Constrained arrays and erasure decoding

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1 Constrained arrays and erasure decoding Utomo, P.H. Published: 06/11/2018 Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publication after peer review. The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication Citation for published version (APA): Utomo, P. H. (2018). Constrained arrays and erasure decoding Eindhoven: Technische Universiteit Eindhoven General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal. Take down policy If you believe that this document breaches copyright please contact us (openaccess@tue.nl) providing details. We will immediately remove access to the work pending the investigation of your claim. Download date: 25. Jan. 2019

2 Constrained arrays and erasure decoding Putranto Hadi Utomo

3 Copyright Putranto Hadi Utomo, Printed by Gildeprint A catalogue record is available from the Eindhoven University of Technology Library ISBN:

4 Constrained arrays and erasure decoding PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir F.P.T. Baaijens, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op dinsdag 6 november 2018 om uur door Putranto Hadi Utomo geboren te Bogor, Indonesië

5 Dit proefschift is goedekeurd door de promotoren en de samenstelling van de promotiecommissie is als volgt: voorzitter: 1e promotor: copromotor : leden: prof.dr. J.J. Lukkien prof.dr. T. Lange dr. G.R. Pellikaan dr.ir. J.H. Weber (Technische Universiteit Delft) prof.dr.ir F.M.J. Willems dr. K.A. Sugeng (University of Indonesia) dr. A. Blokhuis prof.dr. H. Zantema Het onderzoek of ontwerpdat in dit proefschrift wordt beschreven uitgevoerd in overeenstemming met de TU/e Geragscode Wetenschapboefening.

6 Acknowledgment This thesis concluded the four years of research at the Coding Theory and Cryptology group at the Department of Mathematics and Computer Science, Eindhoven University of Technology. This work would not be possible without the help and support from many people around me. First of all, I would express my sincere gratitude for my daily supervisor Ruud Pellikaan. I was very happy when he accepted me as his PhD student four years ago. It was a life changing moment for me to pursue a PhD program/study abroad. He was always there to help me out with any problem I had. It is also by your guide, encouragement, and motivation I could survive these four years of my study. Secondly, I would like to thank my promotor Tanja Lange. Together with Ruud, you always stimulated me to travel abroad, to attend conferences and workshops for the sake of broadening my knowledge and also to present my latest findings. I also very much appreciate your effort to do the proofreading this entire thesis to find mistakes and give suggestion so it became much better to read. I would also like to thank the University of Sebelas Maret for giving me the opportunity to study abroad and also the Directorate of Higher Education, Indonesia for giving me the grant in doing the research. Next, I would like to thanks the other committee members Aart Blokhuis, Kiki Sugeng, Jos Weber, Frans Willems, and Hans Zantema for willingly take the time reading, finding mistakes, and giving comments to furthermore improve this thesis. I would also like to thank Johan Lukkien for chairing my PhD defense ceremony. It would be impossible for me to finish this PhD if I worked all alone. Ruud and Rusidy, thank you for working together so that we manage to have nice papers. I would also like to thank Hans for our discussion about binary puzzles, BDD solvers and the reference to The On-Line Encyclopedia of Integer Sequences. I feel very lucky that I become a member of the Coding Theory and Cryptology group at the Eindhoven University of Technology. With the friendly atmosphere, it feels like home for me. Thank you to Anita who helps me a lot with the bureaucracy and paperwork during my stay at the university. Thanks a lot also to Christine who get a lot of question from me related to Dutch-English translation. I would like to thanks the rest of the group: Dan, Meilof, Thijs, Niels, Jan-Willem, Guus, Benne, Berry, Gustavo, Andy, Chicanouk, Tony, Manos, Jens, and Chloe. I would also thank my parents, Hadi and Dwiana. It is really a bless to be born and raised by the two of you. Thanks to my brother Dwiputra and sister Afifah for the fun we had together. Next, I would like to thanks my mother in law Jamilah and sister in law Anbar. Furthermore, I would like to thank again Ruud and Marion for the great hospitality and caring. Finally, I would like to give my warmest love to my wife Astri and my daughter Alisha. Your patience v

7 vi Acknowledgment and support were really helpful for me to be able to finish this study. Eindhoven, 26 September 2018 Putranto Hadi Utomo

8 Contents Acknowledgment Contents v vii 1 Introduction Background Overview of the thesis Contributions Preliminaries Error-correcting codes Linear codes Isometries and equivalent codes The rate of a code Direct sum and product code Modulation codes The q-ary symmetric channel Decoders Probability of correct decoding Probability of undetected error Probability of correct erasure decoding Enumerators The distance enumerator The nearest distance enumerator Coset leader weight enumerator of the product code [m, m 1, 2] q [n, n 1, 2] q The erasure distance enumerator Average enumerators Probabilities of subcodes Product formulas for the direct sum Constrained systems 61 vii

9 viii Contents 4.1 Constrained sequences Constrained arrays, a study case: the binary puzzle First constraint Second constraint Third constraint All constraints Erasure decoding The hardness of binary puzzles Backtrack-based search Gröbner basis approach Satisfiability solver The 1 st constraint The 2 nd constraint The 3 rd constraint Satisfiability modulo theory solver SMT solver Related theories Combining theory Solving binary puzzles with SMT Binary decision diagram solver Experiments Conclusion 99 Summary 101 Curriculum vitae 103 Bibliography 105

