Cosets of convolutional codes with least possible maximum zero- and one-run lengths

A communication or storage system may use a coset of a binary convolutional code for both symbol synchronization and error control. To facilitate symbol synchronization, the coset must have a short maximum zero-run length L_max. General upper and lower bounds on L_max were given previously by Hole. In this correspondence we use these bounds to identify which convolutional codes have cosets with short L_max. For such a code, we then show how to determine a coset with the least possible L_max among all cosets of the code. Exact expressions for the least possible L_max of convolutional code cosets are given, and examples of such cosets with large free distances are tabulated. Bounds on L_max for cosets of block codes are also provided. It is indicated how to tighten the bounds for block codes satisfying the one-way chain condition. We show that the cosets obtained from traditional high-rate block code constructions have larger L_max than cosets of convolutional codes with approximately the same rates. In some systems the convolutional code cosets must have short maximum one-run lengths as well as short maximum zero-run lengths to avoid loss of symbol synchronization. It is shown how to determine convolutional codes whose cosets with least possible maximum zero-run lengths also have least possible maximum one-run lengths