In this paper we present a new code design technique which produces best equivocation codes (BE_qC) with highest minimum distance for syndrome coding which have better secrecy than the best known error correcting codes (BKC). The best equivocation codes of highest minimum distance have been determined by using a combination of a code design technique based on extensions of the parity check matrix from an optimal set of good equivocation codes of highest minimum distance coupled with a technique of determining the highest minimum distance (dmin) of these extended codes. Code examples are presented that demonstrate that the equivocation rate of these best equivocation codes (BE_qC) exceeds by a large margin the best known error correcting codes having the same parameters.