Nonasymptotic Upper Bounds for Deletion Correcting Codes

Kulkarni, Ankur A.; Kiyavash, Negar

doi:10.1109/tit.2013.2257917

Cited by 94 publications

(124 citation statements)

References 34 publications

Supporting

Mentioning

123

Contrasting

Order By: Relevance

“…Kulkarni and Kiyavash computed the local degree upper bound, or equivalently ϕ An (1) [4]. This shows that κ * (A n ) is at most…”

Section: Application To the Single Deletion Channelmentioning

confidence: 97%

“…1 For x ∈ X, let N A (x) ⊆ Y be the neighborhood of x in the channel graph (the 1 An equivalent approach, taken by Kulkarni and Kiyavash [4], is to represent an error model by a hypergraph. A hypergraph consists of a vertex set and a family of hyperedges.…”

Section: Combinatorial Channelsmentioning

confidence: 99%

“…Levenshtein eventually refined his original asymptotic bound (and the parallel nonbinary bound of Tenengolts) into a nonasymptotic version [19]. Kulkarni and Kiyavash recently proved a better upper bound for an arbitrary number of deletions and any alphabet size [4].…”

Section: Introductionmentioning

confidence: 99%

“…In fact, recently a new bound, explicitly derived via linear programming, was applied by Kulkarni and Kiyavash to find an upper bound on the size of deletion-correcting codes [4]. It was subsequently applied to grain errors [30,31] and multipermutation errors [32].…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, we derive a series of bounds from approximations to packing and covering problems. The local degree bound of [4] is one of the bounds in the series. We associate each bound with an iterative procedure such that the original bound is the result of a single step.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Combinatorial channels from partially ordered sets

Cullina

Kiyavash

2015

2015 IEEE Information Theory Workshop (ITW)

Self Cite

View full text Add to dashboard Cite

A central task of coding theory is the design of schemes to reliably transmit data though space, via communication systems, or through time, via storage systems. Our goal is to identify and exploit structural properties common to a wide variety of coding problems, classical and modern, using the framework of partially ordered sets. We represent adversarial error models as combinatorial channels, form combinatorial channels from posets, identify a structural property of posets that leads to families of channels with the same codes, and bound the size of codes by optimizing over a family of equivalent channels. A large number of previously studied coding problems that fit into this framework. This leads to a new upper bound on the size of s-deletion correcting codes. We use a linear programming framework to obtain spherepacking upper bounds when there is little underlying symmetry in the coding problem. Finally, we introduce and investigate a strong notion of poset homomorphism: locally bijective cover preserving maps. We look for maps of this type to and from the subsequence partial order on q-ary strings.ii Our goal is to identify and exploit structural properties common to a wide variety of coding problems, classical and modern, using the framework of partially ordered sets.In Chapter 2, we introduce the main ideas. We use the notion of a combinatorial channel to represent an adversarial or worst-case error model. We define partially ordered sets (posets), poset homomorphisms, and the other fundamental concepts. We describe a simple method for forming combinatorial channels from posets and identify a structural property of posets that leads to families of channels with the same codes. This leads to ones of our core techniques: bounding the size of codes by optimizing over a family of equivalent channels. In the second half of the chapter, we give examples of previously studied coding problems that fit into this framework. This chapter is based primarily on [1] and also contains some material from [2].In Chapter 3, we apply the methods described in Chapter 2 to the subsequence partial order on q-ary strings. This leads to a new upper bound on the size of s-deletion correcting codes, for fixed s and input length going to infinity. The new bounds improves on the previous best bound whenever the number of deletions is greater than the alphabet size. At the end of this chapter, we look at the partial order on integer compositions. This has many features in common with the subsequence partial order, but many enumerative problems are more tractable. The channels related to the composition partial order also have emerging applications, particularly in DNA storage systems. This chapter is based primarily on [3] and Section 3.8 is based on [1].Chapter 4 covers the linear programming framework for sphere-packing upper bounds. The main contributions are methods for bounding the values 1 of these linear programs when they involve too many variables and constraints to evaluate or even analyze exactly. These methods a...

show abstract

“…Kulkarni and Kiyavash computed the local degree upper bound, or equivalently ϕ An (1) [4]. This shows that κ * (A n ) is at most…”

Section: Application To the Single Deletion Channelmentioning

confidence: 97%

Section: Combinatorial Channelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Combinatorial channels from partially ordered sets

Cullina

Kiyavash

2015

2015 IEEE Information Theory Workshop (ITW)

Self Cite

View full text Add to dashboard Cite

show abstract

Duplication-correcting codes

Lenz

Wachter-Zeh

Yaakobi

2018

Des. Codes Cryptogr.

View full text Add to dashboard Cite

In this work, we propose constructions that correct duplications of multiple consecutive symbols. These errors are known as tandem duplications, where a sequence of symbols is repeated; respectively as palindromic duplications, where a sequence is repeated in reversed order. We compare the redundancies of these constructions with code size upper bounds that are obtained from sphere packing arguments. Proving that an upper bound on the code cardinality for tandem deletions is also an upper bound for inserting tandem duplications, we derive the bounds based on this special tandem deletion error as this results in tighter bounds. Our upper bounds on the cardinality directly imply lower bounds on the redundancy which we compare with the redundancy of the best known construction correcting arbitrary burst insertions.Our results indicate that the correction of palindromic duplications requires more redundancy than the correction of tandem duplications and both significantly less than arbitrary burst insertions.

show abstract

Improved bounds for codes correcting insertions and deletions

Yasunaga

2024

Des. Codes Cryptogr.

View full text Add to dashboard Cite

This paper studies the cardinality of codes correcting insertions and deletions. We give improved upper and lower bounds on code size. Our upper bound is obtained by utilizing the asymmetric property of list decoding for insertions and deletions and can be seen as analogous to the Elias bound in the Hamming metric. Our non-asymptotic bound is better than the existing bounds when the minimum Levenshtein distance is relatively large. The asymptotic bound exceeds the Elias and the MRRW bounds adapted from the Hamming-metric bounds for the binary and the quaternary cases. Our lower bound improves on the bound by Levenshtein, but its effect is limited and vanishes asymptotically.

show abstract

Nonasymptotic Upper Bounds for Deletion Correcting Codes

Cited by 94 publications

References 34 publications

Combinatorial channels from partially ordered sets

Combinatorial channels from partially ordered sets

Duplication-correcting codes

Improved bounds for codes correcting insertions and deletions

Contact Info

Product

Resources

About