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

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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 vectorsalpha_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}wheretheta neq 2(mod 3) ifmis even andphi neq 2(mod 3)ifmis odd. Letbeta_ibe the coefficient vector of the(m-1)-th degree polynomial inx, which representsx^iand letlambda_iandrho^ihave similar meanings with reference toy^iandz^i. Thenalpha_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.

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Information Theory, IRE Transactions on  (Volume:8 ,  Issue: 6 )