10 Chapter 1 Introduction 1.1 Background History proved that the theory of information and communication is very important for the human race. Since the beginning of time, people need to communicate with others as well as to store and to transfer information by means of writing on the wall of a cave, stones, or simply talking or shouting with gestures [25]. One concern in transferring information is the integrity of the message. It is important that the person we are communicating with is receiving the same message as we intended to give. This concern became a serious problem especially when we came to the digital era, where all information is stored or sent in terms of binary data. Imagine that an error occurred so that some bits are flipped. These errors could make the information received by the receiver differ from that sent. This is exactly what Hamming experienced when doing experiments with a punch card computer in around Often the computer got halted when it detected errors in the input [35]. This problem is the heart of a subject we call coding theory [68]. The theory of error correcting codes has flourished over these few decades. People now could see their favorite DVD movie even there is a scratch in the disk. In this vast growing technology in the digital era, it is now possible to store data in form of 2-dimensional format, such as a bar-code on plain paper and holographic recordings in a crystal [4]. From a theoretical point of view, this new format could allow us to store more information per unit area. Now we arrive at the question: How do we define a code in 2-D such that it has good performance, such as the probability of error correction, the rate, the minimum distance, the encoding and decoding algorithm? One way to design a good 2-D code for communication or a storage system is by restricting the way symbols are written in the rows and on the columns, often knows as a constrained 2-D array. As a motivating problem, we consider the binary puzzle which is one example of the constrained 2-D system. This puzzle is registered by Frank Coussement and Peter the Schepper in 2009 as Binairo [90]. It is also called Binero, Bineiro, Takuzu, Brain Snack, or Zernero. A similar game, using crosses and circle, was proposed in 2012 by Aldolfo Zanellat under the name of Tic-Tac-Logic. The binary puzzle is a two-dimensional array that satisfies certain properties. The solved binary puzzle is an n n binary array that satisfies: 1

11 2 Introduction Figure 1.1: Unsolved binary puzzle Figure 1.2: Solved binary puzzle Figure 1.3: Unsolved Sudoku Figure 1.4: Solved Sudoku 1. no three consecutive ones and also no three consecutive zeros in each row and each column, 2. every row and column is balanced, that is the number of ones and zeros must be equal in each row and in each column, 3. every two rows and every two columns must be distinct. Figure 1.1 is an example of a binary puzzle. There is only one solution satisfying all conditions. But there are 3 solutions satisfying (1) and (2). The solution satisfying all conditions is given in Figure 1.2. We can see a collection of n by n binary puzzles as the codewords and the problem of solving the puzzle as an erasure decoding problem. In this thesis, we address some of the problems related to the binary puzzle code, that are the rate, the encoding and decoding scheme, and the probability of error correction. Another puzzle we like to mention is Sudoku, which is a logic-based, combinatorial number-placement puzzle [91]. The objective is to fill a 9 9 grid with digits so that each column, each row, and each of the nine 3 3 sub-grids that compose the grid (also called "boxes", "blocks", "regions", or "sub-squares") contains all the digits from 1 to 9. We call the three rows of boxes the bands and three columns of boxes the stacks. The puzzle setter provides a partially completed grid, which for a well-posed puzzle has a unique solution. A typical sudoku puzzle can be seen in Figure 1.3, and the solution in Figure 1.4.

12 1.2. Overview of the thesis Overview of the thesis The next chapters of this thesis are outlined as follows. Chapter 2 gives a review of the well-known basic theory of error-correcting codes that is used in the thesis. Chapter 3 introduces the notion of enumerators of a possibly non-linear code, such as the distance enumerator, the erasure enumerator and the nearest distance enumerator. This chapter addresses the notion of the (extended) erasure distance enumerator and its relation with the (extended) weight enumerator. Moreover, some result about the weight enumerator of the product code [m, m 1, 2] q [n, n 1, 2] q also presented. Chapter 4 deals with the notion of a constrained system. Some approach in finding the rate and the capacity of the binary puzzle code are discussed in this chapter. Chapter 5 elaborates on several techniques for solving a special case of an erasure decoding problem, that is solving the binary puzzle. In this chapter, we have a experimental result of decoding schemes for the case of binary puzzle code. Chapter 6 concludes with some remark on the further research related to this thesis. 1.3 Contributions This thesis in mainly based on the following publications: Binary puzzles as an erasure decoding problem [82]. This paper outlines the problems of binary puzzles from an erasure decoding point of view, such as the rate of these codes, the erasure decoding probability, the decoding algorithm and their complexity. The coset leader weight enumerator of the product code [m, m 1, 2] q [n, n 1, 2] q [85]. This paper discusses about some structure of the product code [m, m 1, 2] q [n, n 1, 2] q so we could derive a closed formula of the number of coset leaders of a particular weight. We will elaborate more on the details in Chapter 3. Satisfiability modulo theory in finding the distance distribution of binary constrained arrays [79]. In this paper, we utilize the power of the SMT solver to find the distance distribution of a special case 2-D binary constrained systems, that is the binary puzzle. This paper will be presented in Chapter 3 in this thesis. On the rate of constrained arrays [83]. This paper is focused on the problem of finding the rate of a special case of a constrained array, that is the binary puzzle. The result of this paper will be discussed in more detail in Chapter 4. Solving the Binary puzzle [81]. In this paper, we deal with the decoding problem of binary puzzles and discuss several techniques for solving puzzles. The result is given in more detail in Chapter 5.

