Bounds and Constructions for Insertion and Deletion Codes

Insertion and deletion (insdel for short) codes have recently attracted a lot of attention due to their applications in many interesting fields such as DNA storage, DNA analysis, race-track memory error correction and language processing. The present paper mainly studies limits and constructions of insdel codes. The paper can be divided into two parts. The first part focuses on various bounds, while the second part concentrates on constructions of insdel codes. Although the insdel-metric Singleton bound has been derived before, it is still unknown if there are any nontrivial codes achieving this bound. Our first result shows that any nontrivial insdel codes do not achieve the insdel-metric Singleton bound. The second bound shows that every
$[n,k]$
Reed-Solomon code has insdel distance upper bounded by
$2n-4k+4$
and it is known in literature that an
$[n,k]$
Reed-Solomon code can have insdel distance
$2n-4k+4$
as long as the field size is sufficiently large. The third bound shows a trade-off between insdel distance and code alphabet size for codes achieving the Hamming-metric Singleton bound. In the second part of the paper, we first provide a non-explicit construction of nonlinear codes that can approach the insdel-metric Singleton bound arbitrarily when the code alphabet size is sufficiently large. The second construction gives two-dimensional Reed-Solomon codes of length
$n$
and insdel distance
$2n-4$
with field size
$q=O(n^{5})$
. The non-explicit construction of insdel codes is based on constant-weight
$L^{1}$
-codes that are introduced in this paper. We first establish a relation between constant-weight
$L^{1}$
-codes and insdel codes. Based on this relation, we construct constant-weight
$L^{1}$
-codes with reasonable parameters and subsequently give insdel codes approaching the insdel-metric Singleton bound. Via automorphism group of rational function field, we provide a necessary and sufficient condition under which a two-dimensional Reed-Solomon code of length
$n$
has insdel distance
$2n-4$
. Based on this criterion, we present a construction of
$q$
-ary two-dimensional Reed-Solomon codes of length
$n$
and insdel distance
$2n-4$
with
$q=O(n^{5})$
. Though this is worse than the current best field size, we provide a new angle to look into the problem.