Cyclic concatenated codes with constacyclic outer codes

Abstract

The known construction of cyclic concatenated codes is based on the fact that the inner is a cyclic minimal code, the outer code is cyclic, and the lengths of the inner and outer codes are relatively prime. It is shown that if the outer code is a suitably chosen constacyclic code the overall concatenated code is always cyclic regardless of the length of the outer code. Moreover, it follows that any cyclic code of composite length is a direct sum of cyclic concatenated codes with inner cyclic minimal codes and outer constacyclic codes. This description of cyclic codes of composite length leads to the introduction of the concept of a poor-code length (PCL). All but low-rate codes of this length have a poor minimum distance. A PCL is directly related to the existence of irreducible binomials. In the binary case, the first few PCLs are 9, 25, 27, 45, 49, 75, 81 and 99. Arbitrarily long binary cyclic codes that are better than binary BCH codes of primitive length are constructed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>