مرکز منطقه ای اطلاع رساني علوم و فناوري

Abstract :

For the case when $k$ divides $n$ , we introduce a special class of $(n,k)$ $F$ -ary convolutional codes, $F=$ GF $(q)$ a finite field, by considering the input to an $(n,k)$ encoder as a sequence over GF $(q^{k})$ , the output as a sequence over GF $(q^{n})$ (an idea first used by Dym [10]), and then considering encoders which correspond to convolving the input with a fixed sequence $\\Gamma _{0}, \\Gamma _{1}, \\cdots \\Gamma _{m}$ over GF $(q^{n})$ . A means of obtaining an encoder $G(D)$ from the polynomial $\\Gamma (D)=\\Gamma _{0}+\\Gamma _{1}D+\\cdots +\\Gamma _{m}D^{m}$ with respect to a basis for GF $(q^{n})$ over GF $(q)$ is described. A criterion on $\\Gamma (D)$ in order for any $G(D)$ obtained from it to be noncatastrophic is established, which involves computing only the greatest common divisor (gcd) among $s=n/k$ polynomials over GF $(q^{k})$ . This criterion is shown to coincide with that of Massey and Sain when $k=1$ . It is shown that if $\\Gamma (D)$ is noncatastrophic (i.e., if encoders obtained from it are noncatastrophic) and has zero delay, then any encoder $G(D)$ obtained from it is minimal and has a zero-delay feed-forward inverse. The number of zero-delay noncatastrophic polynomials over GF $(q^{n})$ of degree $m$ is shown to be $q^{nm}(q^{n}-1)(q^{n-k}-1)/q^{n-k}(q^{k}-1)$ , a formula which coincides with that of Shusta [11] when $k=1$ . The class of codes just described is shown to form a group under multiplication. If the basis is normal, the class is shown to be dosed under cyclic shifting. When $k=1$ the class of codes described coincides with the class of all $(n,1)$ $F$ -ary convolutional codes; hence we obtain new proofs of certain well-known results about this latter class of codes. Finally, the binary rate $1/2$ convolutional codes obtained from the noncatastrophic divisors of $D^{15}+1$ over GF $(2^{2})$ are studied and optimal codes of constraint lengths $6, 8$ , and $12$ found.