مرکز منطقه ای اطلاع رساني علوم و فناوري - Binary group codes which correct errors in bursts of three for odd redundancy

Abstract :

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 $r = 2m + 1, m \\geq q 4$ , and let $A$ be an $mxr$ matrix whose rows are the $r$ -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: Let $x$ be a primitive element of $GF (2^m)$ , y be a primitive element of $GF (2^{m-1})$ and $z$ be a primitive element of $GF (2^2)$ . Further suppose $1 + x = x^{\\phi}, 1 + y = y^{\\phi}$ where $\\theta \\neq 2$ (mod 3) if $m$ is even and $\\phi \\neq 2(mod 3)$ if $m$ is odd. Let $\\beta _i$ be the coefficient vector of the $(m-1)$ -th degree polynomial in $x$ , which represents $x^i$ and let $\\lambda _i$ and $\\rho^i$ have similar meanings with reference to $y^i$ and $z^i$ . 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 optimum $n$ , it is easily shown that as $m$ increases $n = (2^m -1) (2^{m-1} -1)$ approaches the optimal value.

Keywords :

Burst-correcting codes; Group codes; Cancer; Circuits; Electronic design automation and methodology; Error correction codes; Galois fields; Information theory; Parity check codes; Polynomials; Public healthcare; Redundancy; Research and development; Sufficient conditions;