On Computation of Error Locations and Values in Hermitian Codes

On Computation of Erro r Locations and V alues in Hermitian Codes Rachit Agarwal ∗ (Draft V ersion, Submitted to ITW 2008) November 20, 2007 Abstract W e obtain a techniq ue to reduce the computational complexity associated with de- coding of Hermitian codes. I n pa rticular , we propos e a method to compute the error locations and values using a n uni-variate error locator and an uni-variate error evalu- ator polynomial. T o achieve this, we introduce the notion of Semi-Erasure Decoding of Hermitian codes and prove that decoding of Her mitian codes ca n always be performed using semi-erasure decoding. The central results are: ⋆ Searching for error locations require evaluating an univariate error locator poly- nomial over q 2 points as in Chien search for Reed- Solomon codes. ⋆ Forney’s formula for error value computation in Reed-Solomon codes can di- rectly be applied to compute the e rror values in Hermitian c ode s. The approach develops from the idea tha t transmitti ng a modified form of the infor- mation may be more efficient that the information itself. 1 Introduction The decoding problem in Hermitian codes is sp lit into thr ee main parts, viz. , synd rome computation, computing the e r ror locator p olynomial and computation of locations and error values. The main per s isting problem w ith Hermitian codes is the timing and com- putational complexity associated with the decoding o f the codes. Effic ient computation of error locator polyno mial and error evaluator polynomial have been proposed by Ko etter and O’Sulli van [1]-[5]. Sp e cificall y , the computation of e r ror locator polyn o mials can now be p erformed very ef ficiently [1][2][6]. Besides computing the error evaluator polynomial, most of th e r e search has be en done on computing t he error evaluator polynomial and the values of the errors u s ing Forne y type formula used in Reed-Solomon (RS) codes [7]-[9]. In terms of finding the loca tions of the errors, it has bee n widely believed that given the error locator polynomial, significant impr ovements can not be achieved in terms of finding the error locations. This is d ue to the fact that t he only approach is to find the zeros of the e rror locator p o lynomial by evaluating it at all possible points on the curve. This s earc h p rocess has a high timing and ∗ Rachit Agarwal (ragarwal@cs.ucc.ie) is with Centre for Efficiency-Oriented Languages (CEOL), Depart- ment of Computer Science, University Col l ege Cork, Ireland. 1 computational complexity . Further , te chniques to compute the error values involve either high o rder hasse de rivatives over biva riate polynomials and evaluating those polyno mials at every err or value or to find the err or locator which has a zero of multiplici ty o n e at the error location [10]. This is differ ent to evaluating err or values in R e ed-Solomon (RS) codes where we ge t an univariate error locator polyno mial. In this paper , we pr esent so me further improvements for the computation of err or val- ues from the err or evaluator po lynomial. The proposed technique also gives significant reduction in complexity of computing the error locations given the error locator polyno- mial. I n particular , we obtain a method to e mploy univariate po lynomials which have as roots the locations of the errors and univariate error locator polynomial which allows to use e xactly the F orney’s formula for finding the values o f the errors. T o achieve this, we introduce the no tion of semi-erasur es, which is essentially when the decode r has a partial k nowledge about the e rror locations in the received word. W e give a method to identify partial knowledge about the error locations and show its appli- cabili ty for any random error patt e rn. The p roposed approach is based on an inte resting observation that transmitting a modified form o f the information may r esult in much lower decoding complexity than the ac tual informa tion. The rest of the paper is organized as follows: Section 2 gives the basic d efinitions and code construction. W e derive computability of an auxiliary codeword which we propose to transmit rathe r t han the actual Hermitian codeword in Section 3. W e introduce the notion of semi-erasure decoding in Section 4 and prove its so lvability . W e show the e xistence of an univa riate error locator p olynomial for find ing t he er ror locations in Section 5 and existence of an univaria te error evaluator polynomial in Section 6. Be fore closing the paper , we show that for our construction Forney formula from RS codes can d ir ectly be used fo r Hermitian code s as w ell in Section 5. 