ holding an infinite family of 3-designs and an infinite family of near MDS codes over $ {GF}(2^{2 {s}})$ holding an infinite family of 2-designs. The subfield subcodes of these two families of codes are also studied, and are shown to be dimension-optimal or distance-optimal. [ABSTRACT FROM AUTHOR]

In this paper, locally repairable codes which have optimal minimum Hamming distance with respect to the bound presented by Prakash et al. are considered. New upper bounds on the length of such optimal codes are derived. The new bounds apply to more general cases, and have weaker requirements compared with the known ones. In this sense, they both improve and generalize previously known bounds. Further, optimal codes are constructed, whose length is order-optimal with respect to the new upper bounds. Notably, the length of the codes is super-linear in the alphabet size. [ABSTRACT FROM AUTHOR]

Boolean functions, coding theory and $t$ -designs have close connections and interesting interplay. A standard approach to constructing $t$ -designs is the use of linear codes with certain regularity. The Assmus-Mattson Theorem and the automorphism groups are two ways for proving that a code has sufficient regularity for supporting $t$ -designs. However, some linear codes hold $t$ -designs, although they do not satisfy the conditions in the Assmus-Mattson Theorem and do not admit a $t$ -transitive or $t$ -homogeneous group as a subgroup of their automorphisms. The major objective of this paper is to develop a theory for explaining such codes and obtaining such new codes and hence new $t$ -designs. To this end, a general theory for punctured and shortened codes of linear codes supporting $t$ -designs is established, a generalized Assmus-Mattson theorem is developed, and a link between 2-designs and differentially $\delta $ -uniform functions and 2-designs is built. With these general results, binary codes with new parameters and explicit weight distributions are obtained, new 2-designs and Steiner system $S(2, 4, 2^{n})$ are produced in this paper. [ABSTRACT FROM AUTHOR]

Codebooks with low-coherence have wide utilization in many fields, such as direct spread code division multiple access communications, compressed sensing and so on. There are two major ingredients in this paper. The first is to present a new character sum, the hyper Eisenstein sum and study the properties of this character sum. As an application, the second ingredient is to propose two constructions of codebooks with the hyper Eisenstein sum. The codebooks generated by these constructions asymptotically meet the Welch bound. The parameters of these codebooks are new. [ABSTRACT FROM AUTHOR]

Multicast tree is a key structure for data dissemination from one source to multiple receivers in wireless networks. Minimum length multica modeled as the Steiner tree problem, and is proven to be NP-hard. In this paper, we explore how to efficiently generate minimum length multi wireless sensor networks (WSNs), where only limited knowledge of network topology is available at each node. We design and analyze a simple algorithm, which we call toward source tree (TST), to build multicast trees in WSNs. We show three metrics of TST algorithm, i.e., running and energy efficiency. We prove that its running time is O(\sqrt n\log n) , the best among all existing solutions to our best knowledge. We prove that TST tree length is in the same order as Steiner tree, which give a theoretical upper bound and use simulations to show the ratio be only 1.114 when nodes are uniformly distributed. We evaluate energy efficiency in terms of message complexity and the number of forwarding prove that they are both order-optimal. We give an efficient way to construct multicast tree in support of transmission of voluminous data. [ABSTRACT FROM PUBLISHER]

Fractional repetition (FR) codes are a class of regenerating codes for distributed storage systems with an exact (table-based) repair process that is also uncoded, i.e., upon failure, a node is regenerated by simply downloading packets from the surviving nodes. In this paper, we present the constructions of FR codes based on Steiner systems and resolvable combinatorial designs, such as affine geometries, Hadamard designs, and mutually orthogonal Latin squares. The failure resilience of our codes can be varied in a simple manner. We construct codes with normalized repair bandwidth ( $\beta $ ) strictly larger than one; these cannot be obtained trivially from codes with $\beta = 1$ . Furthermore, we present the Kronecker product technique for generating new codes from existing ones and elaborate on their properties. FR codes with locality are those where the repair degree is smaller than the number of nodes contacted for reconstructing the stored file. For these codes, we establish a tradeoff between the local repair property and the failure resilience and construct codes that meet this tradeoff. Much of prior work only provided lower bounds on the FR code rate. In this paper, for most of our constructions, we determine the code rate for certain parameter ranges. [ABSTRACT FROM AUTHOR]

