Explicit Construction of q-ary 2-deletion Correcting Codes with Low Redundancy

TL;DR

This paper presents an explicit construction of q-ary 2-deletion correcting codes with lower redundancy than existing codes, using binary code transformations and leveraging Guruswami and Hastad's binary 2-deletion code.

Contribution

It introduces a new explicit construction of q-ary 2-deletion codes with minimal redundancy and efficient list-decodability, improving upon prior methods.

Findings

Redundancy of 5 log(n)+10 log(log(n)) + 3 log(q)+O(1) achieved

Construction is efficiently encodable and list-decodable

Redundancy is smaller than existing q-ary 2-deletion codes

Abstract

We consider the problem of efficient construction of q-ary 2-deletion correcting codes with low redundancy. We show that our construction requires less redundancy than any existing efficiently encodable q-ary 2-deletion correcting codes. Precisely speaking, we present an explicit construction of a q-ary 2-deletion correcting code with redundancy 5 log(n)+10log(log(n)) + 3 log(q)+O(1). Using a minor modification to the original construction, we obtain an efficiently encodable q-ary 2-deletion code that is efficiently list-decodable. Similarly, we show that our construction of list-decodable code requires a smaller redundancy compared to any existing list-decodable codes. To obtain our sketches, we transform a q-ary codeword to a binary string which can then be used as an input to the underlying base binary sketch. This is then complemented with additional q-ary sketches that the original q-ary codeword is required to satisfy. In other words, we build our codes via a binary 2-deletion code as a black-box. Finally we utilize the binary 2-deletion code proposed by Guruswami and Hastad to our construction to obtain the main result of this paper.