A SAT+CAS Approach to Finding Good Matrices: New Examples and Counterexamples

We enumerate all circulant good matrices with odd orders divisible by 3 up to order 70. As a consequence of this we find a previously overlooked set of good matrices of order 27 and a new set of good matrices of order 57. We also find that circulant good matrices do not exist in the orders 51, 63, and 69, thereby finding three new counterexamples to the conjecture that such matrices exist in all odd orders. Additionally, we prove a new relationship between the entries of good matrices and exploit this relationship in our enumeration algorithm. Our method applies the SAT+CAS paradigm of combining computer algebra functionality with modern SAT solvers to efficiently search large spaces which are specified by both algebraic and logical constraints

Self-Dual codes from (−1,1)(-1,1)-matrices of skew symmetric type

Previously, self-dual codes have been constructed from weighing matrices, and in particular from conference matrices (skew and symmetric). In this paper, codes constructed from matrices of skew symmetric type whose determinants reach the Ehlich-Wojtas' bound are presented. A necessary and sufficient condition for these codes to be self-dual is given, and examples are provided for lengths up to 52