Ffteenth Internatonal Workshop on Algebrac and Cobnatoral Codng Theory June 18-24, 2016, Albena, Bulgara pp. 35 40 On Syndroe Decodng of Punctured Reed-Soloon and Gabduln Codes 1 Hannes Bartz hannes.bartz@tu.de Vladr Sdorenko vladr.sdorenko@tu.de Insttute for Councatons Engneerng Techncal Unversty of Munch, D-80290 Munch, Gerany Abstract. Beng evaluaton codes, punctured Reed-Soloon (RS) and Gabduln (G) codes over the feld F q wth locators fro the subfeld F q can be represented as nterleavng of correspondent codes over the subfeld F q or can be consdered as vrtual nterleavng of correspondent codes over the feld F q. Usng a probablstc unque syndroe decoder, -nterleaved or vrtually nterleaved codes can be decoded up to the sae radus (d 1), where d s the code dstance n +1 Hang etrc for RS codes and n rank etrc for G codes. We show that the correspondent decoders over the subfeld F q and the feld F q are equvalent and conclude that n practce one should use a decoder over the subfeld snce t has less coplexty. 1 Introducton Reed-Soloon (RS) [1] and Gabduln (G) [2] codes belong to the faly of evaluaton codes and are wdely used for error correcton n any applcatons. An evaluaton code over the fnte feld F q s constructed by evaluatng all polynoals wth coeffcents fro F q of restrcted degree at a set of code locators. By choosng the code locators fro the subfeld F q we obtan a punctured evaluaton code over F q whch can be equvalently nterpreted as an -nterleaved code I over the subfeld F q [3]. It s known that -nterleaved RS and G codes over F q wth dstance d can correct wth hgh probablty up to +1 (d 1) errors n the correspondng etrc [4,5]. In [6,7] t was shown that the sae decodng radus can be acheved by coputng eleent-wse q-powers of the receved word at the decoder. Ths results n a receved word V of a vrtually -nterleaved code V over the large feld F q. Vrtual nterleavng wth usual powers was orgnally proposed n [8] and was odfed to q-powers n [6]. For both decodng schees ether a syndroe- or nterpolaton-based decoder can be used. In ths paper we analyze and copare probablstc unque syndroe-based decodng algorths for nterleaved and vrtually nterleaved RS and G codes. 1 H. Bartz was supported by the Geran Mnstry of Educaton and Research n the fraework of an Alexander von Huboldt-Professorshp. V. Sdorenko s on leave fro the Insttute for Inforaton Transsson Probles, Russan Acadey of Scences.
36 ACCT2016 We show that the syndroe-based decoder of the code I over the subfeld F q s equvalent to the respectve decoder of V n the feld F q. Ths eans, that for the sae nput the decoders return the sae output and shows, that the decodng falure probablty s the sae for both decoders. It allows us to choose the decoder wth the lowest coputatonal coplexty,.e., the respectve decoder over the subfeld F q. The extended verson of the paper wth proofs s avalable onlne at http://goo.gl/nl78p5. 2 Prelnares Let F h be a fnte feld, where h s a power of a pre. Let F q and F q be extensons of F h. By the colun vector a = ( a (0) a (1)... a ( 1))T F q we denote the expanson of an eleent a F q w.r.t. a fxed bass of F q over F q. Gven a vector a of length n over F q, we ntroduce the n expanson atrx over F q as a = (a 0,..., a n 1 ). By F q [x] we denote the rng of all polynoals g(x) = d =0 g x over F q and F q [x] <k s the set of all polynoal fro F q [x] wth degree less than k. A nonzero polynoal of the for p(x) = d =0 p x [], where [] denotes the Frobenus power [] = h, wth p F q, p d 0, s called an h-lnearzed polynoal of h-degree deg h (p(x)) = d. By L q [x] we denote the rng of all h- lnearzed polynoals over F q and L q [x] <k denotes the set of all polynoals fro L q [x] wth h-degree less than k. RS and G codes belong to a class of evaluaton codes, whch are defned as follows. Evaluaton code. Assue 0 < k n and 1 are ntegers. Gven an n-vector of code locators α = (α 0 α 1... α n 1 ) over F q and a set P() of polynoals f(x) over F q, where P() = F q [x] <k or P() = L q [x] <k. We defne f(α) = (f(α 0 ) f(α 1 )... f(α n 1 )). The evaluaton code C ev s the set of all n-words C ev (n, k, α, P()) = {f(α) f P()} (1) obtaned by evaluatng all polynoals f fro P() at the locators α. RS and G codes can be defned as follows. Reed-Soloon