1. Information and Coding Theory Linear Block Codes. Basic definitions and some examples. Some bounds on code parameters. Hemming and Golay codes. Syndrome decoding. Juris Viksna, 2016

2. Transmission over noisy channel

4. Noisy channel In practice channels are always noisy (sometimes this could be ignored). There are several types of noisy channels one can consider. We will restrict attention to binary symmetric channels.

5. Shannon Channel Coding Theorem

7. Codes – how to define them? In most cases it would be natural to use binary block codes that maps input vectors of length k to output vectors of length n. For example: 101101  11100111 110011  01000101 etc. Thus we can define code as an injective mapping from vector space V with dimension k to vector space W with dimension n. Such definition essentially is used in the original Shanon’s theorem. VW

8. Codes – how to define them? We can define code as an injective mapping from “vector space” V with dimension k to “vector space” W with dimension n. Arbitrary mappings between vector spaces are hard either to explicitly define or to use (encode ort decode) in practice (there are almost 2 n2 k of them – already around 1000000000000000000000000000000 for k=4 and n=7). Simpler to define and use are linear codes that can be defined by multiplication with matrix of size k  n (called generator matrix). 0110  0110011 Shanon’s results hold also for linear codes.

9. Codes – how to define them? Simpler to define and use are linear codes that can be defined by multiplication with matrix of size k  n (called generator matrix). What should be the elements of vector spaces V and W? In principle in most cases it will be sufficient to have just 0-s and 1-s, however, to define vector space in principle we need a field – an algebraic system with operations “+” and “  ” defined and having similar properties as we have in ordinary arithmetic (think of real numbers). Field with just “0” and “1” may look very simple, but it turns out that to get some real progress we will need more complicated fields, just that elements of these fields themselves will be regarded as (most often) binary vectors.

10. What are good codes? Linear codes can be defined by their generator matrix of size k  n. Shanon’s theorem tells us that for a transmission channel with a bit error probability p and for an arbitrary small bit error probability p b we wish to achieve there exists codes with rates R = k/n that allows us to achieve p b as long as R<C(p). In general, however, the error rate could be different for different codewords, p b being an “average” value. We however will consider codes that are guaranteed to correct up to t errors for any of codewords.

11. What are good codes? We however will consider codes that are guaranteed to correct up to t errors for any of codewords – this is equivalent with minimum distance between codewords being d and t =  d  1  /2. Such codes will then be characterized by 3 parameters and will be referred to as (n,k,d) codes. For a given k we are thus interested: - to minimize n - to maximize d In most cases for fixed values n and k the larger values of d will give us lower bit error probability p b, although the computation of p b is not that straightforward and depends from a particular code. Note that one can completely “spoil” d value of good code with low p b by including in it a vector with weight 1

12. Vector spaces - definition What we usually understand by vectors? In principle we can say that vectors are n-tuples of the form: (x 1,x 2, ,x n ) and operations of vector addition and multiplication by scalar are defined and have the following properties: (x 1,x 2, ,x n )+(y 1,y 2, ,y n )=(x+y 1,x+y 2, ,x+y n ) a  (x 1,x 2, ,x n )=(a  x 1,a  x 2, ,a  x n ) The requirements actually are a bit stronger – elements a and x i should come from some field F. We might be able to live with such a definition, but then we will link a vector space to a unique and fixed basis and often this will be technically very inconvenient.

13. Vector spaces - definition Definition 4-tuple (V,F,+,  ) is a vector space if (V,+) is a commutative group with identity element 0 and for all u,v  V and all a,b  F: 1) a  (u+v)=a  u+a  v 2) (a+b)  v=a  v+b  v 3) a  (b  v)=(a  b)  v 4) 1  v=v Usually we will represent vectors as n-tuples of the form (x 1,x 2, ,x n ), however such representations will not be unique and will depend from a particular basis of vector space, which we will chose to use (but 0 will always be represented as n-tuple of zeroes (0,0, ,0)).

