### ON SELF-DUAL CONVOLUTIONAL CODES OVER RINGS

#### Abstract

We study the construction of a parity check matrix H(D) \in R(D)^{(n?k)}

of a rate-k/n convolutional code C over a commutative ring R that satisfies the descending chain condition. A (n?k) systematic parity check matrix H(D) is obtained from a standard generator matrix

G(D) \in R(D)^{k} of C. If G(D)=(Ik,A) such that n=2k and A^{?1}=?A^T, then H(D)=(?A^T,Ik) is equivalent to G(D), and consequentlyCis self-dual. New examples of encoders of rate-4/8 self-dual

convolutional codes over the binary field F2 and the integer ring Z4 are presented.

of a rate-k/n convolutional code C over a commutative ring R that satisfies the descending chain condition. A (n?k) systematic parity check matrix H(D) is obtained from a standard generator matrix

G(D) \in R(D)^{k} of C. If G(D)=(Ik,A) such that n=2k and A^{?1}=?A^T, then H(D)=(?A^T,Ik) is equivalent to G(D), and consequentlyCis self-dual. New examples of encoders of rate-4/8 self-dual

convolutional codes over the binary field F2 and the integer ring Z4 are presented.

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