code. If the locators α are parwse dfferent and P() = F q [x] <k, then the code C ev s an [n, k] lnear Reed-Soloon code over F q wth code dstance d = n k + 1 [1] n Hang etrc. Gabduln code. Assue that the locators α are F h -lnearly ndependent. Let P() = L q [x] <k, then C ev s an [n, k] lnear Gabduln code over F q wth code dstance d = n k + 1 [2] n rank etrc. The rank dstance between two n-words v, w over F q s defned as rk(v w) over F h. In general, RS and G codes wth locators fro the feld F q can correct errors of weght up to (d 1)/2 n the correspondent etrc. It s known [3] that t s possble to correct wth hgh probablty ore errors f we puncture the
Bartz, Sdorenko 37 codes and take locators fro the subfeld F q only. Now we ntroduce (proper) punctured codes. Punctured evaluaton code. The evaluaton code (1) over F q s called proper punctured f all locators α belong to the subfeld F q. Later on we consder proper punctured codes only and call the sply punctured. Ths gves us defntons of punctured RS and punctured G codes as well. Interleaved codes over sall feld F q (Schee I). Let us show that a punctured evaluaton code over the large feld F q wth locators α F q s equvalent to nterleavng of evaluaton codes over the subfeld F q [3]. Let f(x) = f x be a polynoal fro P(). By representng each coeffcent f by f we can wrte one polynoal f(x) P() as polynoals f (j) (x) = f (j) x P(1), j [0, 1]. Now every codeword of the punctured evaluaton code (1) can be wrtten over the sall feld F q as c = f(α) f (0) (α) f (0) (α 0 ) f (0) (α n 1 ). =... def = I. (2) f ( 1) (α) f ( 1) (α 0 ) f ( 1) (α n 1 ) Snce f (j) (x) P(1) for all j, every row n the n atrx I over F q n (2) s a codeword of C ev (n, k, α, P(1)). Hence I s obtaned by nterleavng of codewords fro C ev (n, k, α, P(1)). Ths eans that every codeword f(α) F n q of the code C ev (n, k, α, P()) can be wrtten as nterleavng I of codewords fro C ev (n, k, α, P(1)). Vrtual nterleavng over the feld F q (Schee V). Consder a codeword c = (c 0 c 1... c n 1 ) = f(α) of a punctured evaluaton code C ev (n, k, α, P()) wth locators α F q and copute the eleent-wse q-powers c qj = (c qj 0 cqj 1... cqj n 1 ). For f(x) = f x P() denote a bjectve ap f f qj where f qj (x) = f qj x P(). Snce c = f(α ) where α F q for all [0, n 1] and f P(), we have c qj = (f(α )) qj = f qj (α ). Hence c qj C ev (n, k, α, P()) and fro one codeword c we can vrtually create codewords c qj for j [0, 1]. These codewords for an n atrx V over the bg feld F q of the vrtually -nterleaved code c = f(α) f q0 (α). = f q0 (α 0 ) f q0 (α n 1 )... def = V. (3) f q 1 (α) f q 1 (α 0 ) f q 1 (α n 1 ) Notce that n the case of vrtual nterleavng V, we stll transt just one codeword c = f(α) of the orgnal punctured code and receve one word y corrupted by errors. One can thnk that the rest 1 codewords c qj were vrtually transtted as well. The correspondent 1 receved words can be coputed at the recever as y qj. For the Hang etrc t errors n the receved word y wll nduce t erroneous coluns n the vrtually receved atrx
38 ACCT2016 V. If the receved word y s corrupted by an error of rank t then atrx V wll be corrupted by an error of rank t as well. What can we gan usng I or V nterleavng? It s known [4,5] that decodng of an s-nterleaved code wth dstance d can be done up to the radus +1 (d 1) wth hgh probablty. Hence we can ncrease the decodng radus alost twce f we use probablstc decoders I or V nstead of decodng the orgnal punctured code up to radus (d 1)/2. Any known syndroe-based decoder for nterleaved codes can be appled to get ths gan. However, the coplexty of operatons ncreases wth the feld sze. Ths s a dsadvantage of Schee V. Can we gan soethng usng Schee V nstead of I? For exaple, f a syndroe decoder s used wth Schee I t wll fal wth probablty at ost (feld sze) 1 = 1/q. Does t ean that the falure probablty of Schee V over the large feld F q s saller than the one of Schee I as t s claed n [7]? In the next secton we wll descrbe decoders for I and V atrces, analyse and copare the. 3 Syndroe Decodng of Punctured Reed-Soloon and Gabduln Codes Consder a proper punctured evaluaton code C ev (n, k, α, P()) whch s a RS or G code over the feld F q wth locators α fro the subfeld F q. Snce α F q a party check atrx H of the code