Existence of Perfect 4-Deletion-Correcting Codes with Length Six

Abstract

By a T ^*(2, k, v)-code we mean a perfect4-deletion-correcting code of length 6 over an alphabet of size v, which is capable of correcting anycombination of up to 4 deletions and/or insertions of letters that occur in transmission of codewords. Thethird author (DCC Vol. 23, No. 1) presented a combinatorial construction for such codes and prove thata T ^*(2, 6, v)-code exists for all positive integers v≢ 3 (mod 5), with 12 possible exceptions of v. In this paper, the notion of a directedgroup divisible quasidesign is introduced and used to show that a T ^*(2, 6,v)-code exists for all positive integers v ≡ 3 (mod 5), except possiblyfor v ∈ {173, 178, 203, 208}. The 12 missing cases for T ^*(2,6, v)-codes with v ≢ 3 (mod 5) are also provided, thereby the existenceproblem for T ^*(2, 6, v)-codes is almost complete.