14. Groups - definition Consider set G and binary operator +. Definition Pair (G,+) is a group, if there is e  G such that for all a,b,c  G: 1) a+b  G 2) (a+b)+c = a+(b+c) 3) a+e = a and e+a = a 4) there exists inv(a) such that a+ inv(a)= e and inv(a)+a = e 5) if additionally a+b = b+a, group is commutative (Abelian) If group operation is denoted by “+” then e is usually denoted by 0 and inv(a) by  a. If group operation is denoted by “  ” hen e is usually denoted by 1 and inv(a) by a  1 (and a  b are usually written as ab). It is easy to show that e and inv(a) are unique.

15. Vector spaces – dot (scalar) product Let V be a k-dimensional vector space over field F. Let b 1, ,b k  V be some basis of V. For a pair of vectors u,v  V, such that u=a 1 b 1 +...+a k b k and v=c 1 b 1 +...+c k b k their dot (scalar) product is defined by: u·v = a 1 ·c 1 +...+ a k ·c k Thus operator “  ” maps V  V to F. Lemma For u,v,w  V and all a,b  F the following properties hold: 1) u·v = v·u. 2) (au+bv)·w = a(u·v)+b(v·w). 3) If u·v = 0 for all v in V, then u = 0.

16. Vector spaces – dot (scalar) product Let V be a k-dimensional vector space over field F. Let b 1, ,b k  V be some basis of V. For a pair of vectors u,v  V, such that u=a 1 b 1 +...+a k b k and v=c 1 b 1 +...+c k b k their dot (scalar) product is defined by: u·v = a 1 ·c 1 +...+ a k ·c k Two vectors u and v are said to be orthogonal if u·v = 0. If C is a subspace of V then it is easy to see that the set of all vectors in V that are orthogonal to each vector in C is a subspace, which is called the space orthogonal to C and denoted by C .

17. Linear block codes Message source EncoderReceiverDecoderChannel x = x 1,...,x k message x' estimate of message y = c + e received vector e = e 1,...,e n error from noise c = c 1,...,c n codeword Generally we will define linear codes as vector spaces – by taking C to be a k-dimensional subspace of some n-dimensional space V.

18. Linear block codes Let V be an n-dimensional vector space over a finite field F. Definition A code is any subset C  V. Definition A linear (n,k) code is any k-dimensional subspace C ⊑ V.

19. Linear block codes Let V be an n-dimensional vector space over a finite field F. Definition A linear (n,k) code is any k-dimensional subspace C ⊑ V. Example (choices of bases for V and code C): Basis of V (fixed): 001,010,100 Set of V elements:{ 000,001,010,011,100,101,110,111 } Set of C elements:{ 000,001,010,011 } 2 alternative bases for code C: 001,010 001,011 Essentially, we will be ready to consider alternative bases, but will stick to “main one” for representation of V elements.

20. Linear block codes Definition A linear (n,k) code is any k-dimensional subspace C ⊑ V. Definition The weight wt(v) of a vector v  V is a number of nonzero components of v in its representation as a linear combination v = a 1 b 1 +...+a n b n. Definition The distance d(v,w) between vectors v,w  V is a number of distinct components of these vectors. Definition The minimum weight of code C ⊑ V is defined as min v  C,v  0 wt(v). A linear (n,k) code with minimum weight d is often referred to as (n,k,d) code.

21. Linear block codes Theorem Linear (n,k,d) code can correct any number of errors not exceeding t =  (d  1)/2 . Proof The distance between any two codewords is at least d. So, if the number of errors is smaller than d/2 then the closest codeword to the received vector will be the transmitted one However a far less obvious problem: how to find which codeword is the closest to received vector?

22. Linear codes - the main problem A good (n,k,d) code has small n, large k and large d. The main coding theory problem is to optimize one of the parameters n, k, d for given values of the other two.

