On a class of convolutional codes

For the case when divides , we introduce a special class of -ary convolutional codes, GF a finite field, by considering the input to an encoder as a sequence over GF , the output as a sequence over GF (an idea first used by Dym [10]), and then considering encoders which correspond to convolving the input with a fixed sequence over GF . A means of obtaining an encoder from the polynomial with respect to a basis for GF over GF is described. A criterion on in order for any obtained from it to be noncatastrophic is established, which involves computing only the greatest common divisor (gcd) among polynomials over GF . This criterion is shown to coincide with that of Massey and Sain when . It is shown that if is noncatastrophic (i.e., if encoders obtained from it are noncatastrophic) and has zero delay, then any encoder obtained from it is minimal and has a zero-delay feed-forward inverse. The number of zero-delay noncatastrophic polynomials over GF of degree is shown to be , a formula which coincides with that of Shusta [11] when . 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 the class of codes described coincides with the class of all -ary convolutional codes; hence we obtain new proofs of certain well-known results about this latter class of codes. Finally, the binary rate convolutional codes obtained from the noncatastrophic divisors of over GF are studied and optimal codes of constraint lengths , and found.