Asymptotic Minimum Covering Radius of Block Codes

Abstract

Abstract. In this paper, we restudy the covering radius of block codes from an information theoretic point of view by ignoring the combinatorial formulation of the problem. In the new setting, the formula of the statistically defined minimum covering radius, for which the probability mass of uncovered space by M spheres can be made arbitrarily small, is reduced to a minimization of a statistically defined spectrum formula among codeword-selecting distributions. The advantage of the new view is that no assumptions need to be made on the code alphabet (such as finite, countable, etc.) and the distance measure (such as additive, symmetric, bounded, etc.) in the problem transformation, and hence the spectrum formula can be applied in most general situations. We next address a sufficient condition under which uniform codeword-selecting distribution minimizes the spectrum formula. With the condition, the asymptotic minimum covering radius for block codes under J-ary quantized channels and constant weight codes under Hamming distance measure are determined to display the usage of the spectrum formula.

On the Maximum Size of Block Codes Subject to a Distance Criterion

“…• Seek i) a similar formula of the minimum code size subject to a covering radius constraint (cf. [27]) and ii) a formula of maximal code size under a minimum multi-wise distance constraint (cf. [28]).…”