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Open AccessArticle

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

Department of Electronic and Electrical Engineering, Hongik University, Seoul 04066, Korea
Entropy 2019, 21(12), 1202; https://doi.org/10.3390/e21121202
Received: 11 November 2019 / Revised: 29 November 2019 / Accepted: 5 December 2019 / Published: 6 December 2019
(This article belongs to the Special Issue Information Theory and Graph Signal Processing)
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. View Full-Text
Keywords: deletion channel; maximum independent set; mixed integer programming; Varshamov–Tenengolts code deletion channel; maximum independent set; mixed integer programming; Varshamov–Tenengolts code
MDPI and ACS Style

No, A. Nonasymptotic Upper Bounds on Binary Single Deletion Codes via Mixed Integer Linear Programming. Entropy 2019, 21, 1202.

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