s also over the subfeld F q. Let a codeword c C ev be transtted and an n-word y over F q be receved. Then the error vector n the channel s e = y c and the nuber of errors t s the Hang weght of the error e n case of RS code and t = rk(e) for G code. Gven receved word y, the unque decoder should output a codeword or declare a falure. A syndroe decoder frst coputes the syndroe vector s = yh T F n k q. If the syndroe s = 0 then y s a codeword, otherwse for s 0 the followng key equaton (6) ust be solved [9, 10]. Defne the feld autoorphs θ as θ(a) def = { a for RS codes a h for G codes and the reversed syndroes for [0, d 2] as { def s s = for RS codes θ (d 2). (5) (s d 2 ) for G codes Key equaton. t s = σ j θ j (s j ), = [t, d 2]. (6) j=1 To solve the key equaton eans to fnd nu nteger t > 0 such that (6) has a soluton σ = (σ 1,..., σ t ). If the soluton s not unque the decoder fals. (4)
Bartz, Sdorenko 39 Otherwse t fors the error locator polynoal { 1 + σ1 x + + σ t x t F q [x] for RS codes σ(x) = x + σ 1 x h. (7) + + σ t x ht L q [x] for G codes Havng the error locator polynoal t s easy to fnd the error vector e usng known approaches, e.g. n [9, 10], and to copute the codeword c = y e. So, the an part of the decoder s solvng the key equaton, whch can be done by solvng the lnear syste of equatons (6) wth coeffcents fro F q for t = 1, 2,... Ths can be done by standard lnear algebra resultng n a decodng algorth whch always corrects up to d/2 errors, but we would lke to correct ore errors usng a probablstc decoder as follows. Locators α of punctured codes belong to the subfeld F q. Snce the roots of an error locator polynoal belong to the subfeld F q of code locators, the coeffcents of the error locator polynoal σ(x) also belong to F q and we should fnd unknowns σ n (6) fro the subfeld F q. Ths allows to wrte ore equatons and as a result to correct ore errors usng Schees I or V as follows. Key equaton over subfeld, Schee I. We receve the vectort y,.e. the atrx y wth nterleaved words y (l) over F q. The syndroes s (l) can be coputed as s (l) = y (l) H T because H s over F q. Snce error locators are coon for nterleaved words [5, 11], we can wrte the key equaton (6) for each syndroe s (l) wth the coon error locator σ(x) and get the followng syste of equatons over the subfeld F q t ( ) s (l) = σ j θ j s (l) j, = [t, d 2], l = [0, 1]. (8) j=1 Key equaton for vrtual nterleavng, Schee V. The syndroes s ql, l = [0, 1], can be coputed fro vrtually receved words y ql as y ql H T = s ql snce H s over F q. Vrtual error vectors y ql c ql = e ql have the sae weght t and coon error locatons. Hence we can wrte the key equaton (6) for each syndroe s ql wth coon error locator σ(x) and get the followng syste of equatons over the feld F q t = s ql j=1 σ j θ j ( s ql j ), = [t, d 2], l = [0, 1]. (9) Lea 1 Gven a vector s 0 over F q and nteger 0 < t < d 1, a soluton σ I of (8) s unque f and only f (9) has a unque soluton σ V. In ths case σ I = σ V s a vector over F q. Ths eans that for fxed receved word y both decoders I and V wll fnd the sae error locator polynoal n case when (8) and (9) have unque soluton and output the sae result, otherwse both decoders wll fal. Hence the decoders are equvalent and we get the followng theore, where we assue that the errors e of weght t have equal probablty to estate falure probabltes.
40 ACCT2016 Theore 1 For punctured RS and G codes unque probablstc syndroe decoders of Schees I and V are equvalent havng decodng radus t ax = +1 (d 1), and decodng coplexty O(n 2 ) operatons n the feld F q for Schee I and n F q for Schee V. Decodng falure probablty P f (t) γq (+1)(tax t) 1, where t s error weght, γ 3.5 and γ 1 for RS codes. Usng fast operatons, decodng can be further accelerated to sub-quadratc coplexty. The obtaned results are drectly appled to correctng errors and erasures. Wthout establshng equvalence t s not easy to estate the falure probablty for Schee V whch can lead to ncorrect conclusons [7]. References [1] I. S. Reed, A Class of Multple-Error-Correctng Codes and the Decodng Schee, IRE Trans. Inf. Theory, vol. 4, no. 4, pp. 38 49, Sep. 1954. [2] E. M. Gabduln, Theory of Codes wth Maxu Rank Dstance, Probl. Inf. Trans., vol. 21, no. 1, pp. 3 16, 1985. [3] V. Sdorenko, G. