New Linear Codes with Covering Radius 2 and Odd Basis

On the way of generalizing recent results by Cock and the second author, it is shown that when the basis q is odd, BCH codes can be lengthened to obtain new codes with covering radius R=2. These constructions (together with a lengthening construction by the first author) give new infinite families of linear covering codes with codimension r=2k+1 (the case q=3, r=4k+1 was considered earlier). New code families with r=4k are also obtained. An updated table of upper bounds on the length function for linear codes with 24, R=2, and q=3,5 is given.     

Bergman-Hartogs型域的全纯自同构群

我们考虑一类以有界对称域D为底的Bergman-Hartogs型域Ω={(wm(1),...,w(r),z)∈C1×···×Cmr×D:∥w(1)∥2p1+···+∥w(r)∥2prKD(z,z)-q},其中KD(z,z)是D上的Bergman核函数,r 1且为正整数,参数p1,...,pr1和q0为实数.我们给出它的全纯自同构群,并且证明当r=1时此自同构群为最大全纯自同构群;当r1时,若Ω的全纯自同构变换F将(0,z)∈{0}×D映到(0,z*)∈{0}×D,则F在我们给出的全纯自同构群中.     

Exact Bounds on the Sizes of Covering Codes

A code c is a covering code of X with radius r if every element of X is within Hamming distance r from at least one codeword from c. The minimum size of such a c is denoted by c _r(X). Answering a question of Hämäläinen et al. [10], we show further connections between Turán theory and constant weight covering codes. Our main tool is the theory of supersaturated hypergraphs. In particular, for n > n ₀(r) we give the exact minimum number of Hamming balls of radius r required to cover a Hamming ball of radius r + 2 in {0, 1}ⁿ. We prove that c _r(B _n(0, r + 2)) = _{1 i r + 1} ( ^{(n + i – 1) / (r + 1) 2}) + n / (r + 1) and that the centers of the covering balls B(x, r) can be obtained by taking all pairs in the parts of an (r + 1)-partition of the n-set and by taking the singletons in one of the parts.     

Optimal binary covering codes of length 2j

The minimum size of a binary covering code of length n and covering radius r is denoted by K(n,r), and codes of this length are called optimal. For j > 0 and n = 2^j, it is known that K(n,1) = 2 · K(n?1,1) = 2^{n ? j}. Say that two binary words of length n form a duo if the Hamming distance between them is 1 or 2. In this paper, it is shown that each optimal binary covering code of length n = 2^j, j > 0, and covering radius 1 is the union of duos in just one way, and that the closed neighborhoods of the duos form a tiling of the set of binary words of length n. Methods of constructing such optimal codes from optimal covering codes of length n ? 1 (that is, perfect single‐error‐correcting codes) are discussed. The paper ends with the construction of an optimal covering code of length 16 that does not contain an extension of any optimal covering code of length 15. © 2005 Wiley Periodicals, Inc. J Combin Designs     

Infinite highly arc transitive digraphs and universal covering digraphs

A digraph (that is a directed graph) is said to be highly arc transitive if its automorphism group is transitive on the set ofs-arcs for eachs0. Several new constructions are given of infinite highly arc transitive digraphs. In particular, for a connected, 1-arc transitive, bipartite digraph, a highly arc transitive digraphDL() is constructed and is shown to be a covering digraph for every digraph in a certain classD() of connected digraphs. Moreover, if is locally finite, thenDL() is a universal covering digraph forD(). Further constructions of infinite highly arc transitive digraphs are given.The second author wishes to acknowledge the hospitality of the Mathematical Institute of the University of Oxford, and the University of Auckland, during the period when the research for this paper was doneResearch supported by the Australian Research Council     

A New Table of Binary/Ternary Mixed Covering Codes

A table of upper bounds for K3,2(n1,n2;R), the minimum number of codewords in a covering code with n1 ternary coordinates, n2 binary coordinates, and covering radius R, in the range n = n1 + n2 13, R 3, is presented. Explicit constructions of codes are given to prove the new bounds and verify old bounds. These binary/ternary covering codes can be used as systems for the football pool game. The results include a new binary code with covering radius 1 proving K2(13,1) 736, and the following upper bound for the football pool problem for 9 matches: K3(9,1) 1356.     

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.     

On automorphisms of split metacyclic groups

Let D(m, n; k) be the semi-direct product of two finite cyclic groups and , where the action is given by yxy ⁻¹  =  x ^k . In particular, this includes the dihedral groups D _2m . We calculate the automorphism group Aut (D(m, n; k)).     

An intrinsical description of group codes

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism which maps G to the standard basis of . Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space , which does not assume an “a priori” group algebra structure on . As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed–Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes. Research supported by D.G.I. of Spain and Fundación Séneca of Murcia.     

