Upper bounds to the maximum number of codewords in a binary code of length n and asymmetric distance Delta are derived for some values of n and Delta . A method is given in which a code of length n-m and asymmetric distance at least t+1 is constructed by expurgating and puncturing a code of length n and Hamming distance at least 2t+1. Novel asymmetric error-correcting codes are constructed by applying this method to some celebrated symmetric error-correcting codes. a table is presented on the size of optimal asymmetric error-correcting codes of length less than 24 and asymmetric distance less than 6. >

Bounds and constructions for binary codes of length less than 24 and asymmetric distance less than 6

We study single error-correcting codes for the asymmetric channel with input and output alphabets being {0, 1,…, a — 1⩽. From an abelian group G of order N with elements g0 = 0, g1,…, gN—1, Constantin and Rao (1979, Inform. Contr. 40, 20–36) define Vg = {(b1,b2,...,bn−1) ∈ {O, 1,..., a − 1 }N−1 | ∑N−1i−1 bigi 0 g} and show that Vg correct single errors. We give explicit expressions for the size and weight distribution of these codes. We further give a short discussion of some constant weight codes obtained by a similar construction.

On group-theoretic codes for asymmetric channels

This paper studies new bounds and constructions that are applicable to the combinatorial granular channel model previously introduced by Sharov and Roth. The main theme of the paper is that codes capable of correcting grain-errors are related to codes that correct insertions/deletions and codes that correct asymmetric errors. Using this insight, new bounds on the maximum cardinality of a grain-error correcting code are derived and constructions of codes that correct grain-errors are considered. It is also demonstrated that permutations of the classical group codes can correct a single grain-error. In several cases of interest, our results improve upon the currently best known bounds and constructions.

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Correcting grain-errors in magnetic media

The Varshamov-Tenengol'ts codes are binary number-theoretic codes that can correct a single asymmetric error. Levenshtein realised that this same construction can correct a single insertion/deletion error. And yet asymmetric and insertion/deletion errors appear to be fundamentally different. This paper explores this scenario. Constantin and Rao showed that any Abelian group of order n + 1 can correct a single asymmetric error. In this paper, we show that only a subset of Abelian groups can correct a single insertion/deletion error.

E. R. Rodemich

Papers

The Constantin-Rao construction for binary asymmetric error-correcting codes. [Deep Space Network]