Θεωρία Πληροφορίας και Κωδίκων

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1 Θεωρία Πληροφορίας και Κωδίκων Δρ. Νικόλαος Κολοκοτρώνης Λέκτορας Πανεπιστήμιο Πελοποννήσου Τμήμα Επιστήμης και Τεχνολογίας Υπολογιστών Τέρμα Οδού Καραϊσκάκη, Τρίπολη E mail: nkolok@uop.gr Web: Ενότητα 2 η : εισαγωγή στη θεωρία κωδίκων

2 εισαγωγή γρ. κώδικες κλάσεις σύνοψη. Περιεχόμενα Ομιλίας.1.. Ορισμός του προβλήματος.2.. Εισαγωγή σε γραμμικούς τμηματικούς κώδικες Τι είναι γραμμικός κώδικας? Σύνδρομα και ανίχνευση σφαλμάτων Διορθωτική ικανότητα κώδικα Λανθασμένη αποκωδικοποίηση? Μέθοδοι τυπικής αποκωδικοποίησης.3.. Κατηγορίες απλών κωδίκων.4.. Σύνοψη & βιβλιογραφία 2 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

3 εισαγωγή γρ. κώδικες κλάσεις σύνοψη Πρόβλημα Αξιόπιστης Επικοινωνίας Example ( Binary Symmetric Channel BSC) Given input v = (v 0 v n 1 ), the output r = (r 0 r n 1 ) satis ies r i v i with probability p r i = v i with probability 1 p 1 p 0 0 p p p i = 0,..., n 1 and 0 p 1; what happens if p = 1 2? 3 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

4 Τμηματικοί Κώδικες Let the output of the source be a sequence over F 2 = {0, 1}. In block coding: a sequence is segmented into blocks u of ixed length k the encoder transforms each u into an n-tuple v with n > k there should be 1 1 correspondence between u and v Important things to remember: the set C = {v F n 2 : v = f(u), u F k 2} is a (n, k) block code 2 k different codewords v of length n 2 k different message blocks u of length k Unless f : F k 2 F n 2 has a certain special structure, encoding/decoding will be prohibitively complex for large k, n. 4 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

5 Ορισμός Γρ. Τμηματικών Κωδίκων We restrict our attention to linear block codes that can be mechanized in a practical manner Definition A block code C of length n and 2 k codewords is a linear (n, k) code if and only if its codewords form a k dimensional subspace of F n 2. Q: What does this imply? the modulo 2 sum of two codewords is also a codeword it is possible to ind k LI codewords g 0,..., g k 1 of C such that v = u 0 g u k 1 g k 1, v C (1) where u i F 2 5 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

6 Ορισμός Γρ. Τμηματικών Κωδίκων (συν.) if we write (1) in matrix formulation, we get that v = u g 0.. g k 1 v = u G, v C (2) where u = (u 0 u k 1 ) and G is the k n matrix G = g 0. g k 1 g 0,0 g 0,1 g 0,n 1 =... g k 1,0 g k 1,1 g k 1,n 1 the rows of G generate or span the entire (n, k) linear code C G is called a generator matrix for C the encoder only stores the rows of G to process the input u 6 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

7 Παράδειγμα (7, 4) Κώδικα An (7, 4) linear code C is given by message u codeword v (0000) ( ) (1000) ( ) (0100) ( ) (1100) ( ) (0010) ( ) (1010) ( ) (0110) ( ) (1110) ( ) (0001) ( ) (1001) ( ) (0101) ( ) (1101) ( ) (0011) ( ) (1011) ( ) (0111) ( ) (1111) ( ) The codewords marked with are LI; so the matrix G = is a generator matrix for the (7, 4) linear code C Q: Did you notice any structure in C and G? the right part of v C equals u the right part of G equals I 4 7 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

8 Συστηματικοί Κώδικες The above structure, according to which a codeword is divided into two parts (the message part and the redundant checking part), is desirable Definition A linear block code is called a (n, k) linear systematic block code if the message part consists of the k information digits (unaltered) the redundant checking part consists of n k parity check digits (linear sums of the information digits) Then, we can write G = ( P I k ) where P is an k n k matrix, and v = u G = u (P I k ) = ( u P u Ik ) = ( u P u ) The n k equations u P are the parity check equations of the code C 8 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

