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On a class of convolutional codes

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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.

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Information Theory, IEEE Transactions on  (Volume:29 ,  Issue: 2 )