MATH32031: Coding Theory Part 15: Summary
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1 MATH32031: Coding Theory Part 15: Summary 1 The initial problem The main goal of coding theory is to develop techniques which permit the detection of errors in the transmission of information and, if necessary, allow the original message to be reconstructed. For our purposes, a p-ary code C of length n is a subset of F (n p = (F p,..., F p, A p-ary codeword of length n is simply a vector x = (x 1,..., x n with the x i F p (that is, a member of the n-dimensional vector space over F p and the property that xis a member of the code, that is x C F (n p. 2 Hamming distance Definition The Hamming distance d(x, y between two vectors x, y F (n is the number of coefficients in which they differ. Symmetric channels. Suppose that a codeword x is transmitted and y is received. If the nearest neighbour of y in C is x C then it seems reasonable to suppose that x was the original message. To justify this we suppose: Errors in different positions in a word are independent; the occurrence of an error in one position in the word does not affect the probability of error in another position. Each symbol f F has the same probability r of being erroneously transmitted. We also assume that this probability of error is small, r 1/2. If f F is mistransmitted, then we suppose all q 1 remaining symbols are equally likely to be received. Definition. The minimum distance of a code C, denoted d(c, is d(c = min (d(x, y x, y C, x y Theorem 2 (a If, for a code C, then C can detect up to s errors. d(c s + 1 (b If d(c 2t + 1 then the code C can correct up to t errors. i
2 3 Equivalence of codes Definition. Two q-ary codes are called equivalent if one can be obtained from the other by means of the following 2 types of operation: (a a permutation of the positions of the code; (b a permutation of the symbols appearing in a fixed position. Given a code of length n with M elements we can write the elements of C in an M n matrix. For this representation: operation (a corresponds to a permutations of columns; operation (b is just a relabelling of the symbols in a given column. Notice that a permutation of the rows does not alter the code, since the codewords are just permuted. Proposition 5 The distance function is invariant under these operations (and hence under equivalence. The sphere of radius r about u is Lemma 7 S r (u = {x F (n d(x, u r}. (a The number of vectors in F (n 2 of weight i is ( n n! = i i!(n i! (b For r n, and F = F q, S r (u contains exactly S r (u = ( n 0 + ( n 1 ( n (q r (q 1 r = r i=0 ( n i (q 1 i points of F (n. Theorem 8 (Hamming Bound or Sphere Packing Bound If there is a q-ary (n, M, 2t + 1-code, then the following inequality is satisfied: { t ( } n M S t (u q n, or equivalently M (q 1 i q n. i For a linear code of dimension k this becomes S t (u q n k. Perfect codes. A code whose parameters give equality in the sphere packing bound is called a perfect code. (This amounts to saying F (n q is completely covered by the nonintersecting spheres S t (c. ii i=0
3 Linear codes. A code C F n is called an F -linear code if C is an F -vector subspace. Weight. in v. Let v F n. Then the weight of v, w(v, is the number of non-zero co-ordinates Lemma For x, y F n d(x, y = w(x y. Weight of a code. If C is a code then the weight w(c of C is defined as w(c = min{ w(x x C, x 0 }. Theorem If C is a linear code then d(c = w(c. 4 Generator matrices Let r 1,..., r k denote a basis for an F -linear code C F (n. Recall every c C can be written uniquely as c = λ i r i for λ i F. Thus the code C is completely specified by the k n matrix This is called a generator matrix for C, and is usually denoted by G = G(C. r 1. r k. 5 Equivalence of linear codes Definition. Two F -linear codes C, C are (F -linearly equivalent if one can be obtained from the other by means of a combination of transpositions of the following types: (a a permutation of the positions of the code; (b multiplication of symbols in a fixed position by a non-zero scalar in F. Theorem 9 Two k n matrices generate (F -linearly equivalent codes if one can be obtained from the other by a sequence of operations of the following types: (R1 Permutation of rows. (R2 Multiplication of a row by a non-zero scalar. iii
4 (R3 Adding a scalar multiple of one row to another row. (C1 Permutation of columns. (C2 Multiplication of any fixed column by a non-zero scalar. Theorem 10 Let G be the generator matrix of an [n, k] code over F. Then by using operations R1, R2, R3, C1, C2, G can be transformed to a standard form matrix [I k A] where I k = is the (k k identity matrix and A is a (k (n k matrix. 6 Encoding From the point of view of applications, C encodes an arbitrary vector from F (k as a codeword in F (n. Let u = (u 1,..., u k F (k be the message. Produce a codeword x by the (linear transformation u k u i r i = ug C F n. i=1 7 Decoding Consider the cosets F (n /C. A coset leader is a choice of vector of minimal weight in its coset. Write the coset leaders as a i, codewords as c i. Standard array decoding 0 c 1... c q k 1 a 1 a 1 + c 1... a 1 + c q k 1... a i a i + c 1... a i + c q k 1... a q n k 1 a q n k 1 + c 1... a q n k 1 + c q k 1 In terms of the standard array, y = a i + c j for some c j and c j is the decoded vector; hence once y is located, we need only read off the code vector of the top of the column. Proposition 13 (a The received vectors that are decoded as 0 by the algorithm are the coset leaders. (b The algorithm fails to detect an error if and only if y C. (c This is a nearest neighbour decoding. iv