We consider the coded cooperative data exchange problem for general graphs, both undirected and directed. In this problem, given a graph $G=(V,E)$ representing clients in a broadcast network, each of which initially hold a (not necessarily disjoint) set of information packets; one wishes to design a communication scheme in which eventually all clients will hold all the packets of the network. Communication is performed in rounds, where in each round a single client broadcasts a single (possibly encoded) information packet to its neighbors in G$ . The objective is to design a broadcast scheme that satisfies all clients with the minimum number of broadcast rounds. The coded cooperative data exchange problem has seen significant research over the last few years; mostly when the graph G$ is the complete broadcast graph in which each client is adjacent to all other clients in the network, but also on general topologies, both in the fractional and integral setting. In this paper, we focus on the integral setting in general topologies G$ , both undirected and directed. For undirected graphs, we tie the coded cooperative data exchange problem on G is intractable (i.e., NP-hard). We then turn to study efficient data exchange schemes for undirected topologies yielding a number of communication rounds comparable with our intractability result. Last, we tie the coded cooperative data exchange problem for directed topologies to the directed Steiner tree problem, yielding efficient data exchange approximation schemes. Our communication schemes do not involve encoding, and in such yield bounds on the coding advantage in the setting at hand. [ABSTRACT FROM PUBLISHER]

Constant-weight codes play an important role in coding theory. The problem of determining the sizes for optimal quaternary constant-weight codes with length n, weight 4, and minimum Hamming distance 5 ((n,5,4)4 codes) has been investigated in several papers. Although some constructions and several infinite families for such codes with length n\equiv 0,1\pmod4 have been given, the problem is still far from complete. In this paper, we determine the size of an optimal (n,5,4)4 code for each integer n\geq 4 leaving 55 lengths unsolved. Especially, for length n\equiv 0,1\pmod 4, the existence problem of the equivalent combinatorial object, namely the generalized Steiner system, is solved leaving only seven values undetermined. [ABSTRACT FROM AUTHOR]

An optimal constant-composition or constant-weight code of weight w has linear size if and only if its distance d is at least 2w - 1. When d ≥ 2w, the determination of the exact size of such a constant-composition or constant-weight code is trivial, but the case of d = 2w - 1 has been solved previously only for binary and ternary constant-composition and constant-weight codes, and for some sporadic instances. This paper provides a construction for quasicyclic optimal constant-composition and constant-weight codes of weight w and distance 2w - 1 based on a new generalization of difference triangle sets. As a result, the sizes of optimal constant-composition codes and optimal constant-weight codes of weight w and distance 2w - 1 are determined for all such codes of sufficiently large lengths. This solves an open problem of Etzion. The sizes of optimal constant-composition codes of weight w and distance 2w - 1 are also determined for all to ≤ 6, except in two cases. [ABSTRACT FROM AUTHOR]

A complete classification of the perfect binary one-error-correcting codes of length 15 as well as their extensions of length 16 is presented. There are 5983 such inequivalent perfect codes and 2165 extended perfect codes. Efficient generation of these codes relies on the recent classification of Steiner quadruple systems of order 16. Utilizing a result of Blackmore, the optimal binary one-error-correcting codes of length 14 and the (15, 1024, 4) codes are also classified; there are 38 408 and 5983 such codes, respectively. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this correspondence, we study the probability of undetected error for binary constant weight codes. First, we derive a new lower bound on the probability of undetected error for binary constant weight codes. Next, we show that this bound is tight if and only if the binary constant weight codes are generated from certain t-designs in combinatorial design theory. This means that these binary constant weight codes generated from certain t-designs are uniformly optimal for error detection. Along the way, we determine the distance distributions of such binary constant weight codes. In particular, it is shown that binary constant weight codes generated from Steiner systems are uniformly optimal for error detection. Thus, we prove a conjecture of Xia, Fu, Jiang, and Ling. Furthermore, the distance distribution of a binary constant weight code generated from a Steiner system is determined. Finally, we study the exponent of the probability of undetected error for binary constant weight codes. We derive some bounds on the exponent of the probability of undetected error for binary constant weight codes. These bounds enable us to extend the region in which the exponent of the probability of undetected error is exactly determined. [ABSTRACT FROM AUTHOR]

In this correspondence, we study the probability of undetected error for binary constant-weight codes. First, we derive a new formula on the probability of undetected error for binary constant-weight codes. Second, using this new formula and linear programming, we give two new lower bounds on the probability of undetected error for binary constant- weight codes. These two new lower bounds improve on previously known lower bounds in certain cases. Furthermore, we show that these two lower bounds are fight if and only if the binary constant-weight codes are generated from certain t-designs in combinatorial design theory. This means that these binary constant-weight codes generated from certain 2-designs are uniformly optimal for error detection. Along the way, we determine the distance distributions of such binary constant-weight codes. Finally, several examples are given to illustrate the results obtained in this correspondence. [ABSTRACT FROM AUTHOR]

Considers perfect single error-correcting codes or one-perfect codes. Definitions and properties about Steiner triple systems and propelinear codes; Main results about 1-perfect additive codes; Description of coding and decoding algorithms.