A STUDY OF HAMMING CODES AS ERROR CORRECTING CODES

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1 AGU Intenational Jounal of Science and Technology A STUDY OF HAMMING CODES AS ERROR CORRECTING CODES Ritu Ahuja Depatment of Mathematics Khalsa College fo Women, Civil Lines, Ludhiana , Punjab, (India) ABSTRACT Hamming codes ae the eo coecting codes that can coect one eo in a block code of binay symbols. The codes wee developed by Richad Hamming in the late 1950s at the time, when paity checking was being used to detect eos but was unable to coect any eos. The single eo coecting binay hamming codes and thei single eo coecting and double eo detecting extended vesions maked the beginning of coding theoy. Hamming codes emain impotant to the day as they ae still widely used in computing, telecommunications and othe applications including data compession and tubo codes. The pape eviews the popeties of Hamming codes and thei eo coecting capabilities. Keywods: eo coection, block codes, paity check. I.INTRODUCTION Eo-coecting codes ae used in a vaiety of communication systems fo the pupose of identifying and coecting eos in a tansmitted message. Hamming codes ae an impotant class of codes. They ae easy to encode and decode. This pape focuses on popeties of -ay and binay linea codes. In binay code, the messages ae encoded in blocks of bits, called code wods, and any modulo-2 linea combination of code wods is also a code wod. Definition. A [n,k] linea code ove a finite field of ode is just a k-dimensional subspace of V(n,). So, a subset C of V(n,) is a linea code iff u+v, au ɛ GF(). In paticula, a binay code is linea if and only if the sum of any two code wods is a code wod. Example C= {000,011,101,110} is a [3,2] linea code. Definition. A kxn matix G whose ows fom a basis of linea [n,k] code is called geneato matix of the code. G can be tansfomed to the standad fom [I k,a] whee I k is the kxk identity matix and A is a kxn matix. Encoding: Messages can be identified with tuples of V(k,). A message vecto u=u 1 u 2 u k is encoded by post multiplying it by G. X = ug= x 1 x 2...x k x k+1..x n If G is in standad fom, fist K symbols of code wod epesent the message itself: x 1 = u 1, x 2 = u 2 x k =u k. The next n-k symbols ae check symbols. 35 P a g e

2 AGU Intenational Jounal of Science and Technology Definition. The (n-k)xn matix H whose ows ae the paity checks on the code wods is called paity check matix. Given paity check matix H, Hx T = 0 fo all x ɛc So, C= {x ɛ V(n,)} / Hx T = 0} Hence, Paity check matix completely specifies a linea code. we can now define the -ay Hamming code: Definition: (-ay Hamming Code): Given an intege 2, Let n 1 and H be an n matix whose 1 columns ae made of exactly one non-zeo vecto fom each vecto subspace of dimension 1 in v(,). Then, the code having H as its paity check matix is a linea [n, n-] code, called -ay Hamming Code H (,). As is evident fom definition, any two columns of H ae linealy independent. Columns of H may be taken in any ode, the code H(,) is, fo given edundancy, is well-defined upto euivalence of codes i.e. all the -ay Hamming codes of a given length ae euivalent. Binay Case: The columns of a paity-check matix H fo the binay Hamming code H(,2) ae the distinct nonzeo vectos of V(n,2). H is 2-1 matix. Example: 1. Binay H(2,2) code has paity check matix. H , which on educing to standad fom is 1]. So, H(2,2) is just binay tiple epetition code. 2. = 3, Paity check matix fo Hamming Code H(3,2) is in standad fom So, the geneato matix is G It is easily seen that H(3,2) is euivalent to pefect (7, 4, 3) Code , which shows geneato matix = [1 1 which may be educed To establish the minimum distance of Hamming Codes, a fundamental elationship between minimum distance of a linea code and linea independence of columns of paity check matix is as follows: Theoem 1: Fo a [n, k] linea code ove GF() with paity check matix H, the minimum distance of C is d if an only if any d-1 columns of H ae linealy independent but some d columns ae linealy dependent. Poof: It is known that, smallest of the weight of non-zeo codewods is the minimum distance of C. Let x = x- 1x 2 x n be a vecto in V(n,). Then x C <=> xh T = 0 36 P a g e

3 AGU Intenational Jounal of Science and Technology <=> x 1 H 1 + x 2 H x n H n = 0 Whee, H i ae columns of H (i = 1, 2,. N). So, fo evey codewod x of weight d, thee is a set of d linealy independent columns of H Also, if thee exist d 1 linealy dependent columns of H say H i1, H i2,.. H id-1, then thee would exist scalas x i1, x i2, x id-1, not all zeo, such that x i1 H i1 + x i2 H i2 + + x id-1 H id-1 = 0 but then x = (0. 0x i1 0..0x i2.0x id-1 0.0), having x ij in ij th position and Os elsewhee, satisfies xh T = 0 and has weight less than d. Theoem 2: Fo positive intege 2 and n i) has dimensions k = n ii) iii) Poof: 1, the -ay Hamming code H (,). 1 has minimum distance 3, Hence, is a single eo coecting code. is a pefect code. (i) Since the paity check matix H fo H(, ) is an n matix, so, the dimensions of H(, ) is n- (ii) By definition, any two columns of H ae linealy independent. Also H contains columns ( ) t, ( ) t, and ( ) t, which fom linealy dependent set. So, by above theoem, minimum distance of H (, ) is 3 and is a single eo coecting code. (iii) A pefect code is a -ay (n, M, 2t+1) code such that t M k 0 k k n 1 and M = A = (n, 2t+1) H(,) is a (n, M, 3)-code with n 1 and M 1 n With t = 1, left hand side of e. (1) gives, n- (1+n (-1)) = n- ( ) So, H(,) is a pefect code. In othe wods, if n 1 fo some intege 2, then, 1 A (n, 3) = n = n = ight hand side of (1) II.DECODING WITH A Q-ARY HAMMING CODE A Hamming Code is a pefect single eo coecting code. So, the coset leades (diffeent fom 0) ae exactly the vectos of weight 1. The numbe of these coset leades 1 ( 1). n 1 1 ( 1) 1 37 P a g e

