Linear codes with covering radius 3

The shortest possible length of a q-ary linear code of covering radius R and codimension r is called the length function and is denoted by ℓ _q (r, R). Constructions of codes with covering radius 3 are here developed, which improve best known upper bounds on ℓ _q (r, 3). General constructions are given and upper bounds on ℓ _q (r, 3) for q = 3, 4, 5, 7 and r ≤ 24 are tabulated.     

Hadamard matrices and doubly even self-dual error-correcting codes

Let n be an integer with n≡4 (mod 8). For any Hadamard matrices H_n of order n, we give a method to define a doubly even self-dual [2n, n] code C(NH_n). Then we will prove that two Hadamard equivalent matrices define equivalent codes.     

Bounds for binary multiple covering codes

We study codes that are multiple coverings of the Hamming space and discuss lower and upper bounds onK(n, r, ), the minimum cardinality of a binary code of lengthn such that the Hamming spheres of radiusr centered at the codewords cover at least times. We also study the similar problem of multiple coverings containing repeated words. A table of bounds forn16,r4, 4 is given.     

De Bruijn cycles for covering codes

A de Bruijn covering code is a q‐ary string S so that every q‐ary string is at most R symbol changes from some n‐word appearing consecutively in S. We introduce these codes and prove that they can have size close to the smallest possible covering code. The proof employs tools from field theory, probability, and linear algebra. Included is a table of the best known bounds on the lengths of small binary de Bruijn covering codes, up to R = 11 and n = 13, followed by several open questions in this area. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Nonbinary quantum error-correcting codes from algebraic curves

We give a new exposition and proof of a generalized CSS construction for nonbinary quantum error-correcting codes. Using this we construct nonbinary quantum stabilizer codes with various lengths, dimensions, and minimum distances from algebraic curves. We also give asymptotically good nonbinary quantum codes from a Garcia–Stichtenoth tower of function fields which are constructible in polynomial time.     

A new construction of quantum error-correcting codes

In this paper, we present a characterization of (binary and non-binary) quantum error-correcting codes. Based on this characterization, we introduce a method to construct -ary quantum codes using Boolean functions satisfying a system of certain quadratic relations. As a consequence of the construction, we are able to construct quantum codes of minimum distance . In particular, we produce a class of binary quantum -codes for odd length . For , this improves the result by Rains in Quantum codes of minimal distance two, 1999, showing the existence of binary quantum -codes for odd . Moreover, our binary quantum -codes of odd length achieve the Singleton bound asymptotically.

Finally, based on our characterization some propagation rules of quantum codes are proposed and the rules are similar to those in classical coding theory. It turns out that some new quantum codes are found through these propagation rules.

     

Switching of covering codes

Switching is a local transformation of a combinatorial structure that does not alter the main parameters. Switching of binary covering codes is studied here. In particular, the well-known transformation of error-correcting codes by adding a parity-check bit and deleting one coordinate is applied to covering codes. Such a transformation is termed a semiflip, and finite products of semiflips are semiautomorphisms. It is shown that for each code length $n\ge 3$, the semiautomorphisms are exactly the bijections that preserve the set of $r$-balls for each radius $r$. Switching of optimal codes of size at most 7 and of codes attaining $K(8,1)=32$ is further investigated, and semiautomorphism classes of these codes are found. The paper ends with an application of semiautomorphisms to the theory of normality of covering codes.     

Bounds for short covering codes and reactive tabu search

Given a prime power q, c_q(n,R) denotes the minimum cardinality of a subset H in such that every word in this space differs in at most R coordinates from a multiple of a vector in H. In this work, two new classes of short coverings are established. As an application, a new optimal record-breaking result on the classical covering code is obtained by using short covering. We also reformulate the numbers c_q(n,R) in terms of dominating set on graphs. Departing from this reformulation, the reactive tabu search (a variation of tabu search heuristics) is developed to obtain new upper bounds on c_q(n,R). The algorithm is described and conclusions on the results are drawn; they identify the advantages of using the reactive mechanism for this problem. Tables of lower and upper bounds on c_q(n,R), q=3,4, n≤7, and R≤3, are also presented.     

