Decoding Procedure for BCH, Alternant and Goppa Codes defined over Semigroup Ring

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1 Decoding Procedure for BCH, Alternant and Goppa Codes defined over Semigroup Ring Antonio Aparecido de Andrade Department of Mathematics, IBILCE, UNESP, , São José do Rio Preto, SP, Brazil Tariq Shah and Atlas Khan Department of Mathematics, Quaid-i- Azam University Islamabad, Pakistan and Abstract: In this paper we present a decoding principle for BCH, Alternant and Goppa codes constructed through a semigroup ring, which is based on modified Berlekamp-Massey algorithm This algorithm corrects all errors up to the Hamming weight t r, ie, whose minimum Hamming distance is 2r + 1 Key-word: Semigroup rings, BCH codes, Alternant codes, Goppa codes, modified Berlekamp- Massey algorithm 1 Introduction In this work we address the problem of decoding BCH, alternant and Goppa codes constructed through a semigroup ring, as defined by Shah et all [9], by using the modified Berlekamp- Massey algorithm [6] The decoding procedure for these codes consists of four major steps: (1) calculation of the syndromes, (2) calculation of the error-locator polynomial, (3) calculation of the error-location numbers, and (4) calculation of the error magnitudes This paper is organized as follows In section 2, we present basic facts of semigroups and semigroup rings In the section 3, we review the constructions of BCH and alternant codes through semigroup rings and the construction of Goppa codes through semigroup rings instead of polynomial rings and quoted results already established in [9] In section 4, we present an efficient decoding procedure for BCH, alternant and Goppa codes which makes use of the modified Berlekamp-Massey algorithm 2 Basic Results In this section, we review the key properties of commutative semigroup and semigroup ring [5] Let (S, ) be a commutative semigroup and (B, +, ) is a commutative associative ring Let J be the set of all finitely nonzero functions f from S into B Suppose J is a ring with respect to binary operations addition and multiplication defined as; (f + g)(s) = f(s) + g(s) and (fg)(s) = f(t)g(u), where the symbol indicates that the sum is taken over all t u=s pairs (t, u) of elements of S such that t u = s and it is understood that in the situation where s is not expressible in the form t u for any t, u S, then (fg)(s) = 0 J is known as semigroup ring of S over B If S is a monoid, then J is called monoid ring This ring J is represented as B[S] whenever S is a multiplicative semigroup and elements of J are written t u=s 8

2 either as f(s)s or as s S n f(s i )s i The representation of J will be B[X; S] whenever S is an additive semigroup As there is an isomorphism between additive semigroup S and multiplicative semigroup {X s : s S}, so a nonzero element f of B[X; S] is uniquely represented in the canonical form n f(s i )X s i = n f i X s i, where f i 0 and s i s j for i j The concepts of degree and order are not generally defined in semigroup rings but if we consider S to be a totally ordered semigroup, we can define the degree and order of an element of semigroup ring B[X; S] in the following manner; if f = n f i X s i is the canonical form of the nonzero element f R[X; S], where s 1 < s 2 < < s n, then s n is called the degree of f and we write deg(f) = s n and similarly the order of f is written as ord(f) = s 1 Now, if R is an integral domain, then for f, g B[X; S], the deg(fg) =deg(f)+deg(g) and ord(fg) =ord(f)+ord(g) The polynomial ring B[X] = B[X; Z 0 ] is the particular case of semigroup ring Obviously B[X; Z 0 ] B[X; 1 2 Z 0] Furthermore it is noticed that in B[X; 1 2 Z 0] we can define the degree of a pseudo polynomial as the monoid 1 2 Z 0 is totally ordered Goppa codes are the famous subclass of Alternant codes and they can be expressed in terms of polynomials called Goppa Polynomials [4] In [9] we have introduced the construction technique of BCH, Alternant and Goppa codes through a semigroup ring B[X; 1 2 Z 0] instead of a polynomial ring and by this new way we adopted the same lines as in [2] and improved the several results of [2] That is, we considered B as a finite commutative ring with unity and in the same spirit of [2], we fixed a cyclic subgroup of group of units of the factor ring B[X; 1 2 Z 0]/(X n 1) The factorization of X 2s 1 over the group of units of B[X; 1 2 Z 0]/(X n 1) was the main problem as for B[X]/(X n 1), the X s 1 has in [2] 3 BCH, Alternant and Goppa Codes In this section we review the constructs of BCH, alternant and Goppa codes through a semigroup ring instead of a polynomial ring First we discuss the basic properties of Galois extension rings, which is used in the construction of these codes In this section we assume that (B, N) denotes a finite local commutative ring with unity and residue field K = B N = GF (p m ), where p is a prime, m a positive integer The natural projection π : B[X; 1 2 Z 0] K[X; 1 2 Z 0] is defined by π(a(x 1 2 )) = a(x 1 2 ), ie, π( n i=0 a i X 1 2 i ) = n i=0 a i X 1 2 i, where a i = a i + N) Let f(x 1 2 ) be a monic pseudo polynomial of degree t in B[X; 1 2 Z 0] such that π(f(x 1 2 )) is irreducible in K[X; 1 2 Z 0] Since [5, Theorem 72] accommodate B[X; 1 2 Z 0] as B[Z 0 ], therefore f(x 1 2 ) also is irreducible in B[X; 1 2 Z 0], by [7, Theorem XIII7] Take R = B[X; 1 2 Z 0] (f(x 1 2 )) Then R is a finite commutative local factor semigroup ring with unity and again [5, Theorem 72] accommodate our notions to say that it is a Galois ring extension of B with extension degree t Its residue field is K 1 = R = GF (p mt ), where N N 1 is the maximal ideal of R, and K 1 1 is the multiplicative group of K 1 whose order is p mt 1 Let F denotes the multiplicative group of units of R It follows that F is an abelian group, and therefore it can be expressed as a direct product of cyclic groups We are interested in the maximal cyclic group of F, hereafter denoted by G 2s, whose elements are the roots of X 2s 1 for some positive integer s There is only one maximal cyclic subgroup of F having order 2s = 2 ( p mt 1 ) Definition 31 [9] A shortened BCH code C(2n, η) of length 2n 2s is a code over B that has 9