23. Generator matrices Definition Consider (n,k) code C ⊑ V. G is a generator matrix of code C, if C = {vG | v  V} and all rows of G are independent. It is easy to see that generator matrix exists for any code – take any matrix G rows of which are vectors v 1, ,v k (represented as n-tuples in the initially agreed basis of V) that form a basis of C. By definition G will be a matrix of size k  n. Obviously there can be many different generator matrices for a given code. For example, these are two alternative generator matrices for the same (4,3) code:

24. Equivalence of codes Definition Codes C 1,C 2 ⊑ V. are equivalent, if a generator matrix G 2 of C 2 can be obtained from a generator matrix G 1 of C 1 by a sequence of the following operations: 1) permutation of rows 2) multiplication of a row by a non-zero scalar 3) addition of one row to another 4) permutation of columns 5) multiplication of a column by a non-zero scalar (not needed for binary) Note that operations 1-3 actually doesn’t change the code C 1. Applying operations 4 and 5 C 1 could be changed to a different subspace of V, however the weight distribution of code vectors remains the same. In particular, if C 1 is (n,k,d) code so is C 2. In binary case vectors of C 1 and C 2 would differ only by permutation of positions.

25. Generator matrices Definition A generator matrix G of (n,k) code C ⊑ V is said to be in standard form if G = (I,A), where I is k  k identity matrix. Theorem For code C ⊑ V there is an equivalent code C that has a generator matrix in standard form.

26. Hamming code [7,4] Parity bits of H(7,4) No errors - all p i -s correspond to d i -s Error in d 1,...,d 3 - a pair of wrong p i -s Error in d 4 - all pairs of p i -s are wrong Error in p i - this will differ from error in some of d i -s So: - we can correct any single error - since this is unambiguous, we should be able to detect any 2 errors

27. Hamming code [7,4] G - generator matrix A (4 bit) message x is encoded as xG, i.e. if x = 0110 then c = xG = 0110011. Decoding? - there are 16 codewords, if there are no errors, we can just find the right one... - also we can note that the first 4 digits of c is the same as x :)

28. Hamming code [7,4] a = 0001111, b = 0110011 and c = 1010101 H - parity check matrix Why it does work? We can check that without errors yH = 000 and that with 1 error yH gives the index of damaged bit... General case: there always exists matrix for checking orthogonality yH = 0. Finding of damaged bits however isn’t that simple.

29. Hamming codes For simplicity we will consider codes over binary fields, although the definition (and design idea) easily extends to codes over arbitrary finite fields. Definition For a given positive integer r a Hemming code Ham(r) is code a parity check of which as its rows contains all possible non-zero r-dimensional binary vectors. There are 2 r  1 such vectors, thus parity check matrix has size 2 r  1  r and respectively Ham(r) is (n = 2 r  1,n  r) code.

30. Hamming codes Definition For a given positive integer r a Hemming code Ham(r) is code a parity check of which as its rows contains all possible non-zero r-dimensional binary vectors. Example of Hamming code Ham(4): Also not required by definition, note that in this particular case columns can be regarded as consecutive integers 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 written in binary form.

31. Dual codes Definition Consider code C ⊑ V. A dual or orthogonal code of C is defined as C  = {v  V |  w  C: vw = 0}. It is easy check that C  ⊑ V, i.e. C  is a code. Note that actually this is just a re-statement of definition of orthogonal vector spaces we have already seen. There are codes that are self-dual, i.e. C = C .

32. Dual codes - some examples 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: 

33. Dual codes - some examples [Adapted from V.Pless]

34. Dual codes - some examples [Adapted from V.Pless]

35. Dual codes – parity checking matrices Definition Let code C ⊑ V and let C  be its dual code. A generator matrix H of C  is called a parity checking matrix of C. Theorem If k  n generator matrix of code C ⊑ V is in standard form if G = (I,A) then (k  n)  n matrix H = (  A T,I) is a parity checking matrix of C. Proof It is easy to check that any row of G is orthogonal to any row of H (each dot product is a sum of only two non-zero scalars with opposite signs). Since dim C + dim C  = dim V, i.e. k + dim C  = n we have to conclude that H is a generator matrix of C . Note that in binary vector spaces H = (  A T,I) = (A T,I).