13 4 Introduction Satisfiability modulo theory and Binary puzzle [78]. This paper gives an experimental result on solving the binary puzzle with SMT solvers, which we elaborate on more in Chapter 5. Binary puzzle as a SAT problem [84]. This paper is mainly about representing a binary puzzle in terms of a satisfiability problem in order to solve it efficiently. More details are given in Chapter 5. The solver of binary puzzle [80]. This publication contains source code explained in the paper of [81]. The works in [82, 83, 84, 85] are co-authored with Ruud Pellikaan, and the work in [81, 80] is co-authored with Rusydi Makarim.

14 Chapter 2 Preliminaries In this chapter, we will introduce the basic concepts and terminology of error-correcting codes that is well known and fundamental for this thesis and that one can find for instance in the textbook [68]. We will begin with introducing the notion of error-correcting codes and continue with some concepts in decoding probabilities. 2.1 Error-correcting codes The theory of error-correcting codes began with the publication of the seminal paper of Shannon [72]. It deals with the problem of reliable communication. Suppose Bob wants to send a message to Alice over a channel so that the message can be changed by noise. The goal is to correct the error in the message efficiently. To handle this, Bob and Alice agree upon a scheme of encoding the message before transmission. Bob uses an encoder, that is a machine which transforms the message into a codeword, then this codeword is sent over a noisy channel. Alice uses another machine, called a decoder which changes the received word into a codeword and into the associated message, even if the codeword is corrupted by the transmission. The scheme is illustrated in Figure 2.1. Definition Let Q be a set of q symbols called the alphabet. Let Q n be the set of all n-tuples x = (x 1,..., x n ), with entries x i Q. A block code C of length n over Q is a non-empty subset of Q n. The elements of C are called codewords. If C contains M codewords, then M is called the size of the code. We call a code with length n and size M an (n, M) code. If M = q k, then C is called an [n, k] code. For an (n, M) code defined over Q, the value n log q (M) is called the redundancy. message noise codeword recieved word encoder channel decoder message Figure 2.1: A communication model 5

15 6 Preliminaries x y d(y,z) d(x,y) z d(x,z) Figure 2.2: Triangle inequality Definition Let C be a block code in Q n and let M be the message space. An encoder of C is a map: E : M Q n such that E(M) C. Usually the encoder is injective, that is one-to-one, and surjective, that is onto C. If c = E(m), then m is called a message or source word of c. In order to measure the difference between two distinct words and to evaluate the error-correcting capability of the code, we need to introduce an appropriate metric on Q n. A natural metric used in Coding Theory is the Hamming distance. Definition For x = (x 1,..., x n ), y = (y 1,..., y n ) Q n, the Hamming distance d(x, y) is defined as the number of positions where they differ: d(x, y) = {i x i y i }. Moreover, the Hamming weight of x, wt(x) is defined as the number of positions such that it has a non-zero entry: wt(x) = {i x i 0}. Proposition The Hamming distance is a metric on Q n, that means that the following properties hold for all x, y, z Q n : 1. d(x, y) 0 and equality holds if and only if x = y, 2. d(x, y) = d(y, x) (symmetry), 3. d(x, z) d(x, y) + d(y, z) (triangle inequality, as depicted in Figure 2.2). Proof. See [68] Proposition Definition The minimum distance of a code C of length n is defined as d = d(c) = min{ d(x, y) x, y C, x y } if C consists of more than one element, and is by definition n + 1 if C consists of one word. We denote by (n, M, d) a code C with length n, size M and minimum distance d.

16 2.2. Linear codes 7 The main problem of error-correcting codes from Hamming s point view" is to construct for a given length and number of codewords a code with the largest possible minimum distance, and to find efficient encoding and decoding algorithms for such a code. Definition Let x Q n. The ball of radius r around x, denoted by B r (x), is defined by B r (x) = {y Q n d(x, y) r}. The sphere of radius r around x is denoted by S r (x) and defined by S r (x) = {y Q n d(x, y) = r}. Proposition Let Q be an alphabet of q elements and x Q n. Then ( ) n r ( ) n S i (x) = (q 1) i and B r (x) = (q 1) i. i i Proof. See [68] Proposition i=0 Definition Let C be a code in Q n. Let I {1,..., n} where I = p. Let Ī = {i 1,..., i n p } be the complement of I where i 1 < < i n p. Then, C I the punctured code of C at positions I is defined as follows C I = {(c i1,..., c in p ) c C}. Remark that C I can also be seen as the image of π I, where π I : C Q n p given by c (c i1,..., c in p ). 2.2 Linear codes Definition Let Q = F q, the finite field of q elements, then F n q is a vector space of dimension n over F q. An F q -linear subspace of F n q is called an F q -linear code. Remark Let C be an F q -linear code of length n. Then C has size q k, where k is the dimension of C. We denote C as an [n, k] q -code. If moreover C has minimum distance d, we say that C is a linear code over F q with length n, dimension k, and minimum distance d, often denoted by [n, k, d] q -code. Observe also that if C is a linear code with minimum distance d, then d is equal to the minimum weight of nonzero codeword in C, since d = d(x, y) = d(x x, y x) = d(0, z) = wt(z). As the name suggests, any linear code C could be expressed in term of the subspace or, equivalently by a set of equations for which C is the nullspace, which we define as follows: Definition Let C be an [n, k, d] q code, then a generator matrix of C is given by a k n matrix: g 1 g 1,1 g 1,2 g 1,n g G = 2. = g 2,1 g 2,2 g 2,n g k g k,1 g k,2 g k,n where {g 1, g 2,..., g k } is a basis for C. Moreover, an (n k) n matrix H is called a parity check matrix for C if C is the null-space of H, that is the set of all x in F n q such that Hx T = 0.