A Gap in GRM Code Weight Distributions

The weight distribution of GRM (generalized Reed-Muller) codes is unknown in general. This article describes and applies some new techniques to the codes over F3. Specifically, we decompose GRM codewords into words from smaller codes and use this decomposition, along with a projective geometry technique, to relate weights occurring in one code with weights occurring in simpler codes. In doing so, we discover a new gap in the weight distribution of many codes. In particular, we show there is no word of weight 3^m–2 in GRM₃(4,m) for m>6, and for even-order codes over the ternary field, we show that under certain conditions, there is no word of weight d+, where d is the minimum distance and is the largest integer dividing all weights occurring in the code.     

The multicovering radius problem for some types of discrete structures

The covering radius problem is a question in coding theory concerned with finding the minimum radius r such that, given a code that is a subset of an underlying metric space, balls of radius r over its code words cover the entire metric space. Klapper (IEEE Trans. Inform. Theory 43:1372–1377, 1997) introduced a code parameter, called the multicovering radius, which is a generalization of the covering radius. In this paper, we introduce an analogue of the multicovering radius for permutation codes (Des. Codes Cryptogr. 41:79–86, cf. 2006) and for codes of perfect matchings (cf. 2012). We apply probabilistic tools to give some lower bounds on the multicovering radii of these codes. In the process of obtaining these results, we also correct an error in the proof of the lower bound of the covering radius that appeared in (Des. Codes Cryptogr. 41:79–86, cf. 2006). We conclude with a discussion of the multicovering radius problem in an even more general context, which offers room for further research.     

Characterization of the automorphism group of quaternary linear Hadamard codes

A quaternary linear Hadamard code ${\mathcal{C}}$ is a code over ${\mathbb{Z}_4}$ such that, under the Gray map, gives a binary Hadamard code. The permutation automorphism group of a quaternary linear code ${\mathcal{C}}$ of length n is defined as ${{\rm PAut}(\mathcal{C}) = \{\sigma \in S_{n} : \sigma(\mathcal{C}) = \mathcal{C}\}}$ . In this paper, the order of the permutation automorphism group of a family of quaternary linear Hadamard codes is established. Moreover, these groups are completely characterized by computing the orbits of the action of ${{\rm PAut}(\mathcal{C})}$ on ${\mathcal{C}}$ and by giving the generators of the group. Since the dual of a Hadamard code is an extended 1-perfect code in the quaternary sense, the permutation automorphism group of these codes is also computed.     

Cocycle Twisting of E(n)-Module Algebras and Applications to the Brauer Group

We classify the orbits of coquasi-triangular structures for the Hopf algebra E(n) under the action of lazy cocycles and the Hopf automorphism group. This is applied to detect subgroups of the Brauer group BQ(k,E(n)) of E(n) that are isomorphic. For any triangular structure R on E(n) we prove that the subgroup BM(k,E(n),R) of BQ(k,E(n)) arising from R is isomorphic to a direct product of BW(k), the Brauer-Wall group of the ground field k, and Sym_n(k), the group of n × n symmetric matrices under addition. For a general quasi-triangular structure R on E(n) we construct a split short exact sequence having BM(k,E(n),R⁾ as a middle term and as kernel a central extension of the group of symmetric matrices of order r < n (r depending on R). We finally describe how the image of the Hopf automorphism group inside BQ(k,E(n)) acts on Sym_n (k).     

Automorphisms of Constant Weight Codes and of Divisible Designs

We show that the automorphism group of a divisible design is isomorphic to a subgroup H of index 1 or 2 in the automorphism group of the associated constant weight code. Only in very special cases H is not the full automorphism group.     

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     

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.     

Elliptic near-MDS codes over $${\mathbb{F}}_5$$

Let Γ₆ be the elliptic curve of degree 6 in PG(5, q) arising from a non-singular cubic curve of PG(2, q) via the canonical Veronese embedding

Fast decoding of quasi-perfect Lee distance codes

A code D over Z ² _n is called a quasi-perfect Lee distance-(2t + 1) code if d _L(V,W) ≥ 2t + 1 for every two code words V,W in D, and every word in Z ² _n is at distance ≤ t + 1 from at least one code word, where D _L(V,W) is the Lee distance of V and W. In this paper we present a fast decoding algorithm for quasi-perfect Lee codes. The basic idea of the algorithm comes from a geometric representation of D in the 2-dimensional plane. It turns out that to decode a word it suffices to calculate its distance to at most four code words.     

Mappings connected with the gradient of conformal radius

In this paper we prove the following conformity criterion for the gradient of conformal radius ?R(D, z) of a convex domain D: the boundary ?D has to be a circumference. We calculate coefficients K(r) for K(r)-quasiconformal mappings ?R(D(r), z), D(r) ? D, 0 < r < 1, and complete the results obtained by F. G. Avkhadiev and K.-J. Wirths for the structure of boundary elements of quasiconformal mappings of the domain D.     

On linear codes admitting large automorphism groups

Linear codes with large automorphism groups are constructed. Most of them are suitable for permutation decoding. In some cases they are also optimal. For instance, we construct an optimal binary code of length \(n=252\) and dimension \(k=12\) having minimum distance \(d=120\) and automorphism group isomorphic to \(\text {PSL}(2,8)\rtimes \text {C}_{3}\).