3 parity check matrix α 1 α 2 α 2n α1 2 α2 2 α2n 2 α1 2r α2 2r α2n 2r for some r = 2(n k) 1, where η = (α 1, α 2,, α 2n ) = (α k 1, n ) is the locator vector, consisting of distinct elements of G 2s The code C(2n, η), with 2n = 2s, will be called a BCH code Theorem 31 [9] The minimum Hamming distance of a BCH code C(2n, η) satisfies d 2r+1 Example 31 Let B = GF (2)[i] and R = B[X; ( 1Z 2 ) 0], where f(x 1 2 ) = f(x 1 2 ) (1) ( ) X 1 6 ( ) 2 + X is irreducible over B Let X 1 2 = Y, we have f(y ) = Y 6 + Y If α 1 2 is a root of f(y ), then α 1 2 generates a cyclic group G 2s of order 2s = 2 ( ) = 14 Let η = (1, α, α 3 2, α 2, α 5 2, α 3, α 7 2, α 4, α 9 2, α 5, α 11 2, α 6 ) (2) be the locator vector of distinct elements of G 2s If r = 2, then the following matrix 1 α α 3 2 α 2 α 5 2 α 3 α 7 2 α 4 α 9 2 α 5 α 11 2 α 6 1 α 2 α 3 α 4 α 5 α 6 1 α α 2 α 3 α 4 α 5 1 α 3 α 9 2 α 6 α 1 2 α 2 α 7 2 α 5 α 13 2 α α 5 2 α 4 1 α 4 α 6 α α 3 α 5 1 α 2 α 4 α 6 α α 3 is the parity-check matrix of a BCH code C(12, η) of length 12 and, by Theorem 31, the minimum Hamming distance is at least 5 Definition 32 A shortened alternant code C(2n, η, ω) of length 2n 2s is a code over B that has parity check matrix ω 1 ω 2 ω 2n ω 1 α 1 ω 2 α 2 ω 2n α 2n ω 1 α1 2 ω 2 α2 2 ω 2n α2n 2 (4) ω 1 α1 2r 1 ω 2 α2 2r 1 ω 2n α2n 2r 1 where r is a positive integer, η = (α 1, α 2,, α 2n ) = (α k 1, n ) is the locator vector, consisting of distinct elements of G 2s, and ω = (ω 1, ω 2,, ω 2n ) is an arbitrary vector consisting of elements of G 2s Theorem 32 [9] The alternant code C(2n, η, ω) has minimum Hamming distance d 2r + 1 Example 32 Let B = GF (2)[i] and R = B[X; 1 2 Z 0] (3) (f(x 1 2 )), where f(x 1 2 ) = (X 1 2 ) 6 + (X 1 2 ) is irreducible over B Let X 1 2 = Y, we have f(y ) = Y 6 + Y If α 1 2 is a root of f(y ), then α 1 2 generates a cyclic group G 2s of order 2s = 2 ( ) = 14 Let η = (α 13 2, α 6, 1, α 11 2, α 5, α 9 2, α 4, α 7 2, α 3, α 5 2 ) (5) be the locator vector and ω = (α 5, α 9 2, 1, α 4, α 7 2, α 3, α 5 2, α 2, α 3 2, α) If r = 2 then the following matrix α 5 α α 4 α 7 2 α 3 α 5 2 α 2 α 3 2 α α 1 1 α 5 2 α 3 2 α 1 2 α 5 2 α 2 α 1 α 4 α α α 3 α 5 1 α 2 α 1 2 α 6 (6) 1 α α 3 α α 5 2 α 1 2 α 2 1 α

