# An Introduction to Linear and Cyclic Codes Daniel Augot, Emmanuela Orsini, Emanuele Betti

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• Reed-Solomon Codes
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• Scribe Notes
6.440 Essential Coding Theory Feb 19, 2008 Lecture 4 Lecturer: Madhu Sudan Scribe: Ning Xie Today we are going to discuss limitations of codes. More speciﬁcally, we will see rate upper bounds of codes, including Singleton bound, Hamming bound, Plotkin bound, Elias bound and Johnson bound. 1 Review of last lecture n k Let C Σ be an error correcting code. We say C is an (n, k, d)q code if Σ = q, C q and ∆(C) d, ⊆ | | | | ≥ ≥ where ∆(C) denotes the minimum distance of C. We write C as [n, k, d]q code if furthermore the code is a F k linear subspace over q (i.e., C is a linear code). Deﬁne the rate of code C as R := n and relative distance d as δ := n . Usually we ﬁx q and study the asymptotic behaviors of R and δ as n . Recall last time we gave an existence result, namely the Gilbert-Varshamov(GV)→ ∞ bound constructed by greedy codes (Varshamov bound corresponds to greedy linear codes). For q = 2, GV bound gives codes with k n log2 Vol(n, d 2). Asymptotically this shows the existence of codes with R 1 H(δ), which is similar≥ to− Shannon’s result.− Today we are going to see some upper bound results, that≥ is,− code beyond certain bounds does not exist. 2 Singleton bound Theorem 1 (Singleton bound) For any code with any alphabet size q, R + δ 1. ≤ Proof Let C Σn be a code with C Σ k. The main idea is to project the code C on to the ﬁrst k 1 ⊆ | | ≥ | | n k−1 − coordinates.
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• An Introduction to Coding Theory
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• Optimal Codes and Entropy Extractors
Department of Mathematics Ph.D. Programme in Mathematics XXIX cycle Optimal Codes and Entropy Extractors Alessio Meneghetti 2017 University of Trento Ph.D. Programme in Mathematics Doctoral thesis in Mathematics OPTIMAL CODES AND ENTROPY EXTRACTORS Alessio Meneghetti Advisor: Prof. Massimiliano Sala Head of the School: Prof. Valter Moretti Academic years: 2013{2016 XXIX cycle { January 2017 Acknowledgements I thank Dr. Eleonora Guerrini, whose Ph.D. Thesis inspired me and whose help was crucial for my research on Coding Theory. I also thank Dr. Alessandro Tomasi, without whose cooperation this work on Entropy Extraction would not have come in the present form. Finally, I sincerely thank Prof. Massimiliano Sala for his guidance and useful advice over the past few years. This research was funded by the Autonomous Province of Trento, Call \Grandi Progetti 2012", project On silicon quantum optics for quantum computing and secure communications - SiQuro. i Contents List of Figures......................v List of Tables....................... vii Introduction1 I Codes and Random Number Generation7 1 Introduction to Coding Theory9 1.1 Deﬁnition of systematic codes.............. 16 1.2 Puncturing, shortening and extending......... 18 2 Bounds on code parameters 27 2.1 Sphere packing bound.................. 29 2.2 Gilbert bound....................... 29 2.3 Varshamov bound.................... 30 2.4 Singleton bound..................... 31 2.5 Griesmer bound...................... 33 2.6 Plotkin bound....................... 34 2.7 Johnson bounds...................... 36 2.8 Elias bound........................ 40 2.9 Zinoviev-Litsyn-Laihonen bound............ 42 2.10 Bellini-Guerrini-Sala bounds............... 43 iii 3 Hadamard Matrices and Codes 49 3.1 Hadamard matrices.................... 50 3.2 Hadamard codes..................... 53 4 Introduction to Random Number Generation 59 4.1 Entropy extractors...................
