CHAPTER 2: Linear codes ABSTRACT Most of the important codes are special types of so-called linear codes. Linear codes are of very large importance because they have very concise description, very nice properties, very easy encoding And, in principle, easy to describe decoding. Linear codes Linear codes are special sets of words of the length n over an alphabet {0,..,q -1}, where q is a power of prime. Since now on sets of words F[q]^n will be considered as vector spaces V(n,q) of vectors of length n with elements from the set {0,..,q -1} and arithmetical operations will be taken modulo q. The set {0,..,q -1} with operations + and · modulo q is called also the Galois field GF(q). Definition A subset C Í V(n,q) is a linear code if (1) u + v Î C for all u, v Î C (2) au Î C for all u Î C, a Î GF(q) Example Codes C[1], C[2], C[3] introduced in Lecture 1 are linear codes. Exercise Which of the following binary codes are linear? C[1] = {00, 01, 10, 11} C[2] = {000, 011, 101, 110} C[3] = {00000, 01101, 10110, 11011} C[5] = {101, 111, 011} C[6] = {000, 001, 010, 011} C[7] = {0000, 1001, 0110, 1110} Basic properties of linear codes Notation: w(x) (weight of x) denotes the number of non-zero entries of x. Lemma If x, y Î V(n,q), then h(x,y) = w(x - y). Proof x - y has non-zero entries in exactly those positions where x and y differ. Basic properties of linear codes If C is a linear [n,k] -code, then it has a basis consisting of k codewords. Example Code C[4] = {0000000, 1111111, 1000101, 1100010, 0110001, 1011000, 0101100, 0010110, 0001011, 0111010, 0011101, 1001110, 0100111, 1010011, 1101001, 1110100} has the basis {1111111, 1000101, 1100010, 0110001}. How many different bases has a linear code? Advantages and disadvantages of linear codes I. Advantages - big. 1. Minimal distance h(C) is easy to compute if C is a linear code. 2. Linear codes have simple specifications. • To specify a non-linear code usually all codewords have to be listed. • To specify a linear [n,k] -code it is enough to list k codewords. Definition A k ´ n matrix whose rows form a basis of a linear [n,k] -code (subspace) C is said to be the generator matrix of C. Example The generator matrix of the code and of the code 3. There are simple encoding/decoding procedures for linear codes. Advantages and disadvantages of linear codes II. Disadvantages of linear codes are small: 1. Linear q -codes are not defined unless q is a prime power. 2. The restriction to linear codes might be a restriction to weaker codes than sometimes desired. Equivalence of linear codes Definition Two linear codes GF(q) are called equivalent if one can be obtained from another by the following operations: (a) permutation of the positions of the code; (b) multiplication of symbols appearing in a fixed position by a non-zero scalar. Equivalence of linear codes Theorem Let G be a generator matrix of an [n,k] -code. Rows of G are then linearly independent .By operations (a) - (e) the matrix G can be transformed into the form: [ I[k] | A ] where I[k] is the k ´ k identity matrix, and A is a k ´ (n - k) matrix. Encoding with a linear code is a vector ´ matrix multiplication Let C be a linear [n,k] -code over GF(q) with a generator matrix G. Theorem C has q^k codewords. Proof Theorem follows from the fact that each codeword of C can be expressed uniquely as a linear combination of the basis vectors. Corollary The code C can be used to encode uniquely q^k messages. Let us identify messages with elements V(k,q). Encoding of a message u = (u[1], … ,u[k]) with the code C: Uniqueness of encodings with linear codes Theorem If G={w[i]}[i=1]^k is a generator matrix of a binary linear code C of length n and dimension k, then v = uG ranges over all 2^k codewords of C as u ranges over all 2^k words of length k. Therefore C = { uG | u Î {0,1}^k^ } Moreover u[1]G = u[2]G if and only if u[1 ]= u[2]. Proof If u[1]G – u[2]G=0, then And, therefore, since w[i] are linearly independent, u[1] = u[2]. Decoding of linear codes Decoding problem: If a codeword: x = x[1] … x[n] is sent and the word y = y[1] … y[n] is received, then e = y – x = e[1] … e[n] is said to be the error vector. The decoder must decide, from y, which x was sent, or, equivalently, which error e occurred. To describe main Decoding method some technicalities have to be introduced Definition Suppose C is an [n,k] -code over GF(q) and u Î V(n,q). Then the set u + C = { u + x | x Î C } is called a coset (u-coset) of C in V(n,q). Nearest neighbour decoding scheme: Each vector having minimum weight in a coset is called a coset leader. 