مرکز منطقه ای اطلاع رساني علوم و فناوري - On certain projective geometry codes (Corresp.)

Abstract :

Let $V$ be an $(n, k, d)$ binary projective geometry code with $n = (q^{m}-1)/(q - 1), q = 2^{s}$ , and $d \\geq [(q^{m-r}-1)/(q - 1)] + 1$ . This code is $r$ -step majority-logic decodable. With reference to the GF $(q^{m}) = {0, 1, \\alpha , \\alpha ^{2} , \\cdots , \\alpha ^{n(q-1)-1} }$ , the generator polynomial $g(X)$ , of $V$ , has $\\alpha ^{\\nu}$ as a root if and only if $\\nu$ has the form $\\nu = i(q - 1)$ and $\\max _{0 \\leq l < s} W_{q}(2^{l} \\nu) \\leq (m - r - 1)(q - 1)$ , where $W_{q}(x)$ indicates the weight of the radix- $q$ representation of the number $x$ . Let $S$ be the set of nonzero numbers $\\nu$ , such that $\\alpha ^{\\nu}$ is a root of $g(X)$ . Let $C_{1}, C_{2}, \\cdots , C_{\\nu}$ be the cyclotomic cosets such that $S$ is the union of these cosets. It is clear that the process of finding $g(X)$ becomes simpler if we can find a representative from each $C_{i}$ , since we can then refer to a table, of irreducible factors, as given by, say, Peterson and Weldon. In this correspondence it was determined that the coset representatives for the cases of $m-r = 2$ , with $s = 2, 3$ , and $m-r=3$ , with $s=2$ .