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• Optimal Locally Repairable Codes of Distance 3 and 4 Via Cyclic Codes
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CWI's Institutional Repository Optimal locally repairable codes of distance 3 and 4 via cyclic codes Yuan Luo,∗ Chaoping Xing and Chen Yuan y Abstract Like classical block codes, a locally repairable code also obeys the Singleton-type bound (we call a locally repairable code optimal if it achieves the Singleton-type bound). In the breakthrough work of [14], several classes of optimal locally repairable codes were constructed via subcodes of Reed-Solomon codes. Thus, the lengths of the codes given in [14] are upper bounded by the code alphabet size q. Recently, it was proved through extension of construction in [14] that length of q-ary optimal locally repairable codes can be q + 1 in [7]. Surprisingly, [2] presented a few examples of q-ary optimal locally repairable codes of small distance and locality with code length achieving roughly q2. Very recently, it was further shown in [8] that there exist q-ary optimal locally repairable codes with length bigger than q + 1 and distance propositional to n. Thus, it becomes an interesting and challenging problem to construct new families of q-ary optimal locally repairable codes of length bigger than q + 1. In this paper, we construct a class of optimal locally repairable codes of distance 3 and 4 with un- bounded length (i.e., length of the codes is independent of the code alphabet size). Our technique is through cyclic codes with particular generator and parity-check polynomials that are carefully chosen.
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• MAS309 Coding Theory
MAS309 Coding theory Matthew Fayers January–March 2008 This is a set of notes which is supposed to augment your own notes for the Coding Theory course. They were written by Matthew Fayers, and very lightly edited my me, Mark Jerrum, for 2008. I am very grateful to Matthew Fayers for permission to use this excellent material. If you ﬁnd any mistakes, please e-mail me: [email protected]. Thanks to the following people who have already sent corrections: Nilmini Herath, Julian Wiseman, Dilara Azizova. Contents 1 Introduction and deﬁnitions 2 1.1 Alphabets and codes . 2 1.2 Error detection and correction . 2 1.3 Equivalent codes . 4 2 Good codes 6 2.1 The main coding theory problem . 6 2.2 Spheres and the Hamming bound . 8 2.3 The Singleton bound . 9 2.4 Another bound . 10 2.5 The Plotkin bound . 11 3 Error probabilities and nearest-neighbour decoding 13 3.1 Noisy channels and decoding processes . 13 3.2 Rates of transmission and Shannon’s Theorem . 15 4 Linear codes 15 4.1 Revision of linear algebra . 16 4.2 Finite ﬁelds and linear codes . 18 4.3 The minimum distance of a linear code . 19 4.4 Bases and generator matrices . 20 4.5 Equivalence of linear codes . 21 4.6 Decoding with a linear code . 27 5 Dual codes and parity-check matrices 30 5.1 The dual code . 30 5.2 Syndrome decoding . 34 1 2 Coding Theory 6 Some examples of linear codes 36 6.1 Hamming codes . 37 6.2 Existence of codes and linear independence .
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• Optimal Linear and Cyclic Locally Repairable Codes Over Small Fields
Optimal Linear and Cyclic Locally Repairable Codes over Small Fields Alexander Zeh and Eitan Yaakobi Computer Science Department Technion—Israel Institute of Technology [email protected], [email protected] Abstract—We consider locally repairable codes over small This paper is structured as follows. Section II gives ﬁelds and propose constructions of optimal cyclic and linear codes necessary preliminaries on linear and cyclic codes, deﬁnes in terms of the dimension for a given distance and length. LRCs, recalls the generalized Singleton bound, the Cadambe– Four new constructions of optimal linear codes over small ﬁelds with locality properties are developed. The ﬁrst two Mazumdar bound [8] as well as the deﬁnition of availability approaches give binary cyclic codes with locality two. While the for LRCs. The concept of a locality code and the projection ﬁrst construction has availability one, the second binary code is to an additive code are discussed in Section III based on the characterized by multiple available repair sets based on a binary work of Goparaju and Calderbank [9]. Two new constructions Simplex code. of optimal binary codes are given in Section IV and a con- The third approach extends the ﬁrst one to q-ary cyclic codes including (binary) extension ﬁelds, where the locality property struction based on a q-ary shortened cyclic ﬁrst-order Reed– is determined by the properties of a shortened ﬁrst-order Reed– Muller code is given in Section V. The fourth construction Muller code. Non-cyclic optimal binary linear codes with locality in Section VI uses code concatenation and provides optimal greater than two are obtained by the fourth construction.
