A simple algorithm for decoding both errors and erasures of Reed-Solomon codes

A simple algorithm for deco ding b oth errors and era sures of Reed-Solomon co des Sergei V. F edorenko Department of safet y i n in f orma tion systems St.Petersburg State Univ ersity of Aer ospace Instr umentation 1900 0 0, Bol shay a Morsk aia, 67, St.Petersburg, Russia Octob er 200 8 Abstract A simple algo rithm for deco ding b oth errors and erasures of Reed- Solomon co des is describ ed. 1 In tro d u ction In this pa p er, the Ga o algorithm mo dification is giv en. In t he author’s opinion, the suggested algorithm is the simple st f o r algebraic co des with short lengths fo r an y implemen tation. 2 Definiti o ns and notatio n s Let us define the ( n, k , d ) Reed-Solomon co de ov er GF( q ) with length n = q − 1, n um ber of info rmation sym b ols k , designed distance d = n − k + 1, where q is prime p ow er. The message p olynomial of the Reed-Solomon co de is M ( x ) = k − 1 X i =0 m i x i . 1 The comp onen t c i of the co dew ord C ( x ) is computed as c i = M ( α i ) , i ∈ [0 , n − 1] . The receiv ed vec tor is represen t ed as a p olynomial R ( x ) = n − 1 X i =0 r i x i = C ( x ) + E ( x ) = n − 1 X i =0 c i x i + n − 1 X i =0 e i x i , where C ( x ) is the co dew ord, E ( x ) is the error v ector. The error v ector E ( x ) has t errors with a set of error p ositions { i 1 , i 2 , . . . , i t } . Let us define that Z 1 = α i 1 , Z 2 = α i 2 , . . . , Z t = α i t are error lo cations. The error lo cator p olynomial is W ( x ) = t Y i =1 ( x − Z i ) , where t is the n um b er of erro rs, Z i is the erro r lo cation of the erro r v ector E ( x ). The error v ector E ( x ) has l erasures with a set of erasure p ositions S = { j 1 , j 2 , . . . , j l } . X 1 = α j 1 , X 2 = α j 2 , . . . , X l = α j l are erasure lo cations. The erasure lo cator p olynomial is Λ( x ) = l Y i =1 ( x − X i ) , where l is the n um ber o f erasures, X i is the erasure lo cation of the error v ector E ( x ). The inequalit y 2 t + l < d is well known [1]. W e construct an interpolating p o lynomial T ( x ) such that T ( α i ) = r i , i ∈ [0 , n − 1] , where deg T ( x ) < n , and an in terp olating p olynomial T ( x ) such that T ( α i ) = r i , i ∈ [0 , n − 1] \ S, where deg T ( x ) < n − l . 2 3 Existing algorit hms W e describ e here tw o v ersions of the Gao alg orithm [2, 3, 4, 5]. The first v ersion is for deco ding errors only . Let P ( x ) = W ( x ) M ( x ). The k ey equation is      W ( x ) T ( x ) ≡ P ( x ) mo d x n − 1 deg W ( x ) ≤ d − 1 2 maximize deg W ( x ) . (1) The asymptotic complexit y of this algor it hm is O ( n (log n ) 2 ). The second v ersion is for deco ding b oth errors and erasures. The k ey equation is      W ( x ) T ( x ) ≡ P ( x ) mo d x n − 1 Λ( x ) deg W ( x ) ≤ d − l − 1 2 maximize deg W ( x ) . (2) The direct computation by this algorithm has complexit y O ( n 2 ). Next, w e consider the k ey equation deriv ation fo r the T ruong algorithm [6] fo r deco ding b oth errors and erasures. Let Q ( x ) = P ( x )Λ( x ) = W ( x ) M ( x )Λ( x ) . F rom (1) w e hav e W ( x )  ( T ( x )Λ( x )  ≡  P ( x )Λ( x )  mo d x n − 1 and the k ey equation is      W ( x )  ( T ( x )Λ( x )  ≡ Q ( x ) mo d x n − 1 deg W ( x ) ≤ d − l − 1 2 maximize deg W ( x ) . (3) The asymptotic complexit y of this a lg orithm coincides with the complex- it y o f deco ding a lgorithms [2, 3, 4, 5]. 