36. Dual codes – parity checking matrices Theorem If k  n generator matrix of code C ⊑ V is in standard form if G = (I,A) then (k  n)  n matrix H = (  A T,I) is a parity checking matrix of C. So, up to the equivalence of codes we have an easy way to obtain a parity check matrix H from a generator matrix G in standard form and vice versa. Example of generator and parity check matrices in standard form:

37. Vector spaces and linear transformations Definition Let V be a vector space over field F. Function f : V  V is called a linear transformation, if for all u,v  V and all a  F the following hold: 1)af(u) = f(au). 2)f(u)+f(v) = f(u+v). The kernel of f is defined as ker f ={v  V | f(v) = 0}. The range of f is defined as range f ={f(v) | v  V}. It is quite obvious property of f linearity that vector sums and scalar products doesn't leave ker f or range f. Thus ker f ⊑ V and range f ⊑ V.

38. Dimensions of orthogonal vector spaces CCC 0 Proof? We could try to reduce this to “similarly looking” equality dim V = dim (ker f) + dim (range f). However how we can define a linear transformation from dot product? Theorem If C is a subspace of V, then dim C + dim C  = dim V.

39. Dimensions of orthogonal vector spaces Proof However how we can define a linear transformation from dot product? Let u 1, ,u k be some basis of C. We define transformation f as follows: for all v  V: f(v) = (vu 1 ) u 1 +  + (vu k ) u k Note that vu i  F, thus f(v)  C. Therefore we have: ker f = C  (this directly follows form definition of C  ) range f = C (this follows form definition of f) Thus from rank-nullity theorem: dim C + dim C  = dim V. Theorem If C is a subspace of V, then dim C + dim C  = dim V.

40. Dual codes and vector syndromes Definition Let C ⊑ V be an (n,k) code with a parity check matrix H. For each v  V a syndrome of v is n  k dimensional vector syn(v) = vH T. By definition of H we have syn(c) = 0 for all codewords c  C. If some errors have occurred, then instead of c we have received vector y = c + e, where c is codeword and e is error vector. In this case syn(c) = syn(c) + syn(e) = 0 + syn(e). That is, syndrome is determined solely by error vector and in principle the knowing of vector syndrome should us allow to infer which bits have been transmitted incorrectly.

41. Hamming code (7,4) We have already seen a Hemming (7,4) code with generator matrix (in standard form) shown above and have proved that it can correct any single error (or even more – that it is a perfect (7,4,3) code. Can we generalize this to codes of different lengths/dimensions? How simple the decoding procedure for such codes might be? Parity bits of H(7,4)

42. Hamming codes For simplicity we will consider codes over binary fields, although the definition (and design idea) easily extends to codes over arbitrary finite fields. Definition For a given positive integer r a Hemming code Ham(r) is code a parity check of which as its rows contains all possible non-zero r-dimensional binary vectors. There are 2 r  1 such vectors, thus parity check matrix has size 2 r  1  r and respectively Ham(r) is (n = 2 r  1,n  r) code.

43. Hamming codes Definition For a given positive integer r a Hemming code Ham(r) is code a parity check of which as its rows contains all possible non-zero r-dimensional binary vectors. Example of Hamming code Ham(4): Also not required by definition, note that in this particular case columns can be regarded as consecutive integers 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 written in binary form.

44. Hamming codes Hamming code Ham(4): Why we have defined this code in such particular way? 1)Note that if there have been no errors we have syn(y) = 0 for some received vector y. 2)If a single error have occurred we will have syn(y) = [i], where is binary representation of position (column) in which this error has occurred.

45. Hamming codes 1)Note that if there have been no errors we have syn(y) = 0 for some received vector y. 2)If a single error have occurred we will have syn(y) = [i], where is binary representation of position (column) in which this error has occurred. Thus we are able to correct any single error. Also note that any larger number of errors is not distinguishable from 0 or 1 error case. Thus all (binary) Hamming codes have parameters (n = 2 r  1,n  r,3).