17 8 Preliminaries Example As an example, let us consider the class of binary Hamming codes, which have length n = 2 r 1, dimension k = 2 r r 1, and minimum distance 3 where r is an integer greater or equal to 2. These codes can detect up to 2-bits errors and can always correct 1-bit of error. In case r = 3, we have the following generator and parity check matrix for the Hamming code with parameters [7, 4, 3]: G = and H = Remark Let C be an [n, k, d]-code. Observe that d n k + 1 since a codeword with only one non-zero information symbol has weight at most n k + 1. This bound is also called the Singleton bound. Codes with d = n k + 1 are called maximum distance separable or MDS in short. Definition Let C be a linear code in F n q with dimension k. We define the dual code of C as follows: C = {x F n q c x = 0 for all c C}. Proposition Let C be a code in F n q with dimension k with generator matrix G. Then C has dimension n k and G is the parity check matrix for C. Proof. See Proposition of [68]. Definition Let C be a linear code in F n q and I {1,..., n}. Define: C(I) = {c C c i = 0, i Ī} as the subcode of C such that c i = 0 for all i I and l(i) as the dimension of C(I). Proposition Let C be a linear code in F n q with dimension k and I {1,..., n}. Then, Proof. Consider the map π I : C F n p q C I = q k l(ī). as described in Definition Then, the kernel of π I = {c C π I (c) = 0} = {c C c i = 0, i Ī} = C(Ī). Thus C/C(Ī) = C I and so C I = C / C(Ī). Hence, C I = q k l(ī). Definition Let C be a linear code in F n q and x be any word in F n q. The weight of the coset x + C is the minimal weight of an element in that coset, that is min{wt(x + c) c C}. The coset leader of x + C is defined as a choice of one element in the coset x + C with minimal weight.

18 2.2. Linear codes 9 Definition Let C be a linear code in F n q and H be a parity check matrix for code C, that is a matrix where C is the null space of H. Suppose that x be a word in F n q. Then, Hx T is defined as the syndrome for x. Remark Let C be a linear code in F n q and x be any word in F n q. The syndrome of x is in one-to-one correspondence to x + C, since two words are in the same coset if and only if they have the same syndrome. Then wt(x + C) is the minimum number of columns of H such that a linear combination of these columns give xh T, that is the syndrome of x written as a column. Definition Let C be a linear code in F n q. Let α i (C) be the number of coset leaders of C having weight i. Then α C (Z), the coset leader weight enumerator of C, and α C (X, Y ), its homogeneous version are defined by: n n α C (Z) = α i (C)Z i and α C (X, Y ) = α i (C)X n i Y i. i=0 i=0 Definition Let C be a linear code of length n. The covering radius ρ(c) of the C over F q is defined as max{d(x, C) x F n q }. In other words, we have that ρ(c) = max{i α i (C) 0}, as one sees in [68]. Remark that the covering radius may differ after extending the finite field. In case we want to stress that we consider the covering radius of the code C over F q, we denote it by ρ(c, F q ). Definition Let C be a linear code in F n q. Let W i (C) = {c C wt(c) = i}. Then W C (Z), the weight enumerator of C, and W C (X, Y ), its homogeneous version, are defined by: n n W C (Z) = W i (C)Z i and W C (X, Y ) = W i (C)X n i Y i. i=0 i=0 Definition Let C be a linear code over F q with length n. Let F q m be the extension of F q of degree m. The extension by scalars of C to F q m, denoted by C F q m, is defined as the linear subspace in F n q m generated by C. Note that if G is a generator matrix of C, then G is also a generator matrix of C F q m. Definition Let C be a linear code in F n q and I {1,..., n}. Define B I (T ) = T l(i) 1 B i (T ) = B I (T ). I =i The extended weight enumerator of C is defined by: n W C (X, Y, T ) = X n + B i (T )(X Y ) i Y n i. i=0

19 10 Preliminaries Proposition The extended weight enumerator evaluated at T = q m is equal to the weight enumerator of the code C F q m: W C (X, Y, q m ) = W C Fq m (X, Y ). Proof. See Proposition 2.17 of [45]. Theorem (MacWilliams) Let C be a linear code with dimension k and C be its dual. Then the extended weight enumerator of C is determined by the extended weight enumerator of C: W C (X, Y, T ) = T k W C (X + (T 1)Y, X Y, T ). Proof. See Section 8.2 of [45] or Theorem of [68]. 2.3 Isometries and equivalent codes Definition A map ϕ : Q n Q n is called an isometry if it leaves the Hamming metric invariant, that means that d(ϕ(x), ϕ(y)) = d(x, y) for all x, y Q n. Definition Let S n be the symmetric group on n letters that is the group of permutations of {1,..., n}. Let π S n. Define the corresponding permutation map Π : Q n Q n by Π(x 1,..., x n ) = (x π(1),..., x π(n) ). Remark Let π 1,..., π n be permutations of the elements of Q. This gives a map (π 1,..., π n ): Q n Q n defined by (π 1,..., π n )(x 1,..., x n ) = (π 1 (x 1 ),..., π n (x n )). Remark The permutations of S n and the coordinatewise permutations of the elements of Q define isometries of Q n. Conversely, every isometry is the composition of the beforementioned isometries, as shown in Corollary of [3]. Definition Let C and D be codes in Q n. Then C is called equivalent to D if there exists an isometry ϕ of Q n such that ϕ(c) = D. If moreover C = D, then ϕ is called an automorphism of C. The automorphism group with composition as operation of C is the set of all isometries ϕ such that ϕ(c) = C and is denoted by Aut(C). Definition Let C be a code in Q n. Then C is called homogeneous if for all x, y C there exists an automorphism ϕ of C such that ϕ(x) = y. Example Let Q = F q, the finite field with q elements, and let C be an F q -linear code in F n q. Let c C. Define the map ϕ c : F n q F n q by ϕ c (x) = x + c. Then d(ϕ c (x), ϕ c (y)) = d(c + x, c + y) = d(x, y) for all x, y F n q. Hence, ϕ c is an isometry. Furthermore, ϕ c (C) = C, since C is linear and c C. Therefore, C is homogeneous.