3 4 Sugges ted algorith m W e introduce the following lemma. L emm a: T ( x ) ≡ T ( x ) mo d x n − 1 Λ( x ) . Pr o of: F rom Newton’s inte rp olat io n formula we obtain T ( x ) = x n − 1 Λ( x ) U ( x ) + T ( x ) , where U ( x ) is a p olynomial. F rom (2) and the lemma we get a new k ey equation      W ( x ) T ( x ) ≡ P ( x ) mo d x n − 1 Λ( x ) deg W ( x ) ≤ d − l − 1 2 maximize deg W ( x ) . (4) The description of the three algorithms for deco ding b oth errors a nd erasures is in table 1. 5 Conclus ion The suggested algorithm has replaced t he computation using Newton’s in- terp olation formula by the fast computation of the discrete F ourier transform. The algorithm complexit y is less t ha n the T ruong algo rithm [6] complexit y b ecause the suggested algorithm do es not contain some of the in termediate computations. References [1] R . E. Blahut. Algebr aic Co d es on Lines, Planes, and Curves: An Engi- ne eri ng Appr o ach. Cam bridge, U.K.: Cambridge Univers it y Press, 2008. [2] A. Shiozaki, “Deco ding of redundan t residue p olynomial co des using Eu- clid’s algorithm,” IEEE T r ans. Inf. The ory , vol. IT–34, no. 5, pp. 13 51– 1354, Sep. 1988. 4 T able 1 Algorithms for deco ding b oth erro r s and erasures Step Gao’s algorithm T ruong’s alg orithm Suggested alg orithm 0 — Λ( x ) — 1 T ( x ) T ( x ) T ( x ) 2a x n − 1 Λ( x ) T ( x )Λ( x ) x n − 1 Λ( x ) 2b            W ( x ) T ( x ) ≡ P ( x ) mo d x n − 1 Λ( x ) deg W ( x ) ≤ d − l − 1 2 maximize deg W ( x )          W ( x )  ( T ( x )Λ( x )  ≡ Q ( x ) mo d x n − 1 deg W ( x ) ≤ d − l − 1 2 maximize deg W ( x )            W ( x ) T ( x ) ≡ P ( x ) mo d x n − 1 Λ( x ) deg W ( x ) ≤ d − l − 1 2 maximize deg W ( x ) 3 M ( x ) = P ( x ) W ( x ) M ( x ) = Q ( x ) W ( x )Λ( x ) M ( x ) = P ( x ) W ( x ) Complexit y O ( n 2 ) O ( n (log n ) 2 ) O ( n (log n ) 2 ) 5 [3] S. Gao, “A new algor it hm for deco ding Reed-Solomon co des,” in Commu- nic a tions, I nformation and Network Se curity , V. Bhargav a, H. V. Poo r, V. T a rokh, and S. Y o o n, Eds. Norw ell, MA: Klu w er, 2003, v ol.712, pp.55– 68. [4] S. V. F edorenk o, “A simple algorithm fo r decoding Reed-Solomon co des and its relation t o the W elc h-Berlek amp algorithm,” IEEE T r ans. Inf. The ory , v ol. IT–51, no. 3, pp. 1 196–119 8 , Mar. 2 005. [5] S. V. F edorenk o, “Correction to “A simple algorithm for deco ding Reed- Solomon co des and its relation to the W elc h-Berlek amp algorithm,” IEEE T r ans. Inf. The ory , v ol. IT–52, no. 3, p. 12 7 8, Mar. 2006 . [6] T.-C. Lin, P . D. Chen, and T. K. T ruong, “Simplified pro cedure for de- co ding nonsystematic Reed-Solomon co des o v er G F(2 m ) using Euclid’s algorithm a nd the fast F o urier transform,” IEEE T r ans. on Com mun. , accepted for publication, 2008. 6