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

Albert No

doi:10.3390/e21121202

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 ; 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.

Highlights

IntroductionA deletion channel is one of the most important channels in the history of communication
A deletion channel is one of the most important channels in the history of communication.The channel has a deletion error where the symbol is being removed without knowing the position of it
Kanoria and Montanari provided an approximation of the channel capacity of binary deletion channel when p → 0 [1]

Summary

Introduction

A deletion channel is one of the most important channels in the history of communication. The channel has a deletion error where the symbol is being removed without knowing the position of it. Unlike many other channels where the positions of symbols remain the same, the decoder needs to specify the position of each symbol, which is called a synchronization issue. Due to this issue, the deletion channel is surprisingly hard to analyze. Kanoria and Montanari provided an approximation of the channel capacity of binary deletion channel when p → 0 [1]. The channel capacity is unknown in this setting even in binary case

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