9 Κύκλωμα Κωδικοποίησης (n, k) Κώδικα Hence, we have v j = u p t j = k 1 i=0 u i p ij, for all 0 j < n k, which leads to the encoding circuit The encoding complexity is Θ(n), that is linearly proportional to n 9 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

10 Πίνακας Ισοτιμίας Another useful matrix associated with the (n, k) code C is an n k n matrix H called parity check matrix Proposition If C is the (n, k) code generated by G, then v C if and only if Due to the way H is constructed: v H t = 0, v C. (3) the rows of H are LI, that is, rank(h) = n k every vector in the row space of G is orthogonal to the rows of H the code C coincides with the null space of H 10 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

11 Πίνακας Ισοτιμίας (συν.) From (2) and (3) we have (u G) H t = 0 for all u F k 2, which leads to If G = ( P I k ), then H = ( In k P t) Definition ( dual code) G H t = 0. (4) The (n, n k) linear block code generated by (the rows of) H is called the dual code of C and is denoted by C From (4) we obtain (u G) (ũ H) t = 0 for all u F k 2, ũ F n k 2 ; hence: Proposition Let C and C be (n, k) and (n, n k) linear block codes respectively; then C = C if and only if v ṽ t = 0, v C and ṽ C. 11 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

12 . Παράδειγμα (7, 4) Κώδικα (συν.) The parity check matrix (generator matrix of C ) of the (7, 4) linear block code C is given by H = G = The encoding circuit is shown below 12 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

13 Τρόπος Υπολογισμού Συνδρόμου because of the channel noise r v; the vector e = r + v is called error vector the 1 s in e correspond to transmission errors to detect the presence of errors the decoder computes s = r H t (5) which is called the syndrome of r s = 0 if and only if r C (and s 0 if and only if r / C ) if r v and e C, then the error cannot be detected! these error patterns are called undetectable error patterns 2 k 1 undetectable error patterns in this case, the decoder performs a decoding error 13 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

14 Τρόπος Υπολογισμού Συνδρόμου (συν.) Proposition The syndrome s computed via (5) depends only on e, that is s = e H t Proof s = r H t = (v + e) H t = v H t + e H t = e H t Important things to remember: if H = ( I n k P t), then we just have s = r par + r inf P inding the error e from s = e H t is not that simple 2 k different solutions! the decoder tries to ind the most probable error pattern in order to minimize the probability of decoding error this usually leads to assuming a low weight error pattern 14 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

15 Κύκλωμα Υπολογισμού Συνδρόμου (n, k) Κώδικα If C is a systematic (n, k) linear block code, then H = ( I n k syndrome computation circuit is P t) and its due to the above analysis 15 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

16 Παράδειγμα (7, 4) Κώδικα (συν.) Let v = ( ) and r = ( ); then, the receiver computes s = r H t = (111) via the circuit If the channel is a BSC, e = ( ) is the most probable error vector satisfying the preceding equations (has the smallest weight). So, we let v = r + e = ( ) 16 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

17 Ελάχιστη Απόσταση Κώδικα Notation the Hamming weight of vector v F n 2 is de ined as wt(v) = #{0 i < n : v i 0} the Hamming distance of vectors v, w F n 2 is d(v, w) = #{0 i < n : v i w i } = #{0 i < n : v i + w i 0} = wt(v + w) Definition The minimum distance of C, denoted by d min, is de ined as d min = min{d(v, w) : v, w C and v w} 17 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

18 Ελάχιστη Απόσταση Κώδικα (συν.) Theorem If C is a linear block code, then d min = min{wt(v) : v C and v 0} Theorem If C is a linear block code, and v C is such that wt(v) = l, then there exist l columns of H the sum up to 0 (hint: v H t = 0 v C ) Corollary Let C be a linear block code with parity check matrix H if no d 1 or fewer columns of H add to 0, the code has minimum weight at least d the minimum weight (or minimum distance) of C is equal to the smallest number of columns of C that sum to 0 18 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

