Nonbinary M-ary data processed by data entry systems, such as keyboard devices and character recognition systems, often have various types of error, such as symbol-substitution errors, deletion errors, insertion errors, and adjacent-symbol-transposition errors. This paper proposes nonsystematic M-ary codes capable of correcting these errors. The code is defined as a set of codewords that satisfy three conditions required to correct substitution, deletion/insertion, and adjacent-symbol-transposition errors. Since symbol-substitution errors in data entry systems are usually asymmetric, this paper also presents asymmetric-symbol-substitution error correcting codes capable of correcting deletion, insertion, and adjacent-symbol-transposition errors. For asymmetric-symbol-substitution error correction, we employ a mapping derived from the vertex coloring in an error directionality graph. The evaluation shows that the asymmetric codes have three to five times larger number of codewords than the symmetric codes.

Constantin and Rao have given a method for constructing single asymmetric error correcting (SAEC) codes based on the theory of the Abelian group, This paper uses the method of generating function in combinatorics to solve the implementation problems of the SAEC group theoretic codes. The encoding and decoding algorithms of the coding scheme perform simple arithmetic operations recursively. The idea of generating function can also be applied to t symmetric errors and simultaneously detect all unidirectional errors (t-syEC/AUED) codes for 1t3.

This paper presents two new maximum likelihood decoding (MLD) algorithms for linear codes over Z-channel, which are much more efficient than conventional exhaustive algorithms for high rate codes. In the proposed algorithms, their complexities are reduced by employing the projecting set Cs of the code, which is determined by the "projecting" structure of the code. Space and computational complexities of algorithms mainly depend upon the size of Cs which is usually several times smaller than the total number of codewords. It is shown that the upper bounds on computational complexities of decoding algorithms are in proportion to the number of parity bits and the distance between an initial estimate of the codeword and the received word, respectively, while space complexities of them are equal to the size of Cs. Lastly, numerical examples clarify the average computational complexities of the proposed algorithms, and the efficiency of these algorithms for high rate codes is confirmed.