Using integer linear programming and table-lookups, we prove that there is no binary linear [1988, 12, 992] code. As a by-product, the non-existence of binary linear codes with the parameters [324, 10, 160], [356, 10, 176], [772, 11, 384], and [836, 11, 416] is shown. Our work is motivated by the recent construction of the extended dualized Kerdock code K6*, which is a Z4-linear code having a non-linear binary Gray image with the parameters 1988, 212,992. By our result, the code K6* can be added to the small list of Z4-codes for which it is known that the Gray image is better than any binary linear code.

In this correspondence, we describe a heuristic method for the construction of linear codes with given parameters n, k, q, and prescribed minimum distance of at least . Our approach is based on a function estimating the probability that a code of dimension and blocklength over is extendable to a code with the given properties. Combining this evaluation function with a search algorithm, we were able to improve 40 entries in the international tables for the best known minimum distance in the cases and found at least two new optimal linear codes.