20 2.4. The rate of a code 11 Example Consider the Sudoku puzzle as given in Section 1.1. The following operations always transforms a valid puzzle into another valid puzzle: 1. Relabeling symbols, which has 9! possibilities. 2. Band permutations, which has 3! possibilities. 3. Row permutations, which has within a band 3! possibilities. 4. Stack permutations, which has 3! possibilities. 5. Column permutations, which has within a stack 3! possibilities rotation, which has 4 possibilities. 7. Diagonal and antidiagonal reflection, which has 2 possibilities. Hence all above operations are isometries of the Sudoku [92]. Example Consider a solved binary puzzle as given in Section 1.1, that is an n n array satisfying (1) no three consecutive ones and no three consecutive zeros in each row and each column; (2) the number of ones and zeros must be equal in each row and in each column; (3) there are no repeated rows and no repeated columns. The following operations will keep the solution valid: 1. Relabeling symbols, which has 2 possibilities rotation, which has 4 possibilities. 3. Diagonal and antidiagonal reflection, which has 2 possibilities. And hence the above operations give isometries of the binary puzzle. Remark It is an open question whether the operations mentioned in Examples and give all possible isometries to the Sudoku and binary puzzle of size n n, respectively [21]. 2.4 The rate of a code Let Q be a finite alphabet consisting of q elements. Suppose we have a code C Q n. Define the (information) rate of C by R(C) = log q C log q Q n = log q C. n Definition Let C be a code in Q n. The code is called systematic at the k positions (j 1,..., j k ) if for all m Q k there exists a unique codeword c such that c ji = m i for all i = 1,..., k. In that case, the set {j 1,..., j k } is called an information set. Clearly C is systematic at the positions (j 1,..., j k ) if and only if there is an encoding one-to-one map E from Q k onto C such that E(m) ji = m i for all m in Q k and i = 1,..., k. Define a map E : Q k Q n for code C. If for all m Q k there exists at least one codeword c such that c ji = m i for all i = 1,..., k, then E(Q k ) C. So k/n R(C). Therefore, if C is systematic at k positions then the encoding map has C as image and k/n = R(C).

21 12 Preliminaries Example Let C be the binary block code of length n consisting of all words with exactly two ones. This is an (n, n(n 1)/2) code. In this example the number of codewords is not a power of the size of the alphabet. So the code is not systematic. Example Let C n be the binary block code of length n of sequences such that there are no three consecutive ones and also no three consecutive zeros. Let n = 8. Then there is at least one codeword in C 8 if we have the information set at positions (1, 2, 5, 6). More generally, if we have the information set at positions (j 1,..., j k ), where k = 2 n/4 and j 2t 1 = 4t 3 and j 2t = 4t 2 for t = 1,..., k/2, there is at least one codeword in C n containing the information set at the k positions. So, R(C n ) k n = 2 n/4 n 2n 4n = 1 2. Example Let n = 2m, where m is a positive integer. Let B n be the binary code of balanced sequences, that means the number of ones is equal to the number of zeros. Clearly that B n is not systematic. Since we can fill in arbitrary elements at any m-tuple of positions, R(B n ) 1 2. Moreover, the number of codewords of B n is ( ) 2m m. Stirling s formula states that n! ( n ) n 2πn for n, e so B n is approximately equal to (2 n ) 2/πn) as n. Hence, as n goes to, log 2 (2 n ) 2/πn R(B n ). n Therefore, the information rate of B n goes to 1 for n. Example Let q be a positive integer, not necessarily a power of a prime. Let P q be the code on the alphabet Q = {1, 2,..., q} of length q consisting of all codewords of the form (σ(1),..., σ(q)) with σ a permutation of Q. Then the minimum distance of P q is 2, because it is not possible to have two codewords x = (x 1,..., x q ) and y with distance 1, since x and y have q distinct entries and y can be written as (σ(x 1 ),..., σ(x q )). Therefore, x and y must be different in at least 2 positions. Since σ is a permutation of Q with q elements, the number of codewords is equal to the number of permutations of q elements, that is q!. The information rate of P q is equal to 1 q log q q!. Using Stirling s formula, R(P q ) 1 [(q 12 q log q ) log(q) q + 12 ] log(2π) + o(1) for q goes to infinity where log denotes the natural logarithm. We see that as q goes to infinity, R(P q ) will go to 1. In other words, 1 lim q q log q( P q ) = 1 for q. 2.5 Direct sum and product code Definition Let C be a code in Q n and let D be a code in Q m. The direct sum of C and D is defined by C D = {(c, d) c C, d D}. The m-fold direct sum of C is denoted by C C = C m.