1. Design a (Slepian) standard array for an [n,k] -code C - that is a q^n^ ^-^ ^k ´ q^k array of the form: Probability of good error correction What is the probability that a received word will be decoded as the codeword sent (for binary linear codes and binary symmetric channel)? Probability of an error in the case of a given error vector of weight i is p^ ^i (1 - p)^n - i. Therefore, it holds. Theorem Let C be a binary [n,k] -code, and for i = 0,1, … ,n let a[i ]be the number of coset leaders of weight i. The probability P[corr] (C) that a received vector when decoded by means of a standard array is the codeword which was sent is given by Probability of good error detection Suppose a binary linear code is used only for error detection. The decoder will fail to detect errors which have occurred if the received word y is a codeword different from the codeword x which was sent, i. e. if the error vector e = y - x is itself a non-zero codeword. The probability P[undetect] (C) that an incorrect codeword is received is given by the following result. Theorem Let C be a binary [n,k] -code and let A[i ]denote the number of codewords of C of weight i. Then, if C is used for error detection, the probability of an incorrect message being received is Dual codes Inner product of two vectors (words) u = u[1] … u[n], v = v[1] … v[n][] in V(n,q) is an element of GF(q) defined (using modulo q operations) by u × v = u[1]v[1] + … + u[n]v[n]. Example In V(4,2): 1001 × 1001 = 0 In V(4,3): 2001 × 1210 = 2 1212 × 2121 = 2 If u × v = 0 then words (vectors) u and v are called orthogonal. Properties If u, v, w Î V(n,q), l, m Î GF(q), then u × v = v × u, (lu + mv) × w = l (u × w) + m (v × w). Given a linear [n,k] -code C, then dual code of C, denoted by C^^, is defined by C^^ = {v Î V(n,q) | v × u = 0 if u Î C}. Lemma Suppose C is an [n,k] -code having a generator matrix G. Then for v Î V(n,q) v Î C^^ <=> vG^T = 0, where G^T denotes the transpose of the matrix G. Proof Easy. PARITE CHECKS versus ORTHOGONALITY For understanding of the role the parity checks play for linear codes, it is important to understand relation between orthogonality and special parity checks. If words x and y are orthogonal, then the word y has even number of ones (1’s) in the positions determined by ones (1’s) in the word x. This implies that if words x and y are orthogonal, then x is a parity check word for y and y is a parity check word for x. Exercise: Let the word 100001 be orthogonal to a set S of binary words of length 6. What can we say about the words in S? EXAMPLE For the [n,1] -repetition code C, with the generator matrix G = (1,1, … ,1) the dual code C^^ is [n,n - 1] -code with the generator matrix G^^, described by ^ ^^ Parity check matrices Example If ^ ^^ If ^ ^ ^ ^^ Parity check matrices Definition A parity-check matrix H for an [n,k] -code C is a generator matrix of C^^. Theorem If H is parity-check matrix of C, then C = {x Î V(n,q) | xH^T = 0}, and therefore any linear code is completely specified by a parity-check matrix. Syndrome decoding Theorem If G = [I[k] | A] is the standard form generator matrix of an [n,k] -code C, then a parity check matrix for C is H = [-A^T | I[n-k]]. Example Definition Suppose H is a parity-check matrix of an [n,k] -code C. Then for any y Î V(n,q) the following word is called the syndrome of y: S(y) = yH^T. Lemma Two words have the same syndrom iff they are in the same coset. Syndrom decoding Assume that a standard array of a code C is given and, in addition, let in the last two columns the syndrom for each coset be given. When a word y is received, compute S(y) = yH^T, locate S(y) in the “syndrom column”, and then locate y in the same row and decode y as the codeword in the same column and in the first row. KEY OBSERVATION for SYNDROM COMPUTATION When preparing a ”syndrome decoding'' it is sufficient to store only two columns: one for coset leaders and one for syndromes. Example coset leaders syndromes l(z) z 0000 00 1000 11 0100 01 0010 10 Decoding procedure • Step 1 Given y compute S(y). • Step 2 Locate z = S(y) in the syndrome column. • Step 3 Decode y as y - l(z). Hamming codes An important family of simple linear codes that are easy to encode and decode, are so-called Hamming codes. Definition Let r be an integer and H be an r ´ (2^r - 1) matrix columns of