A family of constacyclic ternary quasi-perfect codes with covering radius 3

In this paper a family of constacyclic ternary quasi-perfect linear block codes is presented. This family extends the result presented in a previous work by the first two authors, where the existence of codes with the presented parameters was stated as an open question. The codes have a minimum distance 5 and covering radius 3.     

On codes with covering radius 1 and minimum distance 2

We show that a code C of length n over an alphabet Q of size q with minimum distance 2 and covering radius 1 satisfies |C| ≥ qⁿ⁻¹/(n − 1). For the special case n = q = 4 the smallest known example has |C| = 31. We give a construction for such a code C with |C| = 28.     

Density of constant radius normal binary covering codes

A binary code with covering radius R is a subset C of the hypercube Q_n={0,1}ⁿ such that every x∈Q_n is within Hamming distance R of some codeword c∈C, where R is as small as possible. For a fixed coordinate i∈[n], define to be the set of codewords with a b in the ith position. Then C is normal if there exists an i∈[n] such that for any v∈Q_n, the sum of the Hamming distances from v to and is at most 2R+1. We newly define what it means for an asymmetric covering code to be normal, and consider the worst-case asymptotic densities ν^*(R) and of constant radius R symmetric and asymmetric normal covering codes, respectively. Using a probabilistic deletion method, and analysis adapted from previous work by Krivelevich, Sudakov, and Vu, we show that and , giving evidence that minimum size constant radius covering codes could still be normal.     

Linear covering codes and error-correcting codes for limited-magnitude errors

The concepts of a linear covering code and a covering set for the limited-magnitude-error channel are introduced. A number of covering-set constructions, as well as some bounds, are given. In particular, optimal constructions are given for some cases involving small-magnitude errors. A problem of Stein is partially solved for these cases. Optimal packing sets and the corresponding error-correcting codes are also considered for some small-magnitude errors.     

Some inequalities about the covering radius of Reed-Muller codes

Let R(r, m) be the rth order Reed-Muller code of length 2 ^m , and let (r, m) be its covering radius. We prove that if 2 k m - r - 1, then (r + k, m + k) (r, m + 2(k - 1). We also prove that if m - r 4, 2 k m - r - 1, and R(r, m) has a coset with minimal weight (r, m) which does not contain any vector of weight (r, m) + 2, then (r + k, m + k) (r, m) + 2k(. These inequalities improve repeated use of the known result (r + 1, m + 1) (r, m).This work was supported by a grant from the Research Council of Wright State University.     

Further results on the covering radius of small codes

The minimum number of codewords in a code with t ternary and b binary coordinates and covering radius R is denoted by K(t,b,R). In the paper, necessary and sufficient conditions for K(t,b,R)=M are given for M=6 and 7 by proving that there exist exactly three families of optimal codes with six codewords and two families of optimal codes with seven codewords. The cases M?5 were settled in an earlier study by the same authors. For binary codes, it is proved that K(0,2b+4,b)?9 for b?1. For ternary codes, it is shown that K(3t+2,0,2t)=9 for t?2. New upper bounds obtained include K(3t+4,0,2t)?36 for t?2. Thus, we have K(13,0,6)?36 (instead of 45, the previous best known upper bound).     

The covering radius of extreme binary 2-surjective codes

The covering radius of binary 2-surjective codes of maximum length is studied in the paper. It is shown that any binary 2-surjective code of M codewords and of length has covering radius if M − 1 is a power of 2, otherwise . Two different combinatorial proofs of this assertion were found by the author. The first proof, which is written in the paper, is based on an existence theorem for k-uniform hypergraphs where the degrees of its vertices are limited by a given upper bound. The second proof, which is omitted for the sake of conciseness, is based on Baranyai’s theorem on l-factorization of a complete k-uniform hypergraph.      

A new approach to the covering radius of codes

We introduce a new approach which facilitates the calculation of the covering radius of a binary linear code. It is based on determining the normalized covering radius ϱ. For codes of fixed dimension we give upper and lower bounds on ϱ that are reasonably close. As an application, an explicit formula is given for the covering radius of an arbitrary code of dimension ⩽4. This approach also sheds light on whether or not a code is normal. All codes of dimension ⩽4 are shown to be normal, and an upper bound is given for the norm of an arbitrary code. This approach also leads to an amusing generalization of the Berlekamp-Gale switching game.     

On the covering radius of long binary BCH codes

It is proved that the covering radius of long binary BCH codes with designed distance 2t + 1 is at most 2t.     

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.     

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.      

Lower bound of covering radius of binary irreducible Goppa codes

The lower bound of covering radius of binary irreducible Goppa codes is obtained.     

On a class of binary linear completely transitive codes with arbitrary covering radius

An infinite class of new binary linear completely transitive (and so, completely regular) codes is given. The covering radius of these codes is growing with the length of the code. In particular, for any integer ρ≥2, there exist two codes in the constructed class with d=3, covering radius ρ and lengths and , respectively. The corresponding distance-transitive graphs, which can be defined as coset graphs of these completely transitive codes are described.     

On mixed codes with covering radius 1 and minimum distance 2: (extended abstract)

Let R, S and T be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius 1 and minimum distance 2 is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when r divides s, when $r=s-1$, when s is large, relative to r, when r is large, relative to s, as well as $K(3r,2r,2r;2)$. Finally, a table with bounds on $K(r,s,s;2)$ is given.     

Permutation codes with specified packing radius

Most papers on permutation codes have concentrated on the minimum Hamming distance of the code. An (n, d) permutation code (or permutation array) is simply a set of permutations on n elements in which the Hamming distance between any pair of distinct permutations (or codewords) is at least d. An (n, 2e + 1) or (n, 2e + 2) permutation code is able to correct up to e errors. These codes have a potential application to powerline communications. It is known that in an (n, 2e) permutation code the balls of radius e surrounding the codewords may all be pairwise disjoint, but usually some overlap. Thus an (n, 2e) permutation code is generally unable to correct e errors using nearest neighbour decoding. On the other hand, if the packing radius of the code is defined as the largest value of e for which the balls of radius e are all pairwise disjoint, a permutation code with packing radius e can be denoted by [n, e]. Such a permutation code can always correct e errors. In this paper it is shown that, in almost all cases considered, the number of codewords in the best [n, e] code found is substantially greater than the largest number of codewords in the best known (n, 2e + 1) code. Thus the packing radius more accurately specifies the requirement for an e-error-correcting permutation code than does the minimum Hamming distance. The techniques used include construction by automorphism group and several variations of clique search They are enhanced by two theoretical results which make the computations feasible.     

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.     

The variable radius covering problem

In this paper we propose a covering problem where the covering radius of a facility is controlled by the decision-maker; the cost of achieving a certain covering distance is assumed to be a monotonically increasing function of the distance (i.e., it costs more to establish a facility with a greater covering radius). The problem is to cover all demand points at a minimum cost by finding optimal number, locations and coverage radii for the facilities. Both, the planar and discrete versions of the model are considered. Heuristic approaches are suggested for solving large problems in the plane. These methods were tested on a set of planar problems. Mathematical programming formulations are proposed for the discrete problem, and a solution approach is suggested and tested.