22 2.5. Direct sum and product code 13 Proposition The direct sum of Q n Q m can be identified with Q n+m and In particular R(C m ) = R(C) for all m. R(C D) = nr(c) + mr(d). n + m Proof. By definition, R(C D) = log q C D log q Q n Q m = log q C D log q Q n+m = log q C D n + m. Since C D = {(c, d) c C, d D}, the size of C D is equal to the size of C times the size of D, that is C D = C D. Therefore, log q C D = log q ( C D ) = log q C + log q D. Since R(C) = log q C n and R(D) = log q D m, we have Moreover, R(C D) = R(C m ) = log q( C m ) mn nr(c) + mr(d). n + m = log q C n = R(C). Definition Let C be a code in Q n and let D be a code in Q m. Let Q n m be the collection of all n m arrays with entries in Q. The product code of C and D in Q n m, denoted by C D consists of all arrays such that every column is a codeword in C and every row is a codeword of D. Remark that Q n m can be identified with Q nm. Let C be a code in Q n that is systematic at k positions with encoding map E : Q k Q n such that E(Q k ) = C. Let D be a code in Q n that is systematic at l positions with encoding map F : Q l Q m such that F(Q l ) = D. After permutation of the coordinates we may assume without restricting the generality that C and D are systematic at the first k and l positions, respectively. Let C be systematically encoded by E in the first l columns and let D be systematically encoded by F in the first k rows. Suppose that the encoding maps E and by F commute, that means that the encoding of the last m l columns with E gives the same result as the encoding of the last n k rows with F. Then C D is systematic at kl positions and R(C D) = R(C) R(D). Let Q = F q and C and D are F q -linear codes in F n q and F m q, respectively. Then C and D have encoders that commute, as one sees in the Subsection of [68]. Example Let l and m be a positive integers. Let B 2l 2m be the binary code of balanced 2l 2m arrays, that means every row and every column is balanced. Hence, B 2l 2m = B 2l B 2m. Example Let q be a positive integer. A Latin square of order q, L q, is a q q array with entries from a set Q of q elements, such that every column and every row is a permutation of the symbols in Q. Therefore, L q = P q P q. The exact number L q of Latin squares of order q is not known in general and difficult to determine, but a lower bound is known [87, Theorem 17.2]: L q (q!)2q q q2.

23 14 Preliminaries Consider the code L q on the alphabet Q = {1,... q} of length n = q 2 consisting of all Latin squares of order q. The information rate of L q goes to 1 for q, since R(L q ) = log q( L q ) q 2 log q [ (q!)2q /q q q 2 2 ] = 2 q log q log(q!) 1 and using Stirling s formula, the right side of the inequality will go to 1 as q. Therefore, R(L q ) 1 as q goes to infinity. Example Let s be a positive integer. Let Q be a set of q = s 2 elements. A Sudoku of order s is an s s array of s s arrays with all the q elements of Q as entries in those s 2 arrays and such that every column and every row is a permutation of the symbols of Q. Hence, a Sudoku of order s is a Latin square of order s 2. Let S s be the collection of all Sudoku s of order s. The number S(s) of Sudoku s of order s 2 is not known in general and difficult to determine, but a lower bound is conjectured in [38, p. 716]: log(s(s)) lim s 2s 4 log(s) = 1. If the conjecture holds, then the information rate of S s goes to 1 for s, since the size of a Sudoku s array of order s is s 2 s 2, hence the array space is Q s4. By the definition of rate, we have R(S s ) = log s 2(S(s)) s 4 = log(s(s)) log(s 2 )s 4 = log(s(s)) 2 log(s)s Modulation codes A modulation code, as stated in [40], is a code that is employed to transform or encode arbitrary (binary) sequences into sequences that possess certain "desirable" properties. For example, suppose we want to construct a codeword such that it has "no three consecutive ones and no three consecutive zeros". In other words, the codeword can not contain "000" and "111". Hence, we can create a graph presentation for the code. This shown in Figure 2.3. Constrained systems that can be represented by some finite directed graph is called sofic systems [40] Figure 2.3: Presentation of no three consecutive ones and zeros Now, we will generate a constrained array with respect to (1) no three consecutive ones and no three consecutive zeros, and (2) the number of ones should be equal to the number of zeros.