46. A ternary (13,10,3) Hamming code [Adapted from V.Pless]

47. Golay codes - some history The brief history of the Golay Codes begins in 1949, when M. J. E. Golay published his “Notes on Digital Coding” in the Proceedings of the Institute of Electrical and Electronic Engineers”, ½ page in length. It described the (23,12,7) 2 code (although he evidently did not name it after himself). This inspired a search for more perfect codes. After all, if there was some series of perfect codes, or better yet an algorithm that produces them, much of the rest of coding theory would possibly become obsolete. For any given rate and blocklength, no code with a higher minimum distance or average minimum distance can be constructed, so if it had been determined that perfect codes existed with many rates and many blocklengths, it may have been worthwhile to only search for perfect codes. It soon appeared that such prayers fell on deaf ears, as the existence of perfect codes was disproved in more and more general scenarios. Finally, in 1973, when Aimo Tietäväinen disproved the existence of perfect codes over finite fields in his “Nonexistence of Perfect Codes over Finite Fields” in the SIAM Journal of Applied Mathematics, January 1973. [Adapted from www.wikipedia.org]

48. Golay codes In mathematical terms, the extended binary Golay code consists of a 12- dimensional subspace W of the space V=F 2 24 of 24-bit words such that any two distinct elements of W differ in at least eight coordinates. Equivalently, any non-zero element of W has at least eight non-zero coordinates. The possible sets of non-zero coordinates as w ranges over W are called code words. In the extended binary Golay code, all code words have Hamming weight 0, 8, 12, 16, or 24. Up to relabeling of coordinates W is unique. [Adapted from www.wikipedia.org]

49. Golay codes 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.

50. Golay codes 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.

51. Ternary Golay code The ternary Golay code consists of 3 6 = 729 codewords. Its parity check matrix is: [Adapted from www.wikipedia.org] Any two different codewords differ in at least 5 positions. Every ternary word of length 11 has a Hamming distance of at most 2 from exactly one codeword. The code can also be constructed as the quadratic residue code of length 11 over the finite field F 3. This is a (11,6,5) code.

52. Punctured codes There are certain modifications that can be applied to codes and sometimes allows to obtain codes with better parameters. The usual codes that can be obtained in such a way are punctured, extended and shortened codes. Definition A punctured code of a binary (n,k,d) code C with a generator matrix G is a code C with generator matrix G obtained from G by deleting any one of G columns. By deleting different columns we may obtain different punctured codes. Punctured codes may have parameters (n  1,k,d), (n  1,k,d  1), (n  1,k  1,d) or (n  1,k  1,d  1).

53. Punctured codes - example [Adapted from V.Pless]

54. Punctured codes - example [Adapted from V.Pless]

55. Extended codes Definition An extended code of a binary (n,k,d) code C with a generator matrix G is a code C with generator matrix G obtained from G by adding n+1-st column (a 1, ,a k ), where each a i is the sum of all elements of i-th row of G. Thus for any code C there is just one extended code C. In the case when C contains vectors with odd weight code C has parameters (n+1,k,d) or (n+1,k,d+1). If all vectors of C has even weight code C has parameters (n+1,k,d).

56. Extended codes - example [Adapted from V.Pless]

57. Shortened codes Definition A shortened code of a binary (n,k,d) code C is a code C obtained from C by selecting all vectors with 0 in some fixed position i and removing from these vectors elements in i-th position. By choosing different positions i we may obtain different shortened codes. Shortened codes has parameters (n  1,k  1,d), where d  d.

58. Shortened codes - example [Adapted from V.Pless]

59. Shortened codes - example [Adapted from V.Pless]

60. Hemming inequality and perfect codes We will formulate these results in binary case, although they easily extends to codes over any finite field F. Theorem (Hemming inequality) For any (n,k,d) code C from vector space V over the binary field F the following inequality holds: where t =  (d  1)/2 .