4 is the parity-check matrix of an alternant code C(10, η, ω) of length 10 and, by Theorem 32, the minimum Hamming distance at least 5 Goppa codes are described in terms of polynomials, known as Goppa polynomials In contrast to cyclic codes, where it is difficult to estimate the minimum Hamming distance d from the generator polynomial Goppa codes have the property that d deg(h(x)) + 1, where h(x) is a Goppa polynomial Let α 1 2 be a primitive element of the cyclic group G 2s, where 2s = 2 ( p mt 1 ) Let h(x) = h 0 + h 1 X + h 2 X h 2r X 2r be a polynomial with coefficients in R and h 2r 0 Let T = {α 1, α 2,, α 2n } be a subset of distinct elements of G 2s such that h(α i ) are units from R for i = 1, 2,, 2n Definition 33 [9] A shortened Goppa code C(T, h) of length 2n 2s is a code over B that has parity-check matrix of the form h(α 1 ) 1 h(α 2n ) 1 α 1 h(α 1 ) 1 α 2n h(α 2n ), (7) α1 2r 1 h(α 1 ) 1 α2n 2r 1 h(α 2n) where r is a positive integer, η = (α 1, α 2,, α 2n ) = (α k 1, n ) is the locator vector, consisting of distinct elements of G 2s and ω = ( h(α 1 ) 1,, h(α 2n ) 1) is a vector consisting of elements of G 2s Theorem 33 [9] The Goppa code C(T, h) has minimum Hamming distance d 2r + 1 Example 33 Let B = GF (2)[i] and R = B[X; 1 2 Z 0] (f(x 1 2 )) where f(x 1 2 ) = (X 1 2 ) 6 + (X 1 2 ) is irreducible over B Let X 1 2 = Y, we have f(y ) = Y 6 + Y If α 1 2 is a root of f(y ), then α 1 2 generates a cyclic group G 2s of order 2s = 2 ( ) { = 14 Let h(x) = X 4 + X 3 + 1, T = α, α 5, α 2, 1, α 3, α 7 2, α 4, α 6} and ω = { α 5, α 3, α 3, 1, α 6, 1, α 6, α 5} If r = 2 then the following matrix α 5 α 3 α 3 1 α 6 1 α 6 α 5 α 6 α α 5 1 α 2 α 7 2 α 4 α 4 1 α α α 3 (8) α α 4 α 2 1 α 1 α 4 α 2 is the parity check matrix of a Goppa code over B of length 8 and, by Theorem 33, the minimum Hamming distance at least 5 4 Decoding Procedure In this section we present a decoding algorithm for BCH, alternant and Goppa codes This algorithm is based on the modified Berlekamp-Massey algorithm [6] which corrects all errors up to the Hamming weight t r, ie, whose minimum Hamming distance is 2r +1 The complexity of the proposed decoding algorithm is essentially the same as that these codes are defined over finite fields Let B, R and G 2s as defined in previous section Let α 1 2 be a primitive element of the cyclic group G 2s, where s = 2(p mt 1) Let c = (c 1, c 2,, c 2n ) be a transmitted codeword and b = (b 1, b 2,, b 2n ) be the received vector Thus the error vector is given by e = (e 1, e 2,, e 2n ) = b c Let η = (α 1, α 2,, α 2n ) = (α k 1, n ) be a vector over G 2s Suppose that ν t is the number of errors which occurred at locations x 1 = α i1, x 2 = α i2,, x ν = α iν with values y 1 = e i1, y 2 = e i2,, y ν = e iν 11