2 Code Construction W e conside r codes from a Hermitian curve χ : x q +1 = y q + y over a finite field F q 2 . The sp ace L ( mP ∞ ) consists of all functions o n χ th at have a pole of multipli city at most m only at t he u nique po int at infinity . W e present th e following proposition fr om [11 ][12] without pr oof: Proposition 1: The following set is a basis of L ( mP ∞ ) for each m ≥ 0 h x a y b : aq + b ( q + 1) ≤ m, 0 ≤ a, 0 ≤ b < q i (1) Let y 0 be an e lement of F q 2 that satisfies y 0 + y 0 q = 1 . Then, according to [12], t h e af fine rational points on χ can be represented as ( η , η q +1 y 0 + β j ) , where η ∈ F q 2 and β j , j = 0 , 1 , . . . , q − 1 are the q s olutions in F q 2 for y q + y = 0 . Such a r epr esentation has bee n used in [6] r ecently to develop an e f ficient d e coding algorithm for Hermitian cod e s. A similar , howe ver , a little modified repr esentation of the affine rational p oints has also been used in [13] to d e velop an e fficient encoding algorithm for H ermitian cod es. The 2 latter , shown in (2 ), will play a major role in this paper: P α,β = ( α, α q +1 ( y 0 + β ) + δ ( α ) β ) , (2) where α and β repr esent arbitrary elements in F q 2 and F q respectively and δ is the Kronecker- delta function defined as δ ( α ) =  0 if α 6 = 0 1 if α = 0 Let ǫ be a primi tive e lement in F q 2 and let γ be a primi tive e leme nt in F q . The p o sitions in a codeword are labeled by the corresponding eleme nts α = ǫ i , β = γ j resulting in a code - word as a q × q 2 matrix c . Occasionally we will index elements in this array by element s of t he fields F q 2 and F q , o therwise we index starting with 0 . A H ermitian code C ( m ) may be defin e d as { c ∈ F q 3 q 2 : X α ∈ F q 2 X β ∈ F q c β ,α f ( P α,β ) = 0 , ∀ f ∈ L ( mP ∞ ) } For an in dept h treatment of Hermitian codes, we refer to [11][14][15]. 3 Codeword and T ransmitted W ord The central idea of our approach is that transmitting a modified form of the information may be more efficient (in terms of decoding complexity and err or correcting capability) than t he inf ormation itself. This appr oach, thoug h not very trivial, is fairly easy to formu- late. In t his section, we obtain su ch a modified information for Hermitian cod eword. Le t us call this auxiliary codeword. Let a received word ˜ y = ˜ r + ˜ e be given s uch that all of ˜ y (auxiliary received word), ˜ r (auxilia ry code word) and ˜ e ar e matrix of dimension q × q 2 , and where ˜ e is an error vecto r with Hamming weight t < d ∗ ( C ( m )) / 2 . W e define a “Mapping matrix” M of d imension q × q such that y = M × ˜ y = M × ˜ r + M × ˜ e = r + e . (3) where, r ∈ C ( m ) . Specifically , g iven a q × q 2 matrix ˜ r with columns ˜ r j , we define a q × q 2 matrix r with columns r j as r j = M j × ˜ r j (4) where, M j is chos en from the following array M of matrices of type M and M ′ : M = ( M 0 , M 1 , . . . , M q 2 − 1 ) , M i =  M ′ i = q 2 -1 M otherwise The problem now is to show the ex ist ence of such a mapping matrix and an auxilia ry code- word. Proposition 2 [Computability of A uxiliary Co deword]: Ther e always ex ist Mapp ing Matrices M and M’ . G iven the mapping matrices , ther e always exists an auxiliary codeword for a given Hermitian codeword which sa tisfies (3). 3 Proof: In [13], it is sho wn that given the definition of af fine points (2), the re always exist matrices M − 1 and M ′− 1 which ar e V andermonde matrices. Hence, the y will always have inverses. From [13], t hese inverses of the form (5) and (6) are essentially our mapping matrices. During t he encoding of Hermitian codes using algorithm develop ed in [13], an auxiliary matrix ˜ r is also cons tructed. This proves that such mapping matrices and auxiliary code word always exist. ✷ M =        1 − ( y 0 + 0) q − 1 ( y 0 + 0) q − 2 . . . ( y 0 + 0) 0 1 − ( y 0 + 1) q − 1 ( y 0 + 1) q − 2 . . . ( y 0 + 1) 0 1 − ( y 0 + γ ) q − 1 ( y 0 + γ ) q − 2 . . . ( y 0 + γ ) 0 . . . . . . . . . . . . 