Codes with a Circulant Parity Check Matrix

In this work we study codes characterized by the property that their parity check matrix is circulant, i.e., that rows are obtained as all the distinct cyclic shifts of the rst row. For these codes, we give simple expressions for their dimension and for a lower bound on their minimum distance. We also present an O(nd) algorithm to solve the problem of deciding whether codes satisfying given constraints on their block length n, dimension k and minimum distance d exist in this class. Some long binary codes found using this algorithm are compared against the Varshamov-Gilbert bound. 1 Organization of this Report We start in Section 2 by de ning the class of codes we consider, and giving simple expressions for the parameters (n; k; d) in terms of one row of the check matrix. Then, in Section 3 we present an e cient algorithm to nd a suitable parity check matrix satisfying given constraints, or deciding that those constraints are unsatis able within the class of codes we consider. In Section 4, we present a few codes we found in this class, and in Section 5 we present some conclusions and discuss three topics of interest to us for future work.