Single-Deletion Single-Substitution Correcting Codes

Correcting insertions/deletions as well as substitution errors simultaneously plays an important role in DNA-based storage systems as well as in classical communications. This paper deals with the fundamental task of constructing codes that can correct a single insertion or deletion along with a single substitution. A non-asymptotic upper bound on the size of single-deletion single-substitution correcting codes is derived, showing that the redundancy of such a code of length $n$ has to be at least $2 \log n$. The bound is presented both for binary and non-binary codes while an extension to single deletion and multiple substitutions is presented for binary codes. An explicit construction of single-deletion single-substitution correcting codes with at most $6 \log n + 8$ redundancy bits is derived. Note that the best known construction for this problem has to use 3-deletion correcting codes whose best known redundancy is roughly $24 \log n$.Comment: Paper submitted to International Symposium on Information Theory (ISIT) 202

An Improvement of Non-binary Code Correcting Single b-Burst of Insertions or Deletions

This paper constructs a non-binary code correcting a single $b$-burst of insertions or deletions with a large cardinality. This paper also proposes a decoding algorithm of this code and evaluates a lower bound of the cardinality of this code. Moreover, we evaluate an asymptotic upper bound on the cardinality of codes which correct a single burst of insertions or deletions.Comment: 7 pages, accepted to ISITA 201

Efficient Majority-Logic Decoding of Short-Length Reed--Muller Codes at Information Positions

Short-length Reed--Muller codes under majority-logic decoding are of particular importance for efficient hardware implementations in real-time and embedded systems. This paper significantly improves Chen's two-step majority-logic decoding method for binary Reed--Muller codes $\text{RM}(r,m)$, $r \leq m/2$, if --- systematic encoding assumed --- only errors at information positions are to be corrected. Some general results on the minimal number of majority gates are presented that are particularly good for short codes. Specifically, with its importance in applications as a 3-error-correcting, self-dual code, the smallest non-trivial example, $\text{RM}(2,5)$ of dimension 16 and length 32, is investigated in detail. Further, the decoding complexity of our procedure is compared with that of Chen's decoding algorithm for various Reed--Muller codes up to length $2^{10}$.Comment: 8 pages; to appear in "IEEE Transactions on Communications

Quickest Sequence Phase Detection

A phase detection sequence is a length-$n$ cyclic sequence, such that the location of any length-$k$ contiguous subsequence can be determined from a noisy observation of that subsequence. In this paper, we derive bounds on the minimal possible $k$ in the limit of $n\to\infty$, and describe some sequence constructions. We further consider multiple phase detection sequences, where the location of any length-$k$ contiguous subsequence of each sequence can be determined simultaneously from a noisy mixture of those subsequences. We study the optimal trade-offs between the lengths of the sequences, and describe some sequence constructions. We compare these phase detection problems to their natural channel coding counterparts, and show a strict separation between the fundamental limits in the multiple sequence case. Both adversarial and probabilistic noise models are addressed.Comment: To appear in the IEEE Transactions on Information Theor

Tutorial on algebraic deletion correction codes

The deletion channel is known to be a notoriously diffcult channel to design error-correction codes for. In spite of this difficulty, there are some beautiful code constructions which give some intuition about the channel and about what good deletion codes look like. In this tutorial we will take a look at some of them. This document is a transcript of my talk at the coding theory reading group on some interesting works on deletion channel. It is not intended to be an exhaustive survey of works on deletion channel, but more as a tutorial to some of the important and cute ideas in this area. For a comprehensive survey, we refer the reader to the cited sources and surveys. We also provide an implementation of VT codes that correct single insertion/deletion errors for general alphabets at https://github.com/shubhamchandak94/VT_codes/

An Improvement to Levenshtein's Upper Bound on the Cardinality of Deletion Correcting Codes

We consider deletion correcting codes over a q-ary alphabet. It is well known that any code capable of correcting s deletions can also correct any combination of s total insertions and deletions. To obtain asymptotic upper bounds on code size, we apply a packing argument to channels that perform different mixtures of insertions and deletions. Even though the set of codes is identical for all of these channels, the bounds that we obtain vary. Prior to this work, only the bounds corresponding to the all insertion case and the all deletion case were known. We recover these as special cases. The bound from the all deletion case, due to Levenshtein, has been the best known for more than forty five years. Our generalized bound is better than Levenshtein's bound whenever the number of deletions to be corrected is larger than the alphabet size

Secure, reliable, and efficient communication over the wiretap channel

Secure wireless communication between devices is essential for modern communication systems. Physical-layer security over the wiretap channel may provide an additional level of secrecy beyond the current cryptographic approaches. Given a sender Alice, a legitimate receiver Bob, and a malicious eavesdropper Eve, the wiretap channel occurs when Eve experiences a worse signal-to-noise ratio than Bob. Previous study of the wiretap channel has tended to make assumptions that ignore the reality of wireless communication. This thesis presents a study of short block length codes with the aim of both reliability for Bob and confusion for Eve. The standard approach to wiretap coding is shown to be very inefficient for reliability. Quantifying Eve's confusion in terms of entropy is not solved in many cases, though it is possible for codes with a moderate complexity trellis representation. Using error rate arguments, error correcting codes with steep performance curves turn out to be desirable both for reliability and confusion.Masteroppgave i informatikkINF399MAMN-INFMAMN-PRO

Silence is Golden: exploiting jamming and radio silence to communicate

Jamming techniques require just moderate resources to be deployed, while their effectiveness in disrupting communications is unprecedented. In this paper we introduce several contributions to jamming mitigation. In particular, we introduce a novel adversary model that has both (unlimited) jamming reactive capabilities as well as powerful (but limited) proactive jamming capabilities. Under this powerful but yet realistic adversary model, the communication bandwidth provided by current anti-jamming solutions drops to zero. We then present Silence is Golden (SiG): a novel anti jamming protocol that, introducing a tunable, asymmetric communication channel, is able to mitigate the adversary capabilities, enabling the parties to communicate. For instance, with SiG it is possible to deliver a 128 bits long message with a probability greater than 99% in 4096 time slots in the presence of a jammer that jams all the on-the-fly communications and the 74% of the silent radio spectrum---while competing proposals simply fail. The provided solution enjoys a thorough theoretical analysis and is supported by extensive experimental results, showing the viability of our proposal

Source and Physical-Layer Network Coding for Correlated Two-Way Relaying

In this paper, we study a half-duplex two-way relay channel (TWRC) with correlated sources exchanging bidirectional information. In the case, when both sources have the knowledge of correlation statistics, a source compression with physical-layer network coding (SCPNC) scheme is proposed to perform the distributed compression at each source node. When only the relay has the knowledge of correlation statistics, we propose a relay compression with physical-layer network coding (RCPNC) scheme to compress the bidirectional messages at the relay. The closed-form block error rate (BLER) expressions of both schemes are derived and verified through simulations. It is shown that the proposed schemes achieve considerable improvements in both error performance and throughput compared with the conventional non-compression scheme in correlated two-way relay networks (CTWRNs).Comment: 15 pages, 6 figures. IET Communications, 201

On a Duality Between Recoverable Distributed Storage and Index Coding

In this paper, we introduce a model of a single-failure locally recoverable distributed storage system. This model appears to give rise to a problem seemingly dual of the well-studied index coding problem. The relation between the dimensions of an optimal index code and optimal distributed storage code of our model has been established in this paper. We also show some extensions to vector codes.Comment: A small new section and new references added. A minor error corrected from the previous versio