1 − ( y 0 + γ q − 2 ) q − 1 ( y 0 + γ q − 2 ) q − 2 . . . ( y 0 + γ q − 2 ) 0        (5) M ′ =        1 0 0 . . . 0 − 1 0 − (1) q − 2 − (1) q − 3 . . . − (1) 1 − 1 0 − ( γ ) q − 2 − ( γ ) q − 3 . . . − ( γ ) 1 − 1 . . . . . . . . . . . . . . . 0 − ( γ q − 2 ) q − 2 − ( γ q − 2 ) q − 3 . . . − ( γ q − 2 ) 1 − 1        (6) 4 Semi-Erasure Decoding Decoding by e r rors and by erasures has been stud ied widely in coding the ory literature, both for RS and Hermitian code s. In this section, we introduce the idea of se mi-erasur e decoding for Hermitian codes and show how it helps reducing the d ecoding complexity of Hermitian codes, sp e cificall y computation of error location and error values g iven th e error locator po lynomial. Definition 1 [Semi-Eras ure Decoding]: Given a Hermitian codeword, a semi-era sur e is defined as the condition w hen knowledge of only one of the coordinates is r equir ed to identify the locatio n of an err or . W e call it semi-erasure because it is neither a complete erasure d ecoding nor a complete error only decoding. Notice that in our definition of affine p oints, if we k now the value of e ither of α or β , o nly t he knowledge of t he other is required to completely de fine the location of t h e point. If during our d ecoding pr ocess, we could ide ntify th e location of either of α o r β , we will cal l it a semi-erasur e decoding. W e give a rathe r t r ivial definition which will simplify the furt her discus s ion: Definition 2: [Column E rror Pattern] We define an err or pattern e i,j as a column err or pa ttern if e α 0 ,β =0 6 = 0 ⇒ e α 0 ,β 6 = 0 ∀ β ∈ { 1 , . . . , q − 1 } The following proposition gives a nece s sary cond ition for the fact that decod ing of Her- mitian code s can al ways be performed using semi-erasur e decod ing. 4 Proposition 3 [Semi-Erasure Decoding Can Always be Forced]: Given any err or pattern introd uced by the channel, transmitting the auxilia ry codeword ˜ r will for ce Semi-Erasur e D ecoding. Proof: Assume we always transmit an auxilia ry codeword ˜ r , the computability of which is shown in Proposition 2. Given any random err or pattern, every error in ˜ r whe n operated over (4) will be converte d into a column error as defined in Definition 2. Hence, we know β for all e rrors o ccurring ( β = 0 , 1 , . . . , q − 1 ). Since β is known, only the knowled ge of α is requir ed to complete ly determine the error location, which is Semi-Erasur e Decoding as defined in Definition 3. ✷ Given that s emi-erasure de coding can always be forced, we ne ed the following theorem, which presents the relation between error weights in ˜ r and r . W e only sk etch the proof here. Theorem 1 [Solvability of Semi-Era sure Dec oding]: The error w eights for the auxiliary receiv ed w ord and corr espo nding Hermitian r eceived w ord are equal. In other words, given a q × q 2 auxiliary rece ived word ˜ y with t uc uncorr ectable erro rs, the err ors can be corre cted by the corr esponding Hermitian code give n that: t uc ≤ d H (7) wher e, d H is th e decoding capabi lity of the Hermitia n code. Proof: (Sketch) In semi-erasure decod ing, only one of the coor dinates has to be solved for . It suf fices to pr ove that the value o f the unknown coor dinate c 0 , c 1 , . . . , c k − 1 can be found, given the error correcting capabil ity is k . The intuition behind the the orem is fairly clear . Given that t h e error correcting capability of the Hermitian code is k , in normal de coding process, we solve equations for 2 k va riables, α s i and β s i , whereas in this case, there are o n ly k variables given th at either all of α s i or β s i ar e k nown. Given the error corr ecting capability of t he code is k , we can always solve for 2 k variabl es in se mi-erasur e de coding. He nce the claim. ✷ This in itself is an interesting result since this g ives us a hint as to why some o f the e rror patterns requir e lesser check symbols for d ecoding of Hermitian codes. This is one of the eff ects no ticed by O’Sulli van [16]. 5 Error Location Computation Once an e rror locator polynomial has been computed in Koe tter ’s decoding algorithm, t he only known way to compute the error locations is t hrough Chien Se arch, which implies computing the value of the error locator polynomial at all pos s ible points on the curve. The er ror locations ar e the n the zeros of the error locator polynomial. 