24 2.7. The q-ary symmetric channel 15 First, we begin to create a sequence using the graph in Figure 2.3 such that the walks begins with the square vertex and end also at the square vertex. To impose the balancedness properties, the length of the walk must be even. Furthermore, the number of passes in edge 11 should be the same as in edge 00. Now, suppose we have a balanced constrained vector a = (a 1, a 2,..., a n ). Put a as the first row and shift cyclically one position to the right as we copy down for the 2 nd row. Similarly, we copy the 2 nd row to the third row and shift cyclically one position to the right, until we have an n n array satisfying our required condition. In matrix notation, suppose A is the desired array. Thus, we let (a 1,1, a 1,2,..., a 1,n ) = a, and a 1,1 a 1,2 a 1,3 a 1,n 1 a 1,n a A = 1,n a 1,1 a 1,2 a 1,n 2 a 1,n a 1,2 a 1,3 a 1,3 a 1,n a 1,1 Since we forbid 11 and 00 at the beginning and at the end of the initial vector, the process of shifting will not lead to 111 or 000 and hence every row is satisfying the conditions. Notice also that the last column is equal to the first row in reversed direction, and every two consecutive columns are a cyclic shift of another. Therefore every column satisfies both conditions (1) and (2). 2.7 The q-ary symmetric channel Let C be a code in Q n. Consider a channel where a codeword c of C is sent and a word r Q n is received. The crossover probability p is the probability that a symbol i is sent and another symbol is received. In a q-ary symmetric channel (qsc) this crossover probability is the same for all symbols and the probability that i is sent and j is received is the same for all i Q and j Q with j i. So that it equal to p/(q 1). Moreover, in a qsc it is assumed that the crossover probability is memoryless, that means that it does not depend on the position of the symbol in the codeword. Finally, it is assumed that the probability P (c) that the codeword c is sent is same for all codewords. Hence, we have that P (c) = 1 C for all c C. The crossover probability is illustrated in Figure 2.4a for the q-ary symmetric channel and in Figure 2.4b for the binary symmetric channel (BSC). Proposition The probability that r is received given that c is sent over a q-ary symmetric channel with crossover probability p is given by is given by ( ) p d(r,c) P (r c) = (1 p) n d(r,c). q 1 Proof. Suppose c is sent and r is the received word. Since the channel is memoryless, P (r c) = n i=1 P (r i c i ). In a symmetric channel, there are only two possibilities of P (r i c i ) for all i, which is equal to 1 p if c i = r i, and p q 1 if c i r i. Since the number of different positions is equal to d(r, c), we have ( ) p d(r,c) P (r c) = (1 p) n d(r,c). q 1

25 16 Preliminaries i 1 p i 1 p 0 0 p p q 1 j i 1 p 1 p 1 (a) q-ary symmetric channel (b) BSC Figure 2.4: Crossover with probability p 2.8 Decoders Definition Let C be a code in Q n of minimum distance d. If c is a transmitted codeword and r is the received word, then {i r i c i } is the set of error positions and the number of error positions is called the number of errors of the received word. Remark If r is the received word and t = d(c, r) is the distance of r to the code C, then there exists a nearest codeword c such that t = d(c, r). If the number of errors t is at most (d 1)/2, then we are sure that c = c. In other words, the nearest codeword to r is unique when r has distance at most (d 1)/2 to C. An illustration of codewords having sphere of radius (d 1)/2 is given in Figure 2.5. c 5 c 4 c 8 c 3 c 1 c 2 c 10 c 7 c 6 c 9 Figure 2.5: Illustration of codewords in space Definition The number e(c) = (d(c) 1)/2 is called the error-correcting capability decoding radius of the code C. Definition A decoder D for the code C is a map D : Q n Q n {?} such that D(c) = c for all c C and will return? if D fail to find a codeword. Remark It is allowed that the decoder gives as outcome the symbol? in case it fails to find a codeword. This is called a decoding failure. If c is the codeword sent, r is the received word and D(r) = c

26 2.8. Decoders 17 is a codeword not equal to c, then this is called a decoding error. If D(r) = c, then r is decoded correctly. Notice that a decoding failure is noted on the receiving end, whereas there is no way that the decoder can detect a decoding error without a retransmission. Definition A complete decoder is a decoder that always gives a codeword in C as outcome. A nearest neighbor decoder, also called a minimum distance decoder, is a complete decoder with the property that D(r) is a nearest codeword. A decoder D for a code C is called a t-bounded distance decoder or a decoder that corrects t errors if D(r) is a nearest codeword for all received words r with d(c, r) t errors. A decoder for a code C with error-correcting capability e(c) decodes up to half the minimum distance if it is an e(c)-bounded distance decoder, where e(c) = (d(c) 1)/2 is the error-correcting capability of C. Remark If D is a t-bounded distance decoder, then it is not required that D gives a decoding failure as outcome for a received word r if the distance of r to the code is strictly larger than t. In other words: D is also a t -bounded distance decoder for all t t. A t-bounded distance decoder D is called strict if D(r) =? for all r such that d(r, C) > t. Definition Let d(y, C) denote the Hamming distance between y Q n to code C so d(y, C) = min c C {d(y, c)}. The covering radius of code C, denote by ρ(c), is the largest distance from C to any word in Q n, that is ρ(c) = max d(y, C) y Qn and the Newton radius of code C, denoted by ν(c) is the largest distance from any word r in Q n to C such that r is uniquely decodeable. Remark A nearest neighbor decoder is a t-bounded distance decoder for all t ρ(c), where ρ(c) is the covering radius of the code. 2. A ρ(c)-bounded distance decoder is a nearest neighbor decoder, since d(c, r) ρ(c) for all received words r. Definition Suppose C is a code in Q n. A maximum-likelihood decoder of C transforms any received word r Q n to codeword c C such that the probability P {r received c transmitted} attains the maximum value. Definition For every decoding scheme and channel one defines three probabilities P cd (p), P de (p) and P df (p), that are the probability of correct decoding, decoding error and decoding failure, respectively. Then, P cd (p) + P de (p) + P df (p) = 1 for all 0 p 1 2. So it suffices to find formulas for two of these three probabilities. The error probability, also called the error rate is defined by P err (p) = 1 P cd (p). Hence, P err (p) = P de (p) + P df (p).

