Bounds on the sizes of constant weight covering codes

Abstract

Abstract. Motivated by applications in universal data compression algorithms we study the problem of bounds on the sizes of constant weight covering codes. We are concerned with the minimal sizes of codes of length n and constant weight u such that every word of length n and weight v is within Hamming distance d from a codeword. In addition to a brief summary of part of the relevant literature, we also give new results on upper and lower bounds to these sizes. We pay particular attention to the asymptotic covering density of these codes. We include tables of the bounds on the sizes of these codes both for small values ofn and for the asymptotic behavior. A comparison with techniques for attaining bounds for packing codes is also given. Some new combinatorial questions are also arising from the techniques.

Upper bounds on the general covering number Cλ(v, k, t, m)

“…Some of the entries in this table correspond to ðv; k; tÞ coverings, previously found in [4,12,24,25]. Several entries in the table are based on known results.…”

Covering arbitrary point patterns

“…More specifically, in the latter problem, we seek a constant-weight-dT /∆ code C such that, for any weight-λT binary vector, there exists an element of C within Hamming distance (1 − 2b + d/∆)λT from the vector. This problem has been considered in [13].…”

Exploiting orbits in symmetric ILP

“…, v and the order of the group is v!. The optimal value for the (10, 5, 4)-covering design was found by Etzion et al [13], without optimality proof. The branch-and-cut of [22] proved optimality and generated 40 non-isomorphic optimal solutions.…”