Nonasymptotic Upper Bounds on Binary Single Deletion Codes via Mixed Integer Linear Programming

Abstract

The size of the largest binary single deletion code has been unknown for more than 50 years. It is known that Varshamov–Tenengolts (VT) code is an optimum single deletion code for block length n ≤ 10 ; however, only a few upper bounds of the size of single deletion code are proposed for larger n. We provide improved upper bounds using Mixed Integer Linear Programming (MILP) relaxation technique. Especially, we show the size of single deletion code is smaller than or equal to 173 when the block length n is 11. In the second half of the paper, we propose a conjecture that is equivalent to the long-lasting conjecture that “VT code is optimum for all n”. This equivalent formulation of the conjecture contains small sub-problems that can be numerically verified. We provide numerical results that support the conjecture.

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“…As to the cardinality of VT codes, for about 50 years, a lower bound of size of the best class of VT codes can be achieved, but an upper bound is rarely provided even in binary case. The author in [ 23 ] used Mixed Integer Linear Programming (MILP) relaxation technique to obtain the tighter upper bound of the binary VT code, for example, with the length n = 11, the maximum size of one deletion code was calculated as 173. Moreover, the conjecture about maximum size of VT code for all n was also provided.…”