On a class of convolutional codes
Gérald E. Séguin · IEEE Transactions on Information Theory · 1983
For the case whenkdividesn, 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 encoderG(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 anyG(D)obtained from it to be noncatastrophic is established, which involves computing only the greatest common divisor (gcd) amongs=n/kpolynomials over GF(q^{k}). This criterion is shown to coincide with that of Massey and Sain whenk=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 encoderG(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 degreemis shown to beq^{nm}(q^{n}-1)(q^{n-k}-1)/q^{n-k}(q^{k}-1), a formula which coincides with that of Shusta [11] whenk=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. Whenk=1the 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 rate1/2convolutional codes obtained from the noncatastrophic divisors ofD^{15}+1over GF(2^{2})are studied and optimal codes of constraint lengths6, 8, and12found.