19 Κατανομή Βάρους Κωδικών Λέξεων Let C be an (n, k) linear code, and de ine the numbers A i = #{v C : wt(v) = i}, 0 i n. (6) Then {A 0,..., A n } is called the weight distribution of C. It may also be represented in polynomial form as follows A(z) = A 0 + A 1 z + + A n z n = n A i z i. A(z) is called the weight enumerator for the (n, k) linear code C. i=0 The weight enumerator B(z) of the dual C satis ies A(z) = 2 (n k)( 1 + z ) ( ) n 1 z B 1 + z which is well known as the MacWilliams identity. (7) 19 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

20 Παράδειγμα (7, 4) Κώδικα (συν.) An (7, 4) linear code C is given by message u codeword v (0000) ( ) (1000) ( ) (0100) ( ) (1100) ( ) (0010) ( ) (1010) ( ) (0110) ( ) (1110) ( ) (0001) ( ) (1001) ( ) (0101) ( ) (1101) ( ) (0011) ( ) (1011) ( ) (0111) ( ) (1111) ( ) The weight distribution of C is A 0 = 1 A 1 = A 2 = 0 A 3 = A 4 = 7 A 5 = A 6 = 0 A 7 = 1 or equivalently A(z) = 1 + 7z 3 + 7z 4 + z 7 as a polynomial d min = 3 20 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

21 Ανιχνευτική Ικανότητα Κώδικα Any pair of codewords of an (n, k) block code C differs in at least d min places. Therefore: error patterns e such that wt(e) = l < d min cannot change v into another codeword r these are called detectable error patterns ( 2 n 2 k ) Property ( error detection) The random error detecting capability of a block code with minimum distance d min is d min 1 Error patterns e that can be detected: all e with wt(e) < d min many e with wt(e) d min 21 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

22 Διορθωτική Ικανότητα Κώδικα Any pair of codewords of an (n, k) block code C differs in at least d min places. Therefore: a received word satis ies d(r, v) + d(r, w) d(v, w) d min correct decoding is performed if d(r, v) < d(r, w), w C with w v (assuming v was transmitted) Property ( error correction) The random error correcting capability of a block code with minimum distance d min is t = (d min 1)/2 Error patterns e that can be corrected: all e with wt(e) t many e with wt(e) > t (a total of 2 n k ) 22 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

23 Πιθανότητα Μη Ανιχνεύσιμου Σφάλματος The probability of an undetected error Pr u (E) occurs only when the error pattern e is identical to a codeword of C = C \ {0}; hence Pr u (E) = Pr ( e C ) = = v C Pr ( e = v ) n A i p i( 1 p ) n i ( ) n = 1 p i=1 n ( p A i 1 p i=1 ) i from which we conclude that Pr u (E) = ( 1 p ) n [ ( p ) ] A 1 1 p (8) where p is the transition probability of the BSC 23 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

24 Πιθανότητα Μη Ανιχνεύσιμου Σφάλματος (συν.) Proposition It holds E C [Pr u (E)] 2 (n k)[ 1 (1 p) n ] Proof Let C be the ensemble of all systematic (n, k) linear block codes; if the code C C is chosen uniformly at random, then E C [Pr u (E)] = n Pr(C ) Pr u (E C ) = p i( 1 p ) n i 1 C C C i=1 n ( ) n 2 (n k) p i( 1 p ) n i i i=1 C C A C i which concludes our proof Since [ 1 (1 p) n ] 1, we also have E C [Pr u (E)] 2 (n k) 24 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

25 Μέθοδος Αποκωδικοποίησης Definition ( decoding) A decoding scheme is a rule to partition the 2 n possible received vectors r into 2 k disjoint subsets D 1,..., D 2 k, so that if r D i r v i where v i C is the only one codeword contained in the subset D i, for 1 i 2 k Solution (standard array: D i = v i + D 1 ) e 1 = 0 = v 1 v 2 v 2 k e 2 e 2 + v 2 e 2 + v 2 k. e 2 n k.. e 2 n k + v 2 e 2 n k + v 2 k 25 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