4 AGU Intenational Jounal of Science and Technology The syndome of such coset leade (0 0 b0 0) having enty say b at j th place is S (0 0b0.0) = (0 0b0 0)H T = bhj T whee Hj denotes the j th column of H decoding scheme is as follows: 1. Given a eceived vecto y, calculate syndome S (y) = yh T 2. If S(y) = 0, assume no eos in eceived vecto. 3. If S(y) 0, It is a non zeo vecto of V (, ). i.e. S(y) = bh j T fo some b an dj. In this case, assumed single eo is coected by subtacting b fom the j th enty of y. Example 1: A paity check matix fo H (3, 2) in binay ascending fom (i.e. lexicogaphical ode) is H The eceived vecto has syndome [1 1 0], which coesponds to sixth column of H, so that single eo should have occued at sixth position, Thus tansmitted codewod is Fo non binay Hamming Code, computed syndome should be compaed with all non-zeo multiples of the columns of Paity Check matix. Example 2: Suppose = 5 and H Let the eceived vecto is y = Then syndome S(y) = (2,3) = (2,8) = 2(1,4) which means an eo of 2 has occued at the sixth position and decode vecto as x = y a = = Example 3: Suppose = 3 and a paity check matix H (2, 3) is H If the vecto y = 2011 is eceived, it has syndome S(y) = [2 2] = 2[1 1]. So, assume single eo of 2 has occued in the thid position and the decoded vecto will be x = y e = = 2021 Extended Binay Hamming Code It is possible to extend existing codes by one o moe digits to obtain an extended code. The extended code may have stonge eo detection/coection capabilities. Def. The extended binay Hamming Code, H, 2 is the code obtained fom H (, 2) by adding an oveall paity check. Popeties of extended binay Hamming Code 1. Extended H, 2 is a binay [2, 2 1, 4] linea code 2. Let H be a paity check matix fo H (, 2). Then a paity check matix H fo extended code H (, 2) is 38 P a g e

5 AGU Intenational Jounal of Science and Technology 0. H. H Poof: 1) We know that if C is an [n, M, 2t + 1] binay code, then afte adding a paity check digit, we get [n, M, 2t + 2] binay code C. So, in this case, minimum distance is inceased fom 3 to 4. Moeove, lineaity of C implies that C is also linea. 2) Let G be a geneato matix of H (, 2), so, GH T = 0 Addition of a paity check digit gives a geneato matix G [ G, C] of H (,2), Clealy, H has the ight size as a paity check matix fo H (,2) and its ows ae linealy independent. Also, t t H t t t G H [ G, C] [ GH, G1 ] [0 0] Since, all the ows of Decoding G have even weights Having even minimum distance 4, H (, 2) is suited fo incomplete decoding fo it can simultaneously coect any single eo and detect any double eo. Taking columns of H in lexicogaphical ode, a neat incomplete decoding algoithm fo n = 3 follows. Example: A paity check matix fo H (3, 2) is H The syndome of eo vecto (with 1 in the ith place) is just the tanspose of the ith column of H 1. Calculate syndome S(y) = y H say S(y) = (t 1, t 2, t 3, t 4 ) 2. If t 4 = 0 and (t 1, t 2, t 3 ) = 0, assume no eo. 3. If t 4 = 0 and (t 1, t 2, t 3 ) 0, atleast two eos have occued. 4. If t 4 = 1 and (t 1, t 2, t 3 ) = 0, a single eo at the fouth position is assumed. 5. If t 4 = 1 and (t 1, t 2, t 3 ) 0, assume single eo in jth position whee j coesponds to the numbe with binay epesentation as (t 1, t 2, t 3 ) III.SUMMARY Hamming Code ove a -alphabet wee intoduced and thei popeties eviewed. Hamming Codes ae linea codes uniuely defined upto euivalence by thei paametes. A constuction of paity check matix of -ay 39 P a g e

6 AGU Intenational Jounal of Science and Technology linea code of edundancy and minimum distance d by finding a set of vecto of V(,) such that any d-1 of them ae L.I. was done. Though it is easy to wite such set of N vectos fo d = 3 of any size, we desie, upto a maximum value of 1, finding the maximum possible numbe of vectos in V (,) such that any d 1 ae 1 linealy independent is not easy fo d 4 and is vey little known except fo cases 4. The discussion of extended Hamming Code is applicable only to the binay case as in geneal, an oveall paity check cannot be added to a -ay code, so as to suely esult in an incease in the minimum distance. REFERENCES [1.] Raymond Hill A fist couse in coding theoy Claendon pess Oxfod [2.] Shu Lin, Castello Daniel J. Eo Contol Coding [3.] [4.] [5.] U.K. Kuma, Umashanka B.S. Impoved hamming code fo eo detection and coection. Wieless- Pevasive Computing, 2007, ISWPC 07, 2 nd Intenational Symposium. 40 P a g e