5 1 BCH code C(2n, η) The syndrome values s l of an error vector e = (e 1, e 2,, e 2n ) is defined as 2n e j α l j, l 1 Since s = (s 1, s 2,, s 2r ) = bh t = eh t, then the first 2r syndrome values s l can be calculated from the received vector b as 2n e j α l j = 2n b j α l j, l = 1, 2,, 2r 2 Alternant code C(η, ω) The syndrome values s l of an error vector e = (e 1, e 2,, e 2n ) is defined as 2n e j w j α l j, l 0 Since s = (s 0, s 1,, s 2r 1 ) = bh t = eh t, then the first 2r syndrome values s l can be calculated from the received vector b as 2n e j w j α l j = 2n b j w j α l j, l = 0, 1, 2,, 2r 1 3 Goppa code C(T, h) The syndrome values s l of an error vector e = (e 1, e 2,, e 2n ) is defined as 2n e j h(α j ) 1 α l j, l 0 Since s = (s 0, s 1,, s 2r 1 ) = bh t = eh t, then the first 2r syndrome values s l can be calculated from the received vector b as 2n e j h(α j ) 1 α l j = 2n b j h(α j ) 1 α l j, l = 0, 1, 2,, 2r 1 The elementary symmetric functions σ 1, σ 2,, σ ν of the error-location numbers x 1, x 2,, x ν are defined as the coefficients of the polynomial ν ν σ(x) = (x x i ) = σ i x ν i, i=0 where σ 0 = 1 Thus, the decoding algorithm being proposed consists of four major steps: Step 1 - Calculation of the syndrome vector s from the received vector Step 2 - Calculation of the elementary symmetric functions σ 1, σ 2,, σ ν from s, using the modified Berlekamp-Massey algorithm [6] Step 3 - Calculation of the error-location numbers x 1, x 2,, x ν from σ 1, σ 2,, σ ν, that are roots of σ(x) Step 4 - Calculation of the error magnitudes y 1, y 2,, y ν from x i and s, using Forney s procedure [3] 12

6 Now, each step of the decoding algorithm is analyzed There is no need to comment on Step 1 since the calculation of the vector syndrome is straightforward The set of possible errorlocation numbers is a subset of {α 0, α 1,, α 2s 1 } In Step 2, the calculation of the elementary symmetric functions is equivalent to finding a solution σ 1, σ 2,, σ ν, with minimum possible ν, to the following set of linear recurrent equations over R s j+ν + s j+ν 1 σ s j+1 σ ν 1 + s j σ ν = 0, j = 0, 1, 2,, (2r 1) ν, (9) where s 0, s 1,, s 2r 1 are the components of the syndrome vector We make use of the modified Berlekamp-Massey algorithm to find the solutions of Equation (9) The algorithm is iterative, in the sense that the following n equations (called power sums) s n σ (n) 0 + s n 1 σ (n) s n ln σ (n) = 0 s n 1 σ (n) 0 + s n 2 σ (n) s n ln 1σ (n) = 0 s ln+1σ (n) 0 + s ln σ (n) s 1 σ (n) = 0 are satisfied with as small as possible and σ (0) 0 = 1 The polynomial σ (n) (x) = σ (n) 0 + σ (n) 1 x + + σ(n) x n represents the solution at the n-th stage The n-th discrepancy is denoted by d n and defined by d n = s n σ (n) 0 + s n 1 σ (n) s n ln σ (n) The modified Berlekamp- Massey algorithm for commutative rings with identity is formulated as follows The inputs to the algorithm are the syndromes s 0, s 1,, s 2r 1 which belong to R The output of the algorithm is a set of values σ i, i = 1, 2,, ν, such that Equation (9) holds with minimum ν Let σ ( 1) (x) = 1, l 1 = 0, d 1 = 1, σ (0) (x) = 1, l 0 = 0 and d 0 = s 0 be the a set of initial conditions to start the algorithm as in Peterson [8] The steps of the algorithm are: 1 n 0 2 If d n = 0, then σ (n+1) (x) σ (n) (x) and +1 and to go 5) 3 If d n 0, then find m n 1 such that d n yd m = 0 has a solution y and m l m has the largest value Then, σ (n+1) (x) σ (n) (x) yx n m σ (m) (x) and +1 max{, l m +n m} 4 If +1 = max{, n + 1 } then go to step 5, else search for a solution D (n+1) (x) with minimum degree l in the range max{, n + 1 } l < +1 such that σ (m) (x) defined by D (n+1) (x) σ (n) (x) = x n m σ (m) (x) is a solution for the first m power sums, d m = d n, with σ (m) 0 a zero divisor in R If such a solution is found, σ (n+1) (x) D (n+1) (x) and +1 l 5 If n < 2r 1, then d n = s n + s n 1 σ (n) s n ln σ (n) 6 n n + 1; if n < 2r 1 go to 2); else stop The coefficients σ (2r) 1, σ (2r) 2,, σ ν (2r) satisfy Equation (9) At Step 3, the solution to Equation (9) is generally not unique and the reciprocal polynomial ρ(z) of the polynomial σ (2r) (z) (output by the modified Berlekamp-Massey algorithm), may not be the correct error-locator polynomial (z x 1 )(z x 2 ) (z x ν ), where x j = α k i, for j = 1, 2,, ν and i = 1, 2,, 2n, are the correct error-location numbers Thus, the procedure for the calculation of the correct error-location numbers is the following: compute the roots z 1, z 2,, z ν of ρ(z); 13