Classification of binary covering codes

The minimum size of a binary covering code of length n and covering radius r is denoted by K (n, r) and corresponding codes are called optimal. In this article a classification up to equivalence of all optimal covering codes having either length at most 8 or cardinality at most 4 is completed. Moreover, we prove that K (9, 2) = 16. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 391–401, 2000     

A random construction for permutation codes and the covering radius

We analyse a probabilistic argument that gives a semi-random construction for a permutation code on n symbols with distance n − s and size Θ(s!(log n)^1/2), and a bound on the covering radius for sets of permutations in terms of a certain frequency parameter.      

Constructing covering codes by tabu search

The problem of finding good covering codes in Hamming spaces is considered. Many different local search methods have been used to find packing codes (the dual problem), whereas practically all published results on searches for covering codes are based on simulated annealing. In this article tabu search is evaluated and compared against the simulated annealing method. A novel neighborhood function is also presented. The combination of a new optimization method and a new neighborhood function turns out to speed up the search for covering codes remarkably compared to the traditional simulated annealing approach. Using the new approach, the best known upper bound for the football pool problem for 9 matches is improved to 1341. © 1997 John Wiley & Sons, Inc.     

Linear error-block codes

A linear error-block code is a natural generalization of the classical error-correcting code and has applications in experimental design, high-dimensional numerical integration and cryptography. This article formulates the concept of a linear error-block code and derives basic results for this kind of code by direct analogy to the classical case. Some problems for further research are raised.     

On 2-dimensional insertion-deletion Reed-Solomon codes with optimal asymptotic error-correcting capability

Reed-Solomon codes have gained a lot of interest due to its encoding simplicity, well structuredness and list-decoding capability [6] in the classical setting. This interest also translates to other metric setting, including the insertion and deletion (insdel for short) setting which is used to model synchronization errors caused by positional information loss in communication systems. Such interest is supported by the construction of a deletion correcting algorithm of insdel Reed-Solomon code in [22] which is based on the Guruswami-Sudan decoding algorithm [6]. Nevertheless, there have been few studies [3] on the insdel error-correcting capability of Reed-Solomon codes.In this paper, we discuss a criterion for a 2-dimensional insdel Reed-Solomon codes to have optimal asymptotic error-correcting capabilities, which are up to their respective lengths. Then we provide explicit constructions of 2-dimensional insdel Reed-Solomon codes that satisfy the established criteria. The family of such constructed codes can then be shown to extend the family of codes with asymptotic error-correcting capability reaching their respective lengths provided in [3, Theorem 2] which provide larger error-correcting capability compared to those defined in [25].     

Linear codes for high payload steganography

Steganography is concerned with communicating hidden messages in such a way that no one apart from the sender and the intended recipient can detect the very existence of the message. We study the syndrome coding method (sometimes also called the “matrix embedding method”), which uses a linear code as an ingredient. Among all codes of a fixed block length and fixed dimension (and thus of a fixed information rate), an optimal code is one that makes it most difficult for an eavesdropper to detect the presence of the hidden message. We show that the average distance to code is the appropriate concept that replaces the covering radius for this particular application. We completely classify the optimal codes in the cases when the linear code used in the syndrome coding method is a one- or two-dimensional code over . In the steganography application this translates to cases when the code carries a high payload (has a high information rate).     

Linear codes and modular curves

Results of recent investigations at the juncture of coding theory, the theory of computability, and algebraic geometry over finite fields are presented. The basic problems of the asymptotic theory of codes and Goppa's construction of codes on the basis of algebraic curves are presented, and a detailed algorithmic analysis is given of the codes arising on the modular curves of elliptic modules of V. G. Drinfel'd.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 25, pp. 209–257, 1984.