Explicit Construction of q-Ary 2-Deletion Correcting Codes With Low Redundancy

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
$5\log n+10\log \log n + 3\log q+O(1)$
where
$q$
is assumed to be a constant with respect to
$n$
. 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 Håstad to our construction to obtain the main result of this paper.