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• Constructions of MDS-Convolutional Codes Ring H, with RNk Q@Haak
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 5, JULY 2001 2045 Constructions of MDS-Convolutional Codes ring h, with rnk q@hAak. For the purpose of this correspon- dence, we define the rate kan convolutional code generated by q@hA Roxana Smarandache, Student Member, IEEE, as the set Heide Gluesing-Luerssen, and n k Joachim Rosenthal, Senior Member, IEEE g a fu@hAq@hA P @hAju@hA P @hAg and say that q@hA is a generator matrix for the convolutional code g q@hA qH@hA Abstract—Maximum-distance separable (MDS) convolutional codes are . If the generator matrices and both generate the same characterized through the property that the free distance attains the gener- convolutional code g then there exists a k k invertible matrix @hA alized Singleton bound. The existence of MDS convolutional codes was es- with qH@hAa @hAq@hA and we say q@hA and qH@hA are equiva- tablished by two of the authors by using methods from algebraic geometry. lent encoders. This correspondence provides an elementary construction of MDS convo- lutional codes for each rate and each degree . The construction is Because of this, we can assume without loss of generality that the based on a well-known connection between quasi-cyclic codes and convo- code g is presented by a minimal basic encoder q@hA. For this, let lutional codes. #i be the ith-row degree of q@hA, i.e., #i amxj deg gij @hA. In the # Index Terms—Convolutional codes, generalized Singleton bound, max- literature [7], the indexes i are also called the constraint length for the imum-distance separable (MDS) convolutional codes.
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• Bounds on Error-Correction Coding Performance
Chapter 1 Bounds on Error-Correction Coding Performance 1.1 Gallager’s Coding Theorem The sphere packing bound by Shannon [18] provides a lower bound to the frame error rate (FER) achievable by an (n, k, d) code but is not directly applicable to binary codes. Gallager [4] presented his coding theorem for the average FER for the ensemble of all random binary (n, k, d) codes. There are 2n possible binary combinations for each codeword which in terms of the n-dimensional signal space hypercube corresponds to one vertex taken from 2n possible vertices. There are 2k codewords, and therefore 2nk different possible random codes. The receiver is considered to be composed of 2k matched ﬁlters, one for each codeword and a decoder error occurs if any of the matched ﬁlter receivers has a larger output than the matched ﬁlter receiver corresponding to the transmitted codeword. Consider this matched ﬁlter receiver and another different matched ﬁlter receiver, and assume that the two codewords differ in d bit positions. The Hamming distance between the two = k codewords is d. The energy per transmitted bit is Es n Eb, where Eb is the energy σ 2 = N0 per information bit. The noise variance per matched ﬁltered received bit, 2 , where N0 is the single sided noise spectral density. In the absence√ of noise, the output of the matched ﬁlter receiver for the transmitted codeword√ is n Es and the output of the other codeword matched ﬁlter receiver is (n − 2d) Es . The noise voltage at the output of the matched ﬁlter receiver for the transmitted codeword is denoted as nc − n1, and the noise voltage at the output of the other matched ﬁlter receiver will be nc + n1.
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• Lecture 16: Plotkin Bound 1 Plotkin Bound
Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall2007) Lecture 16: Plotkin Bound October 2, 2007 Lecturer:AtriRudra Scribe:NathanRussell&AtriRudra 1 In the last lecture we proved the GV bound, which states that for all δ with 0 ≤ δ ≤ 1− q , there exists a q-ary code of distance δ and rate at least 1 − Hq(δ) − ε, for every ε> 0. In fact we proved that with high probability, a random linear code C lies on the GV bound. We picked a generator matrix G at random and proved the latter result. At this point, we might ask what happens if G does not have full rank? There are two ways to deal with this. First, we can show that with high probability G does have full rank, so that |C| = qk. However, the proof from last lecture already showed that, with high probability, the distance is greater than zero, which implies that distinct messages will be mapped to distinct codewords and thus |C| = qk. 1 Further, the proof required that δ ≤ 1 − q because it is needed for the volume bound – Hq(δ)n V olq(0, δn) ≤ q – to hold. It is natural to wonder if the above is just an artifact of the 1 proof or, for example, is it possible to get R > 0 and δ > 1 − q ? In today’s lecture, we will show that this cannot be the case by proving the Plotkin bound. 1 Plotkin Bound We start by stating the Plotkin bound. Theorem 1.1 (Plotkin bound). The following holds for any C ⊆ [q]n with distance d: 1 1.
