The existence of directed BIBDs

AbstractFor any positive integers k⩾3 and λ, let cd(k,λ) denote the smallest integer such that the necessary conditions 2λ(v−1)≡0(modk−1) and λv(v−1)≡0(mod(k2)) for the existence of a DB(k,λ;v) are also sufficient for every v⩾cd(k,λ). In this article we provide an estimate for cd(k,λ) when k≡0(mod4) and any λ. Combined with the results in (Discrete Math. 222 (2000) 27–40), we completely give an estimate of cd(k,λ) for any integers k⩾3 and λ

Construction of I-Deletion-Correcting Ternary Codes

Finding large deletion correcting codes is an important issue in coding theory. Many researchers have studied this topic over the years. Varshamov and Tenegolts constructed the Varshamov-Tenengolts codes (VT codes) and Levenshtein showed the Varshamov-Tenengolts codes are perfect binary one-deletion correcting codes in 1992. Tenegolts constructed T codes to handle the non-binary cases. However the T codes are neither optimal nor perfect, which means some progress can be established. Latterly, Bours showed that perfect deletion-correcting codes have a close relationship with design theory. By this approach, Wang and Yin constructed perfect 5-deletion correcting codes of length 7 for large alphabet size. For our research, we focus on how to extend or combinatorially construct large codes with longer length, few deletions and small but non-binary alphabet especially ternary. After a brief study, we discovered some properties of T codes and produced some large codes by 3 different ways of extending some existing good codes