5 This metho d of computing the error locations requir e evaluating a bivariate err or loca- tor polynomial at q 3 points of the curve. E ve n for mod erate number of errors, the computa- tional complexity of this op e ration is very high. In th is s ection, we show how to reduce this complexity to that of R eed-Solomon code s (computing the polynomials at q 2 locations). Theorem 2: [Uni-variate E rror Locator P olynomial] For semi-erasur e deco ding, th e zeros of σ ( x, y ) are equivalent to the zer os of an uni-variate polyno- mial ψ ( x ) . In genera l, ψ ( x ) is of the form: ψ ( x ) = f ( x ) Y k : e k 6 =0 ( x − x k ) (8) wher e f ( x ) 6 = 0 at any error location. Proof: Recall that for semi-erasure decod ing, one o f the coordinates d efining the error po- sitions will be kno wn. For our construction, we know fr om pr oof of Pr oposition 3 th at the values of β for all errors ar e kno wn. Given any bivaria te error locator polynomial σ ( x, y ) (fr om Koett er ’s decoding algorithm, say), we know that zeros of σ ( x, y ) are equivalent to roots o f σ ( x, x q +1 ( y 0 + β ) + δ ( x ) β ) using the definition of our affine points, where β is known. This is an u n ivariate polynomial in x . The zer os o f t his polynomial are the x-coordinate of th e error po sitions, say x 1 , x 2 , . . . , x t . The following statements can be made for ψ ( x ) : • ψ ( x ) must have a term Q k : e k 6 =0 ( x − x k ) , for x 1 , x 2 , . . . , x t ar e t he so lutions. • ψ ( x ) may have a term f ( x ) which ca n not be zero at any error location. • ψ ( x ) can not have a t erm of t h e form ( x − x r ) s for any x and any s ≥ 2 . The firs t two st atements are trivial. W e only s how the third one. A s sume that ψ ( x ) has a term of the form ( x − x r ) s for some s ≥ 2 . T his will have multiple solutions at x = x r . Given such a ψ ( x ) , there will always exist a ψ ′ ( x ) which has a solution at x = x r and will be of the form ψ ′ ( x ) = ψ ( x ) / ( x − x r ) s − 1 and hence, lower pole order than ψ ( x ) . Since decoding algorithms compute an error locator with the lowes t possible p ole order , s uch a ψ ′ ( x ) is not possible. T h e n, by contradiction, ψ ( x ) can not have a te rm of the for m ( x − x r ) s for s ome s ≥ 2 . ✷ Having discuss ed the uni-variate nature o f our error locator p olynomial, the follow- ing t heorem is a direct r esult o f the fact that the x-coordinate in our affine points satisfy x ∈ F ( q 2 ) : Proposition 4: [Reduced Com p lexity Error Location Computation] The C hien search for Semi-erasur e decoding r equir es evaluating the ψ ( x ) poly nomial only at q 2 points. Given the complexity of error locator polynomial in e r ror-only decoding of Hermitian 6 codes (pole order as high as t + 2 g + q − 1 ), this is a s ignificant reduction in computational steps for finding the e rror locations. This is due t o the fact that we now need to evaluate the polynomial only at q 2 points (rather than q 3 for error -only decoding) and also, we ar e working w ith only one variable for evaluating the polyno mial. 6 Error Evaluation The decoding complexity of Hermitian codes can be further r educed by using semi-erasure decoding. This improvement can be done in evaluating the values of the errors. In p ar - ticular , semi-erasure decod ing allows us to use Forney ’s formula d ir ectly for computing error values in Hermitian codes. Lemma 1: Given an erro r locator poly nomial ψ ( x ) of the form as in Theor em 2, the deriv ative ψ ′ ( x ) with r espect to x can n ot be zero at any of the err or lo cations. Proof: If ψ ( x ) is of the form as shown in (8), its derivative with r espect to x can be written as: ψ ′ ( x ) = ( f ( x ) X k : e k 6 =0 Y j 6 = k ( x − x j )) + f ′ ( x ) Y k : e k 6 =0 ( x − x k ) For any err or location x k , ψ ′ ( x k ) = f ( x k ) X k : e k 6 =0 Y j 6 = k ( x k − x j ) (9) which is non-zer o. ✷ Given the syndromes and the error locator polyno mial σ ( x, y ) , it is kno w n how to com- pute the error evaluator polyno mial ω ( x, y ) [3][10][4][9]. Be fore de riving the appli cabili ty of Forne y’s formula in e rror value computation in Hermitian codes, we will need a simple Lemma. Theorem 3: [Uni-variate E rror Evaluator P olynomial] For semi-erasur e decoding, an err or evaluator polynomial