Binary group codes which correct errors in bursts of three for odd redundancy
A. Gross · IEEE Transactions on Information Theory · 1962
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. Letr = 2m + 1, m \geqq 4, and letAbe anmxrmatrix whose rows are ther-place binary vectors\alpha_1, \alpha_2, \cdots \alpha_n. 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: Letxbe a primitive element ofGF (2^m), y be a primitive element ofGF (2^{m-1})andzbe a primitive element ofGF (2^2). Further suppose1 + x = x^{\phi}, 1 + y = y^{\phi}where\theta eq 2(mod 3) ifmis even and\phi eq 2(mod 3)ifmis odd. Let\beta_ibe the coefficient vector of the(m-1)-th degree polynomial inx, which representsx^iand let\lambda_iand\rho^ihave similar meanings with reference toy^iandz^i. Then\alpha_i = (\beta_i, \lambda_i, \rho_i), i = 0, 1, \cdots (2^m-l) (2^{m--1}-1)-l. Although this method does not yield an optimumn, it is easily shown that asmincreasesn = (2^m -1) (2^{m-1} -1)approaches the optimal value.