Binary group codes which correct errors in bursts of three for odd redundancy

Abramson and Bose and Chakravarti have constructed, simultaneously and independently, binary group codes which correct errors in bursts of three or less for even redundancy. In this paper the corresponding problem when the redundancy is odd is considered. Let , and let be an matrix whose rows are the -place binary vectors . We show that A satisfies the necessary and sufficient condition for being the parity check matrix of the required code if it is obtained in the following manner: Let be a primitive element of , y be a primitive element of and be a primitive element of . Further suppose where (mod 3) if is even and if is odd. Let be the coefficient vector of the -th degree polynomial in , which represents and let and have similar meanings with reference to and . Then l. Although this method does not yield an optimum , it is easily shown that as increases approaches the optimal value.