26 Μέθοδος Αποκωδικοποίησης (συν.) The decoding based on the standard array is the minimum distance decoding (i.e., the maximum likelihood decoding) Theorem All the 2 k n tuples of a coset have the same syndrome; the syndromes for different cosets are different Algorithm ( syndrome decoding or table lookup decoding) compute the syndrome r H t of r locate the coset leader e l whose syndrome is equal to r H t then el is assumed to be the error pattern caused by the channel decode the received vector r into the codeword v = r + e l 26 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

27 Κύκλωμα Αποκωδικοποίησης (n, k) Κώδικα 27 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

28 Παράδειγμα (7, 4) Κώδικα (συν.) The systematic (7, 4) linear block code C of our example has the following standard array ( ): e v 0 v 1 v 14 v 15 s ( ) ( ) ( ) ( ) ( ) (000) ( ) ( ) ( ) ( ) ( ) (100) ( ) ( ) ( ) ( ) ( ) (010) ( ) ( ) ( ) ( ) ( ) (001) ( ) ( ) ( ) ( ) ( ) (110) ( ) ( ) ( ) ( ) ( ) (011) ( ) ( ) ( ) ( ) ( ) (111) ( ) ( ) ( ) ( ) ( ) (101) The number of correctable errors (due to d min = 3) equals the total number of correctable errors (given by 2 n k ): ( 70 ) + ( 71 ) = = 8 28 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

29 Παράδειγμα (7, 4) Κώδικα (συν.) The decoding circuit for the systematic (7, 4) linear block code C will be e 0 = s 0 s 1 s 2, e 1 = s 0 s 1 s 2, e 2 = s 0 s 1 s 2, e 3 = s 0 s 1 s 2, e 4 = s 0 s 1 s 2, e 5 = s 0 s 1 s 2, e 6 = s 0 s 1 s / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

30 Πιθανότητα Λανθασμένης Αποκωδικοποίησης For any t error correcting block code C, the probability Pr(E) that the decoder commits an erroneous decoding is upper bounded by Pr(E) Pr ( wt(e) > t ) = = n i=t+1 n i=t+1 Pr ( wt(e) = i ) ( ) n p i( 1 p ) n i. (9) i An alternative formula is derived if the weights of the coset leaders are known; indeed, if {a 0,..., a n } is the weight distribution, then n Pr(E) = 1 a i p i( 1 p ) n i i=0 (10) where p is the transition probability of the BSC 30 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

31 εισαγωγή γρ. κώδικες κλάσεις σύνοψη Κώδικες Απλής Ισοτιμίας Definition A single parity check (SPC) code C is an (n, n 1) linear block code with one parity check digit Property generator matrix G = ( 1 t n 1 I n 1 ) parity check matrix H = ( 1 1 n 1 ) minimum distance d min = 2 (all codewords have even weight) rate R = 1 1/n Can only be used for simple error detection; errors of even weight are not detectable 31 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

32 εισαγωγή γρ. κώδικες κλάσεις σύνοψη Κώδικες Επανάληψης Definition A repetition code C of length n is an (n, 1) linear block code with two codewords: 0 n and 1 n Property generator matrix G = ( 1 n 1 1 ) parity check matrix H = ( I n 1 ) 1 t n 1 minimum distance d min = n (usually choose n odd) rate R = 1/n It is the dual of the (n, n 1) SPC code 32 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

33 εισαγωγή γρ. κώδικες κλάσεις σύνοψη Αυτο Δυϊκοί Κώδικες Definition An (n, k) linear block code C is self dual if and only if C = C Property generator matrix G = ( ) P I n/2 parity check matrix H = ( I n/2 P t) we have G G t = 0, and for systematic codes P P t = I n/2 rate R = 1/2 n must be even, since k = n k k = n/2 33 / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )

34 εισαγωγή γρ. κώδικες κλάσεις σύνοψη Προτεινόμενη Βιβλιογραφία T. M. Cover and J. A. Thomas Elements of Information Theory John Wiley & Sons, 2006 R. Gallager Information Theory and Reliable Communication John Wiley & Sons, 1968 S. Lin and D. Costello Error Control Coding, 2nd ed. Prentice Hall, / 34 Δρ. Νικόλαος Κολοκοτρώνης Θεωρία Πληροφορίας και Κωδίκων ( Προπτυχιακό )