Design Methods for Maximum Minimum-Distance Error-Correcting Codes

In error-correcting codes for combating noisy transmission channels, a central concept i s the notion of minimum distance. If a code can be constructed with minimum distance between code points of 2m-k 1 I then any number of errors per code word which does not exceed m can be corrected, thus increasing the reliability of transmission above that to be expected with no redundancy i n the code. An upper bound on minimum distance is derived which depends on g (the number of code points or messages required) and n (the number of binary symbols per code point). This bound i s complementary to a bound due to Hamming and uses an argument which i s essentially due to Plotkin. Construction methods are presented for codes which actually achieve the upper bound on minimum distance for any g and an infinite class of integers n which depend on g. Sixteen code types are described: three for g = 2 h-l I six for g=2hl and seven for g = 2 k .