5 Theorem 14 Let C be a [n, k] F 2 -code transmitted by a binary symmetric channel with symbol error rate r. Let α i be the number of coset leaders of weight i and let A i be the number of codewords of weight i. Then P corr (C = n α i r i (1 r n i. i=0 P undetect (C = n A i r i (1 r n i. i=1 8 The dual code Given a linear code C V we define the dual code (or orthogonal vector space C as Proposition 15 C is a linear code. (C = C. C = { v V v c = 0 for every c C }. Lemma 16 Let C be an [n, k] F -code in V with generator matrix G; then x C if and only if xg T = 0. Theorem 17 Suppose C is an [n, k]-code, then C is an [n, n k]-code. Definition. A parity check matrix H for C is a generator matrix for the dual code C. Proposition 20 C = {x V xh T = 0}. Theorem 21 Let C be [n, k] F -code, and suppose C has a generator matrix G = [I k A] in standard form; then H = [ A T I n k ] is a parity check matrix for C. 9 Syndrome decoding Let C V be an F linear code with parity check matrix H. For y V the syndrome of y, denoted S(y, is S(y = yh T. Lemma 22 S(y = S(z if and only if y z C. For a given received vector y: Step 1. Determine S(y. Step 2. Locate the a i with S(a i = S(y; by the above it follows that y a i + C. Step 3. (Recall that a i is taken as the error. Decode y as y a i. v
6 10 Incomplete decoding Step 1. Given a received vector y, calculate the syndrome S(y. Step 2. Find the unique coset leader a i with S(a i = S(y. (Recall that a i is taken as the error. So if w(a i t, then we take x = y a i as the (unique nearest neighbour decode of y. (Note: this decode of y will be correct if y contains t errors, since x is the (unique nearest neighbour. Step 3. If w(a i > t, then y may well have several nearest neighbours, so we seek re-transmission. 11 Minimal distance and the parity check matrix Theorem 23 (Distance Theorem for Linear Codes Let C be a linear code with parity check matrix H. Then d(c = d if and only if every set of d 1 columns of H is linearly independent and some set of d columns of H is linearly dependent. 12 Hamming codes We introduce a relation on K (n \ {0} by saying This is an equivalence relation. We define v u if and only if v = λu for some λ K. P n 1 (K = (K (n \ {0}/ (the set of equivalence classes. Geometrically P n 1 (K should be thought of as the set of (undirected lines in K (n. Lemma 24 P n 1 (F q = qn 1 q 1. Let s 1 be given. We let Ham(s, q denote the F q -linear code whose parity check matrix has the (representative vectors of P s 1 (F q as its columns. Theorem 25 (a Ham(s, q is an F q -code with parameters [n, n s, 3] where n = qs 1 q 1. (b Ham(s, q is a perfect, single error correcting code. vi
7 13 Decoding a Hamming code For a Hamming code we may proceed as follows: 1. Calculate S(y. 2. Check S(y against the columns of H and find S(y = λh j for some j, some λ F. We then know by ( that S(y = S(a for a the coset leader a = (0,..., 0, j λ, 0,..., 0, and so to decode just subtract λ from the j-th co-ordinate of y. 14 Construction of codes with large distance Theorem 27 Let n > k, d n k and suppose d 2 ( n 1 (q 1 i < q n k. i i=0 Then there exists an [n, k, d] F q -code. 15 Rings of polynomials and factor rings Proposition 28 Let f be a polynomial of degree deg f = m. Then R f has { 1, x,..., x m 1 } as an F p -basis. In particular R f has p m elements. 16 Cyclic codes With the previous notation we set f n = x n 1, and we write R n for Definition of cyclic codes. then belongs to C. R fn = F p [x] /(f n. An F p -linear code in F (n p (a 0, a 1,..., a n 1 C (a n 1, a 0, a 1,..., a n 2 is called cyclic if whenever Theorem 29 A code C R n is a cyclic code if and only if C is an R n -ideal. Theorem 30 Let C be an ideal in R n. Then there exists a unique monic polynomial g C such that: (a C = gr n, (b g (x n 1 (divisibility in F p [x]. We call such a g the generator of the code C. vii
8 17 Generator matrices Theorem 31 Let g(x = g g r 1 x r 1 + x r be the generator of a cyclic code C in R n. Then dim Fp (C = n r and if g f n then C has an (n r n generator matrix given by g 0 g 1... g r g G = g r g g r Given a cyclic code C = gr n as above, we write gh = f n (in F p [x]. Definition. h is called the check polynomial of the code C. Proposition 32 An element c R n is a codeword in gr n if and only if ch = 0 (in R n. Theorem 33 With the above notation, set k = n r and write h = h h k 1 x k 1 + x k. (a If h f n then a parity check matrix for the code gr n is 1 h k 1... h h H = k 1... h h 0. (b C is the cyclic code generated by ĥ(x = 1 + h k 1 x + + h 0 x k. 18 The u u + v construction If C 1 is an (n, M 1, d 1 F -code and C 2 is an (n, M 2, d 2 F -code, let C 1 C 2 = { u u + v F (2n u C 1, v C 2 }. Theorem 34 C 1 C 2 is a (2n, M 1 M 2, min{2d 1, d 2 } F -code. 19 Boolean Functions Theorem 35 The square-free monomials in the v i s form a basis for the Boolean functions on V m. viii
9 20 Reed-Muller Codes Definition For 0 r m, the rth order binary Reed-Muller code R(r, m is the set of all vectors f that correspond to Boolean functions f(v 1,..., v m in m variables that have degree at most r. It has length 2 m. Lemma The dimension of R(r, m is k = 1 + ( m 1 + ( m ( m r. Theorem 36 R(r + 1, m + 1 = R(r + 1, m R(r, m. Theorem 37 d(r(r, m = 2 m r. Lemma R(m 1, m is the even weight code. Theorem 38 R(m r 1, m = R(r, m for 0 r m 1. ix
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