Two new optimal ternary two-weight codes and strongly regular graphs (Q1910570)

Let \(C\) be a linear \(\text{GF}(q)\)-code with generator \(k\times n\) matrix \(G\), and let \(d(C)\) be the minimal (nonzero) Hamming distance of \(C\). Let \(d_q(n, k)\) be a maximal \(d(C)\) over the linear \((n, k) \text{GF}(q)\)-codes. A linear \((n, k) \text{GF}(q)\)-code \(C\) is called optimal, if \(d(C)= d_q(n, k)\) (optimal \((n, k, d(C))\) code). Quasi-cyclic codes are a generalization of cyclic codes whereby a cyclic shift of a codeword by \(p\) positions results in a codeword. In this paper optimal \((84, 6, 54)\) and \((98, 6, 63)\) quasi-cyclic two-weight codes over \(\text{GF}(3)\) are constructed by a simple greedy heuristic combinatorial optimization algorithm. As a corollary two new strongly regular graphs with parameters \((729, 168, 27, 42)\) and \((729, 196, 43, 56)\) are obtained.