61. Hemming inequality and perfect codes Definition An (n,k,d) code C ⊑ V is perfect if balls with radius t =  (d  1)/2  around vectors of C cover the whole vector space V. Proposition A binary (n,k,d) code C ⊑ V is perfect if and only if the following equality holds: It is easy to see that all “full-space” (n, n,1) codes and all binary repetition (2k+1,1,2k+1) codes are perfect. These are trivial perfect codes.

62. Hemming inequality and perfect codes [Adapted from J.MacCay]

63. Hemming inequality and perfect codes Do good perfect codes exist? Theorem The only non-trivial perfect codes are the following: all Hamming codes binary (23,12,7) Golay code ternary (12,6,5) Golay code. We will not prove this result here. However it is very simple to show that these particular codes are perfect, but the proof that these are the only ones is difficult.

64. Singleton bound and MDS codes Theorem (Singleton inequality) For any (n,k,d) code C the inequality n  k  d  1 holds. Proof (version 1) All codewords in C are distinct. If we delete first d  1 components, they are still distinct. New code has length n  d+1 and still has size q k, thus n  d +1  k. Named after Richard Collom Singleton

65. Singleton bound and MDS codes Theorem (Singleton inequality) For any (n,k,d) code C the inequality n  k  d  1 holds. Proof (version 2) Observe that rank H = n  k. Any dependence of s columns in parity check matrix yields a codeword of weight s (any set of non-zero components in codeword  dependence relation in H). Thus n  k  d  1.

66. Singleton bound and MDS codes Definition An (n,k,d) code is called maximum distance separable (or MDS code) if n  k = d  1. There exist only trivial binary MDS codes. However for codes over other finite fields this is not necessarily so. MDS conjecture: [Adapted from G.Seroussi]

67. Some positive results For simplicity we again will state just the binary case. Theorem (Gilbert-Varshamow bound) In n dimensional vector space V over the binary field F there exists a code C with parameters (n,k,d) code for at least one value d  k = n  m, if the following inequality holds:

68. Some positive results Theorem (Gilbert-Varshamow bound) In n dimensional vector space V over the binary field F there exists a code C with parameters (n,k,d) code for at least one value d  k = n  m, if the following inequality holds: Proof We start with d and m and attempt to construct parity check matrix, so that no d  1 columns of it is dependent (implying that minimal weight will be at least d). The number of columns will be n. We start with an arbitrary column and continue to add other columns whilst number of linear combinations from d  2 of them doesn’t exceed number of all m-tuples.

69. Sizes of some known codes [Adapted from V.Pless]

70. Cosets Definition Let C ⊑ V be an (n,k) code. For each a  V a set a+C ={v+c | c  C} is called a coset. Note that this is the same definition we used in proof of Lagrange theorem, if we consider C as an additive group. So, by the same simple arguments we can show that: 1) all cosets have the same number of elements (equal to |C|); 2) two cosets are either completely disjoint or equal; 3) every vector belongs to some coset. Also note, that if error vector has been e, then received message y = c + e is in coset e+C.

71. Cosets Definition Let C ⊑ V be an (n,k) code. For each a  V a set a+C ={v+c| c  C} is called a coset. Note, that if error vector has been e, then received message y = c + e is in coset e+C. Thus for all y  e+C we have syn(y) = syn(e). How do we decode y? It seems reasonable to assume that the number of errors has been minimal, so we could chose error e as a vector from e+C with minimal weight (such vector can be non-unique). A vector x  a+C with a minimal weight is called coset leader.

72. Syndrome decoding Syndrome decoding: 1) precompute array of syndromes and their coset leaders, 2) if vector y is received, compute syn(y), locate its coset and coset leader e, 3)decode y as x = y  e.

73. Syndrome decoding - example [Adapted from V.Pless]

74. Syndrome decoding Complexity?? “Brute force” method (precompute all codewords (2 k of them) and for the received vector find the closest codeword): Time: ~ 2 k n Memory: ~ 2 k (k+n) Syndrome decoding: Time (precomputing):~ 2 n (k+n) Memory: ~ 2 n  k n Time (decoding):~ (k+n) n Depends... but for a range of parameters syndrome decoding has advantages.