Existence of directed BIBDs with block size 7 and related perfect 5-deletion-correcting codes of length 7.

Wang and Yin have established the existence of a directed BIBD with block size 7 and index 1 (a DBIBD( v, 7, 1)) for all v ≡ 1, 7 (mod 21) except for v = 22 and possibly for 68 other cases. In this paper, we reduce the number of possible exceptions to 4, namely v = 274, 358, 400, 526. Correspondingly, for all such v, our results establish the existence of a T(2, 7, v)-code or equivalently a perfect 5-deletion-correcting code with words of length 7 over an alphabet of size v, where all the coordinates must be different. In the process, we also reduce the possible exceptions for ( v, 7, 2)-BIBDs to 2 cases, v = 274 and 358 (in addition to the non-existent (22, 7, 2)-BIBD). [ABSTRACT FROM AUTHOR]