which are non-zero distinct words from V(r,2). The code having H as its parity-check matrix is called binary Hamming code and denoted by Ham(r,2). Example Hamming codes - decoding Decoding algorithm for the case the columns of H are arranged in the order of increasing binary numbers the columns represent. • Step 1 Given y compute syndrome S(y) = yH^T. • Step 2 If S(y) = 0, then y is assumed to be the codeword sent. • Step 3 If S(y) ¹ 0, then assuming a single error, S(y) gives the binary position of the error. Example For the Hamming code given by the parity-check matrix and the received word y = 110 1011, we get syndrome S(y) = 110 and therefore the error is in the sixth position. Hamming code was discovered by Hamming (1950), Golay (1950). 1 It was conjectured for some time that Hamming codes and two so called Golay codes are the only non-trivial perfect codes. Comment Hamming codes were originally used to deal with errors in long-distance telephon calls. ADVANTAGES of HAMMING CODES Let a binary symmetric channel is used which with probability q correctly transfers a binary symbol. If a 4-bit message is transmitted through such a channel, then correct transmission of the message occurs with probability q^4. If Hamming (7,4,3) code is used to transmit a 4-bit message, then probability of correct decoding is q^7^ + 7(1 - q)q^6. In case q = 0.9 the probability of correct transmission is 0.651 in the case no error correction is used and 0.8503 in the case Hamming code is used - an essential improvement. IMPORTANT CODES • Hamming (7,4,3) -code. It has 16 codewords of length 7. It can be used to send 2^7 = 128 messages and can be used to correct 1 error. GOLAY CODES - DESCRIPTION Golay codes G[24] and G[23] were used by Voyager I and Voyager II to transmit color pictures of Jupiter and Saturn. Generation matrix for G[24] has the form G[24] is (24,12,8) –code and the weights of all codewords are multiples of 4. G[23] is obtained from G[24] by deleting last symbols of each codeword of G[24]. G[23] is (23,12,7) –code. GOLAY CODES - CONSTRUCTION Matrix G for Golay code G[24 ]has actually a simple and regular construction. The first 12 columns are formed by a unitary matrix I[12], next column has all 1’s. Rows of the last 11 columns are cyclic permutations of the first row which has 1 at those positions that are squares modulo 11, that is 0, 1, 3, 4, 5, 9.[] SINGLETON BOUND If C is a linear [n,k,d] -code, then n - k ³ d - 1 (Singleton bound). To show the above bound we can use the following lemma. Lemma If u is a codeword of a linear code C of weight s,then there is a dependence relation among s columns of any parity check matrix of C, and conversely, any dependence relation among s columns of a parity check matrix of C yields a codeword of weight s in C. Proof Let H be a parity check matrix of C. Since u is orthogonal to each row of H, the s components in u that are nonzero are the coefficients of the dependence relation of the s columns of H corresponding to the s nonzero components. The converse holds by the same reasoning. REED-MULLER CODES Reed-Muller codes form a family of codes defined recursively with interesting properties and easy decoding. If D[1] is a binary [n,k[1],d[1]] -code and D[2] is a binary [n,k[2],d[2]] -code, a binary code C of length 2n is defined as follows C = { u | u + v |, where u Î D[1], v Î D[2]}. Lemma C is [2n,k[1][ ]+ k[2], min{2d[1],d[2]}] -code and if G[i] is a generator matrix for D[i], i = 1, 2, then is a generator matrix for C. Reed-Muller codes R(r,m), with 0 £ r £ m are binary codes of length n = 2^m. R(m,m) is the whole set of words of length n, R(0,m) is the repetition code. If 0 < r < m, then R(r + 1,m + 1) is obtained from codes R(r + 1,m) and R(r,m) by the above construction. Theorem The dimension of R(r,m) equals The minimum weight of R(r,m) equals 2^m^ ^-^^r. Codes R(m - r - 1,m) and R(r,m) are dual codes. Singleton Bound Singleton bound: Let C be a q-ary (n, M, d)-code. Then M £ q ^n-d+1 . Proof Take some d − 1 coordinates and project all codewords to the resulting coordinates. The resulting codewords are all different and therefore M cannot be larger than the number of q-ary words of length n−d−1. Codes for which M = q ^n−d+1 are called MDS-codes (Maximum Distance Separable). Corollary: If C is a q-ary linear [n, k, d]-code, then k + d £ n + 1. Shortening and puncturing of linear codes Let C be a q-ary linear [n, k, d]-code. Let D = {(x[1], ... , x[n-1]) | (x[1], ... , x[n-1], 0)ÎC}. Then D is a linear [n-1, k-1, d]-code – a shortening of the code C. Corollary: If there is a q-ary [n, k, d]-code, then shortening yields a q-ary [n−1, k−1, d]-code. Let C be a q-ary [n, k, d]-code. Let E = {(x[1], ... , x[n-1]) | (x[1], ... , x[n-1], x)ÎC, for some x £ q}, then E is a linear [n-1, k, d-1]-code – a puncturing of the code C. Corollary: If there is a q-ary [n, k, d]-code with d >1, then there is a q-ary [n−1, k, d-1]-code. Lengthening of Codes – Constructions X and XX Construction X Let C and D be q-nary linear codes with parameters [n, K, d] and [n, k, D], where D > d, and K > k. Assume also that there exists a q-nary code E with parameters [l, K − k, δ ]. Then there is a ”longer” q-nary code with parameters [n + l, K, min(d + δ, D)]. The lengthening of C is constructed by appending φ(x) to each word x ∈ C, where φ : C/D → E is a bijection – a well known application of this construction is the addition of the parity bit in binary codes. Construction XX Let the following q-ary codes be given: a code C with parameters [n, k, d]; its sub-codes C[i ], i = 1,2 with parameters [n, k − k[i ], d[i]] and with C[1] ∩ C[2] of minimum distance ≥ D; auxiliary q-nary codes E[i ], i = 1,2 with parameters [l[i ], k[i ], δ[i]]. Then there is a q-ary code with parameters [n + l[1] + l[2 ], k, min{D, d[2] + δ[1], d[1] + δ[2 ], d + δ[1] + δ[2]}]. Strength of Codes • Strength of codes is another important parameter of codes. It is defined through the concept of the strength of so-called orthogonal arrays - an important concepts of combinatorics. • An orthogonal array QA[λ](t, n, q) is an array of n columns, λq ^t rows with elements from F[q] and the property that in the projection onto any set of t columns each possible t-tuple occurs the same number λ of times. t is called strength of such an orthogonal array. • For a code C, let t(C) be the strength of C - if C is taken as an orthogonal array. • Importance of the concept of strength follows also from the following Principle of duality: For any code C its minimum distance and the strength of C^^ are closely related. Namely d(C) = t(C^^) + 1. Dimension of Dual Linear Codes If C is an [n, k]-code, then its dual code C^⊥ is [n, n − k] code. A binary linear [n, 1] repetition code with codewords of length n has two codewords: all-0 codeword and all-1 codeword. Dual code to [n, 1] repetition code is so-called sum zero code of all binary n-bit words whose entries sum to zero (modulo 2). It is a code of dimension n − 1 and it is a linear [n, n − 1, 2] code Reed-Solomon Codes An important example of MDS-codes are q-ary Reed-Solomon codes RSC(k, q), for k ≤ q. They are codes generator matrix of which has rows labelled by polynomials X ^i, 0 ≤ i ≤ k − 1, columns by elements 0, 1, . . . , q − 1 and the element in a row labelled by a polynomial p and in a column labelled by an element u is p(u). RSC(k, q) code is [q, k, q − k + 1] code. Example Generator matrix for RSC(3, 5) code is Interesting property of Reed-Solomon codes: RSC(k, q)^⊥ = RSC(q − k, q). Reed-Solomon codes are used in digital television, satellite communication, wireless communication, barcodes, compact discs, DVD,... They are very good to correct burst errors - such as ones caused by solar energy. Trace and Subfield Codes • Let p be a prime and r an integer. A trace tr is mapping from F[p]r into F[p] defined by tr(x) = • Trace is additive (tr(x[1] + x[2]) = tr(x[1]) + tr(x[2])) and F[p]-linear (tr(λx) = λtr(x)). • If C is a linear code over F[p]r and tr is a trace mapping from F[p]r to F[p], then trace code tr(C) is a code over F[p] defined by (tr(x[1]), tr(x[2]), . . . , tr(x[n])) where (x[1], x[2], . . . , x[n]) ÎÎ C. • If F^n[p]rÉ C is a linear code of strength t, then strength of tr(C) is at least t. • Let C be a linear code. The subfield code C[F]p consists of those codewords of C all of whose entries are in F[p]. • Delsarte theorem If C is a linear code. Then tr(C)^^ = (C^^)[F]p . Soccer Games Betting System Ternary Golay code with parameters (11, 729, 5) can be used to bet for results of 11 soccer games with potential outcomes 1 (if home team wins), 2 (if guests win) and 3 (in case of a draw). If 729 bets are made, then at least one bet has at least 9 results correctly guessed. In case one has to bet for 13 games, then one can usually have two games with pretty sure outcomes and for the rest one can use the above ternary Golay code.