7 among the x i = α k j, j = 1, 2, 2n, select those x i s such that x i z i are zero divisors in R The selected x i s will be the correct error-location numbers and each k j, for j = 1, 2,, 2n, indicates the position j of the error in the codeword At Step 4, the calculation of the error magnitude is based on Forney s procedure [3] The error magnitude is given by ν 1 l=0 y j = σ jls ν 1 l E ν 1 j l=0 σ jlx ν 1 l, (10) j for j = 1, 2,, ν, where the coefficients σ jl are recursively defined by σ j,i = σ i + x j σ j,i 1, i = 0, 1,, ν 1, starting with σ 0 = σ j,0 = 1 The E i = 1 for BCH code, E j = w ij for alternant code and E j = h(x i ) 1 for Goppa code, for i = 1, 2,, ν, are the corresponding location of errors in the vector w It follows from [1, Theorem 7] that the denominator in Equation (10) is always a unit in R Example 41 As in Example 4, if the received vector is given by b = (0, i, 0, 0, 0, 0, 0, 0), then the syndrome vector is given by s = bh t = (iα 3, iα, iα 6, iα 4 ) Applying the modified Berlekamp- Massey algorithm, we obtain that σ (4) (z) = 1 + α 5 z The root of ρ(z) = z + α 5 (the reciprocal of σ (4) (z)) is z 1 = α 5 Among the elements of G 2s we have x 1 = α 5 is such that x 1 z 1 = 0 is a zero divisor in R Therefore, x 1 is the correct error-location number, and k 2 = 5 indicates that one error has occurred in the second coordinate of the codeword The correct elementary symmetric function σ 1 = α 4 is obtained from x x 1 = x σ 1 = x α 4 Finally, applying Forney s method to s and σ 1, gives y 1 = i Therefore, the error pattern is given by e = (0, i, 0, 0, 0, 0, 0, 0) Referências [1] AA Andrade and R Palazzo Jr, A note on units of a local finite rings, Revista de Matemática e Estatística, 18 (2000) [2] AA Andrade, R Palazzo Jr, Linear codes over finite rings, Tema - Tend Mat Apl Comput, 6, No 2 (2005) [3] GD Forney Jr, On decoding BCH codes, IEEE Trans Inform Theory, IT-11 (1965) [4] VD Goppa, A new class of linear error-correcting codes, Probl Peredach Inform, 6, No 3 (1970) [5] R Gilmer, Commutative semigroup rings, University Chicago Press, Chicago and London, 1984 [6] JC Interlando, R Palazzo Jr and M Elia, On the decoding of Reed-Solomon and BCH codes over integer residue rings, IEEE Trans Inform Theory, IT-43 (1997) [7] B R McDonald, Finite rings with identity, Marcel Dekker, New York, 1974 [8] WW Peterson and EJ Weldon Jr, Error Correcting Codes, MIT Press, Cambridge, Mass, 1972 [9] Tariq Shah, Atlas Khan and A A Andrade, Encoding through generalized polynomial codes, (Sumitted) 14

Alternant and BCH codes over certain rings

Computational and Applied Mathematics Vol. 22, N. 2, pp. 233 247, 2003 Copyright 2003 SBMAC Alternant and BCH codes over certain rings A.A. ANDRADE 1, J.C. INTERLANDO 1 and R. PALAZZO JR. 2 1 Department