Two-Error Correcting Bose-Chaudhuri Codes are Quasi-Perfect

Bose and Chaudhuri (1960) have introduced a class of binary errorcorrecting codes of block length 2 ~ 1, m = 2,3,• • which we refer to here as B-C codes. An efficient decoding scheme has been devised for them by Peterson (1960), and generalized to the natural extension of the B-C Codes to codes in pm symbols by D. Gorenstein and N. Zierler (1960). Two further properties of B-C codes are established in this note. Let us recall tha t a quasi-perfect t-error correcting binary code of block length n has as coset leaders all binary n-tuples of norm ( = number of ones) t or less, while no n-tuple of norm more than t + i is a coset leader. Theorem 1 asserts tha t every 2-error correcting B-C code (binary, of block length 2 ~ 1) is quasi-perfect and therefore opt imum. Define the weight of a coset to be the norm of its leaders. Theorem 2 states as a consequence tha t every B-C 3-error correcting code has a coset of weight at least 5, and suggests tha t no other nontrivial B-C codes are quasiperfect. Others are sometimes opt imum, however, as is, for example, the 3-error correcting code of block length 15 (see Peterson, 1960).