Integer \(t\)-shift codes and factoring abelian groups (Q2920211)

In this paper the author studies certain classes of perfect error correcting codes. The main results makes it easier to classify such codes by relating them to factorizations of abelian groups.NEWLINENEWLINELet \(\mathbb Z_p\) denote the ring of integers modulo a prime \(p\), and let \(E = {\pm 1, \pm 2,\dots, \pm t}\) where \(t\) is also prime. We consider a code \(C \subset Z_p^n\) of dimension \(n-1\) with codewords \(a = (a_1,\dots, a_n)\). A single substitution error occurs if the received codeword is equal to \(a\), except that \(a_i\) is replaced with \(a_i + e\) for some \(e \in E\). We say that \(C\) is a perfect single substitution error correcting code if every element of \(\mathbb Z_p^n\) is either in \(C\) or is a single substitution error for exactly one codeword. A single shift error occurs if the received codeword is equal to \(a\), except that \(a_i\) is replaced with \(a_i - e\) and \(a_{i+1}\) is replaced with \(a_{i+1} + e\).NEWLINENEWLINETwo finite subsets \(A,B\) of an abelian group \(G\) give a factorization of \(G\) if every \(g \in G\) is expressible in a unique way as \(a+b\) where \(a \in A\) and \(b \in B\). Given \(A\) and \(G\), it is generally difficult to determine whether there exists a subset \(B\) giving a factorization of \(G\). A case of particular interest that is easier to solve is when we restrict to asking for such a set \(B\) that is a subgroup of \(G\).NEWLINENEWLINEThe main result of this paper simplifies the search for perfect 1-error correcting \(t\)-substitution and t-shift codes by relating them to finding certain kinds of factorizations of \(\mathbb Z_p^*\) modulo the subgroup of order two generated by \(-1\), and showing that these factorizations must come from subgroups.