[Paper Review] Criss-Cross Deletion Correcting Codes

Rawad Bitar, Ilia Smagloy+3 more|arXiv (Cornell University)|Apr 30, 2020

DNA and Biological Computing20 references3 citations

TL;DR

This paper introduces $(1,1)$-criss-cross deletion correcting codes for $n \times n$ arrays, where one row and one column may be deleted. It establishes an asymptotic lower bound on redundancy of $2n - 2 + 2\log n$ bits and presents a code construction with explicit decoding that achieves redundancy within $2\log n + 9 + 2\log e$ bits of this bound.

ABSTRACT

This paper studies the problem of constructing codes correcting deletions in arrays. Under this model, it is assumed that an $n imes n$ array can experience deletions of rows and columns. These deletion errors are referred to as $(t_\mathrm{r},t_\mathrm{c})$-criss-cross deletions if $t_\mathrm{r}$ rows and $t_\mathrm{c}$ columns are deleted, while a code correcting these deletion patterns is called a $(t_\mathrm{r},t_\mathrm{c})$-criss-cross deletion correcting code. The definitions for criss-cross insertions are similar. Similar to the one-dimensional case, it is first shown that the problems of correcting criss-cross deletions and criss-cross insertions are equivalent. Then, we mostly investigate the case of $(1,1)$-criss-cross deletions. An asymptotic upper bound on the cardinality of $(1,1)$-criss-cross deletion correcting codes is shown which assures that the asymptotic redundancy is at least $2n-2+2\log n$ bits. Finally, a code construction with an explicit decoding algorithm is presented. The redundancy of the construction is away from the lower bound by at most $2 \log n+9+2\log e$ bits.

Motivation & Objective

• To address the problem of correcting deletions in two-dimensional arrays where entire rows and columns may be lost.
• To establish theoretical limits on the size of codes capable of correcting $(t_r, t_c)$-criss-cross deletions, particularly for the $(1,1)$ case.
• To develop an explicit code construction with efficient decoding for $(1,1)$-criss-cross deletion correction.
• To analyze the redundancy of the proposed code and compare it to the theoretical lower bound.

Proposed method

• Proves equivalence between the problems of correcting criss-cross deletions and criss-cross insertions in the array model.
• Derives an asymptotic upper bound on the cardinality of $(1,1)$-criss-cross deletion correcting codes using combinatorial and information-theoretic arguments.
• Constructs a code based on parity-check constraints designed to detect and correct the loss of one row and one column.
• Designs an explicit decoding algorithm that identifies the positions of the deleted row and column using redundancy symbols.
• Analyzes the redundancy of the construction and shows it is within $2\log n + 9 + 2\log e$ bits of the theoretical lower bound.

Experimental results

Research questions

• RQ1 What is the minimum redundancy required for a code to correct $(1,1)$-criss-cross deletions in an $n \times n$ array?
• RQ2 How do the problems of correcting criss-cross deletions and insertions relate in the two-dimensional array model?
• RQ3 Can an explicit code construction be designed with efficient decoding for $(1,1)$-criss-cross deletions?
• RQ4 How close can the redundancy of such a code be to the theoretical lower bound?

Key findings

• The asymptotic redundancy of $(1,1)$-criss-cross deletion correcting codes is bounded below by $2n - 2 + 2\log n$ bits.
• The proposed code construction achieves redundancy within $2\log n + 9 + 2\log e$ bits of this lower bound.
• An explicit decoding algorithm is provided that correctly identifies and recovers the deleted row and column.
• The problems of correcting criss-cross deletions and insertions are shown to be equivalent in the array model.
• The code construction is shown to be asymptotically optimal up to a small additive term in redundancy.