Dual codes of systematic group codes over abelian groups (Q1352429)

In this paper the class of self-dual codes and dual codes over finite abelian groups are characterized. An \((n,k)\) systematic group code over an abelian group \(G\) is a subgroup of \(G^n\) with order \(|G|^k\) described by \(n-k\) homorphisms \(\Phi_j\), \(j=1,2,\dots,n-k\) of \(G^k\) onto \(G\). Its codewords are \((x_1,\dots,x_k,x_{k+1},\dots,x_n),\) where \[ x_{k+j}=\Phi_j(x_1,\dots,x_k)=\sum_{l=1}^k\Phi_j(e,\dots,e,x_l,e,\dots,e), \] and \(e\) is the identity element of group \(G.\) The authors generalize the result for linear codes over finite fields. It is proved that the dual code of a systematic code over a finite abelian group is a systematic code. In terms of generator matrices: If \([I\mid\Phi]\) is a generator matrix of a systematic group code, then its dual has the generator matrix \([(\Phi^d)^{tr}\mid I]\), where \([\Phi^d]\) is the matrix obtained by replacing each entry of \([\Phi]\) by its dual. There is given a necessary sufficient condition for a \((2k,k)\) group code to be self-dual. The special cases of group codes over cyclic group and elementary abelian group are also discussed.