27 18 Preliminaries Probability of correct decoding Suppose C is a code in Q n. Let D be a nearest neighbor decoder of C. Hence, d(r, D(r)) = d(r, C) for every r Q n. The probability of correct decoding of a nearest neighbor decoder D of a code C is the probability that the output of the decoder is equal to codeword that was sent. Proposition The probability of correct decoding of the code C with respect to a nearest neighbor decoder D is given by P cd (C, p) = c C P (c) P (r c), r Q n D(r)=c where P (c) is the probability that codeword c is sent and P (r c) is the probability that r is received given that c is sent. Proof. Suppose a codeword c C is sent with probability P (c) over the noisy channel. The received word can be any word in Q n. Suppose the received word is r which has probability P (r c). The decoder will return the correct codeword if and only if D(r) = c. Hence, the probability of correct decoding of codeword c is equal to the summation of P (c) P (r c) for all r Q n such that D(r) = c, which is equal to P (c) r,d(r)=c P (r c). Therefore, the probability of correct decoding of code C is equal to c C P (c) r,d(r)=c P (r c). Proposition The probability of correct decoding of code C with respect to a nearest neighbor decoder D over a q-ary symmetric channel with crossover probability p is given by P cd (C, p) = 1 ( ) p d(r,d(r)) (1 p) n d(r,d(r)). C q 1 r Q n Proof. In a qsc with crossover probability p, we have P (c) = 1 c for all c C, and P (r c) = (1 p) n d(r,c) ( p q 1) d(r,c) by Proposition Hence, P cd (C, p) = 1 C ( ) p d(r,c) (1 p) n d(r,c). q 1 c C r Q D(r)=c Since a nearest neighbor decoder always returns a codeword, that is D(r) C, we have P cd (C, p) = 1 ( ) p d(r,d(r)) (1 p) n d(r,d(r)). C q 1 r Q n Proposition The probability of correct decoding of a decoder that corrects up to t errors and fails to decode all error patterns with more than t errors with 2t + 1 d of a code C of minimum distance d on a q-ary symmetric channel with cross-over probability p is given by t ( ) n P cd (p) = p w (1 p) n w. w w=0

28 2.8. Decoders 19 Figure 2.6: Capacity of the binary symmetric channel with respect to p Proof. Every codeword has the same probability of transmission. So Now P cd (p) = c C P (c) d(c,r) t P (r c) = d(c,r) t t w=0 P (r c) = 1 C c C d(c,r) t P (r c). ( ) ( ) n p w (q 1) w (1 p) n w, w q 1 since P (r c) depends only on the distance between r and c by Proposition and by Proposition So this last sum does not depend on the chosen c C. Hence, P cd (p) = t w=0 ( ) ( ) n p w (q 1) w (1 p) n w. w q 1 Clearing the factor (q 1) w in the numerator and the denominator gives the desired result. Example Consider the binary n-fold repetition code. Let t = (n 1)/2. Use the decoding algorithm correcting all patterns of t errors. Then, P err (p) = n i=t+1 ( ) n p i (1 p) n i. i Hence the error probability becomes arbitrarily small for increasing n. The price one has to pay is that the information rate R = 1/n tends to 0. The remarkable result of Shannon states that for a fixed rate R < C(p), where C(p) = 1 + p log 2 (p) + (1 p) log 2 (1 p) is the capacity of the binary symmetric channel, one can devise encoding and decoding schemes such that P err (p) becomes arbitrarily small. The capacity of the binary symmetric channel with respect to p is given in Figure 2.6. The main problem of error-correcting codes from Shannon s point view" is to construct efficient encoding and decoding algorithms of codes with the smallest error probability for a given information rate and cross-over probability.

29 20 Preliminaries Probability of undetected error Consider the q-ary symmetric channel where the receiver checks whether the received word r is a codeword or not, and asks for retransmission in case r is not a codeword. Now it may occur that r is again a codeword but not equal to the codeword that was sent. This is called an undetected error. The probability of undetected error of the code C on a q-ary symmetric channel with cross-over probability p is given by P ue (C, p) = c C P (c) r C r c P (r c), where P (c) is the probability that codeword c is sent and P (r c) is the probability that r is received given that c is sent. Proposition The probability of undetected error of code C with respect to a nearest neighbor decoder over a q-ary symmetric channel with crossover probability p is given by P ue (C, p) = 1 C ( ) p d(r,c) (1 p) n d(r,c). q 1 c C r C r c Proof. The proof is similar to the proof of Proposition for P cd (C, p) Probability of correct erasure decoding Let C be a code in Q n. We define erasure decoding as follows: Let c C be sent over a noisy q-ary symmetric channel. The decoder will receive word r ˆQ n. Suppose x is the i th symbol in c. The only thing that can happen to x is that it is changed into an erasure with probability p or that it does not change with probability 1 p where 0 p 1, as as we see in Figure 2.7. Similarly, as for the q-ary symmetric channel it is assumed that this probability p is the same for all symbols and that it is memoryless, it does not depend on the position of the symbol in the codeword. Finally, it is assumed that the probability P (c) that the codeword c is sent is the same for all codewords. Hence, we have that P (c) = 1 C for all c C. Suppose the decoder receives the word r given that c is sent. Now d(r, c), the Hamming distance between r and c is equal to the number of blanks in r, since it is assumed that there are only erasures, that is r i = c i or r i = for all i. Let r Ĉ and D(r) be a closest codeword to r. Then d(r, C) = d(r, D(r)) and is equal to the number of erasures of r, since r Ĉ and only erasures are corrected and no errors. x 1 p p x Figure 2.7: Crossover probability for erasure channel