ω ( x, y ) corr espondi ng to σ ( x, y ) is equiv- alent to an uni-variate polynomial Ω( x ) corr esponding to the un i-varia te err or locator polynomial ψ ( x ) , i.e., satisfies: Ω( x ) = ψ ( x ) ( S e | β = β j ) (10) Further , Ω( x ) is of the form: Ω( x ) = f ( x ) X k : e k 6 =0 e k g ( x, x k ) Y j 6 = k ( x − x j ) (11) wher e f ( x ) 6 = 0 at any error location and g ( x, x k ) | x = x k = 1 at all err or locati ons. 7 Proof: T o prove this, we use an alternative definition of syndrome polynomial for an e r ror vector e : S e = n X k =1 e k h k (12) where h k = 1 x − x k y q + y − y q k − y k y − y k (13) W e say that polynomials σ, ω ∈ F q 2 [ x, y ] satisfy the k ey e quation if: ω = σ S e Since this is true for all points , without loss of ge nerality we can say that for any y = x q +1 ( y 0 + β ) + δ ( x ) β : ω | y = ( σ | y )( S e | y ) Since β is known for semi-erasure decoding, we can reduce it to: ω | β = ( σ | β )( S e | β ) which is equivalent to: Ω( x ) = ψ ( x ) S e | β = β j (14) T o prove the second statement , w e expand t he fractions in (13): h k = 1 x − x k ( y q − 1 + 1 + y k y q − 2 + · · · + y q − 2 k y + y q − 1 k ) (15) Notice that for any e r ror position with y-coordinate y k and for fields of characteristic 2, it is e asy to show that: h k = 1 x − x k which implies that substituting th e explicit form of y-coordinate from (2 ), y = x q +1 ( y 0 + β ) + δ ( x ) β , we get: h k = 1 x − x k g ( x, x k ) where g ( x ) = 1 , when we substitut e the value of x -coo rdinate. Plugging this in (12), we get: S e | β = β j = n X k =1 e k g ( x, x k ) x − x k (16) Plugging ψ ( x ) from (8) and S e | β = β j fr om (16) in (14), we get : Ω( x ) = f ( x ) Y k : e k 6 =0 ( x − x k ) n X k =1 e k g ( x, x k ) x − x k ⇒ Ω( x ) = f ( x ) X k : e k 6 =0 e k g ( x, x k ) Y j 6 = k ( x − x j ) (17) such that f ( x ) 6 = 0 at any err or location and g ( x, x k ) | x = x k = 1 for all error locations. 8 ✷ Finally we sho w that errors can be evaluated using a Reed-So lomon t ype error evalua- tion. Theorem 4: [Forney’ s Formula for Hermitian Codes] For an y err or location at point P k with x-coord inate x k , error value can be cal culated using the fo rmula: e x k = Ω( x k ) ψ ′ ( x k ) (18) wher e ψ and Ω are uni-variate err or locator and error evaluato r polynomials and ψ ′ ( x ) is derivat ive with resp ect to x. Proof: Subst ituting the values for Ω( x ) and ψ ′ ( x ) from (11) and (9) respectively , the R.H.S. of t he above e q u ation (18) becomes: f ( x ) P k : e k 6 =0 e k g ( x, x k ) Q j 6 = k ( x − x j ) f ( x ) P k : e k 6 =0 Q j 6 = k ( x − x j ) + f ′ ( x ) Q k : e k 6 =0 ( x − x k ) This fraction at point x = x k and us ing constraint that g ( x, x k ) | x = x k = 1 , w e get : Ω( x k ) ψ ′ ( x k ) = f ( x k ) P k : e k 6 =0 e k Q j 6 = k ( x k − x j ) f ( x k ) P k : e k 6 =0 Q j 6 = k ( x k − x j ) = e k Hence t h e proof. ✷ 7 Discussion and Future D i r ections In this paper , we have s hown that by car efully constructing the codes and t ransmitting a modified informa tion cod eword, we can s ignificantly reduce the decod ing complexity in Hermitian code s. Of particular interests are r esults concerning fo r mation o f univariate error locator and error evaluator polynomials to locate and evaluate the errors. It h as bee n shown that the Chien searc h for Hermitian codes can be r educed to evalua ting an uni- variate polyno mial over q 2 points as opp o sed to a bivaria te polyno mial evaluation o n q 3 points. F inally , it has been s hown t h at F orney formula from Ree d-Solomon codes can be used for evaluating the error values in Her mitian cod es. Though it sound s a little not so obvious, but proof of Th e orem 1 does indicate that us ing this con s truction, it might be pos- sible to decod e Hermitian code s using a single modified Berlekamp Massey algorithm. An answer to this might r esult in highly reduced complexity of decoding of Hermitian codes, thereby solving a long standing pr oblem o f enhancing the applicabi lity of the s t rongest competitors of ubiquitous Ree d-Solomon codes. 9 References [1] R. K ¨ otter , A Fast Parallel Implementation o f Be rlekamp-Massey algorithm for Algebraic-Geometric Cod es, IEEE T rans. In formation Theory , V ol. 44, No. 4, July 199 8. [2] M. 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