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• MT361/MT461/MT5461 Error Correcting Codes
MT361/MT461/MT5461 Error Correcting Codes Mark Wildon, [email protected] I Please take a handout, the preliminary problem sheet, and the sheet of challenge problems. I All handouts and problem sheets will be put on the Moodle page for MT361. Everyone has access to this page. If you are not yet registered for the course, bookmark this link: moodle.rhul.ac.uk/course/view.php?id=381 I Lectures are at: Monday 3pm in MFLEC, Tuesday 3pm in BLT2, Thursday 10am in BLT2. I There is an additional lecture at Thursday noon for MT4610 and MT5461 in ABLT3. Correction: in week TWO only, this lecture will be moved to Friday 2pm in C219. I Oﬃce hours: Tuesday 11am, Thursday 2pm and Friday 11am. (This week as normal.) But in the modern world we want to store and process huge amounts of data. Think of a ‘bit’ as a single piece of information: on or oﬀ, 0 or 1. I A small QR-code: 441 bits I Text on ﬁrst page of handout: 7144 bits I First 14 pages of handout: I Compact disc of Beethoven 9th: 700 MB I Bluray disc of 3 hour ﬁlm: 50 GB I Large Hadron Collider data, per second: 300 GB Why Coding Theory? We want to communicate reliably in the presence of noise. With small amounts of data, we can do this quite easily. I Text on ﬁrst page of handout: 7144 bits I First 14 pages of handout: I Compact disc of Beethoven 9th: 700 MB I Bluray disc of 3 hour ﬁlm: 50 GB I Large Hadron Collider data, per second: 300 GB Why Coding Theory? We want to communicate reliably in the presence of noise.
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• Introduction to Coding Theory Lecture Notes∗
Introduction to Coding Theory Lecture Notes∗ Yehuda Lindell Department of Computer Science Bar-Ilan University, Israel January 25, 2010 Abstract These are lecture notes for an advanced undergraduate (and beginning graduate) course in Coding Theory in the Computer Science Department at Bar-Ilan University. These notes contain the technical material covered but do not include much of the motivation and discussion that is given in the lectures. It is therefore not intended for self study, and is not a replacement for what we cover in class. This is a ﬁrst draft of the notes and they may therefore contain errors. ∗These lecture notes are based on notes taken by Alon Levy in 2008. We thank Alon for his work. i Contents 1 Introduction 1 1.1 Basic Deﬁnitions ........................................... 1 1.2 A Probabilistic Model ........................................ 3 2 Linear Codes 5 2.1 Basic Deﬁnitions ........................................... 5 2.2 Code Weight and Code Distance .................................. 6 2.3 Generator and Parity-Check Matrices ............................... 7 2.4 Equivalence of Codes ......................................... 9 2.5 Encoding Messages in Linear Codes ................................ 9 2.6 Decoding Linear Codes ........................................ 9 2.7 Summary ............................................... 12 3 Bounds 13 3.1 The Main Question of Coding Theory ............................... 13 3.2 The Sphere-Covering Lower Bound ................................. 14 3.3 The Hamming (Sphere Packing) Upper Bound .......................... 15 3.4 Perfect Codes ............................................. 16 3.4.1 The Binary Hamming Code ................................. 16 3.4.2 Decoding the Binary Hamming Code ............................ 17 3.4.3 Extended Codes ....................................... 18 3.4.4 Golay Codes ......................................... 18 3.4.5 Summary - Perfect Codes .................................. 20 3.5 The Singleton Bound and MDS Codes ..............................
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