On the packing radius and the covering radius of equal-weight codes

On the covering radius of codes

Designing a good error-correcting code is a packing problem. The corresponding covering problem has received much less attention: now the codewords must be placed so that no vector of the space is very far from the nearest codeword. The two problems are quite different, and with a few exceptions good packings, i.e. codes with a large minimal distance, are usually not especially good coverings. ...

On the Covering Radius Problem for Codes I . Bounds on Normalized Covering Radius

In this two-part paper we introduce the notion of a stable code and give a new upper bound on the normalized covering radius ofa code. The main results are that, for fixed k and large n, the minimal covering radius t[n, k] is realized by a normal code in which all but one of the columns have multiplicity l; hence tin + 2, k] t[n, k] + for sufficiently large n. We also show that codes with n _-<...

On the covering radius of Reed-Muller codes

Cohen, G.D., S.N. Litsyn, On the covering radius of Reed-Muller codes, Discrete Mathematics 106/107 (1992) 147-155. We present lower and upper bounds on the covering radius of Reed-Muller codes, yielding asymptotical improvements on known results. The lower bound is simply the sphere covering one (not very new). The upper bound is derived from a thorough use of a lemma, the ‘essence of Reed-Mul...

On the Covering Radius of Long Goppa Codes

On the covering radius of small codes versus dual distance

Tietäväinen’s upper and lower bounds assert that for block-length-n linear codes with dual distance d, the covering radius R is at most n2 − ( 2 − o(1)) √ dn and typically at least n2 − Θ( √ dn log nd ). The gap between those bounds on R − n2 is an Θ( √ log nd ) factor related to the gap between the worst covering radius given d and the sphere-covering bound. Our focus in this paper is on the c...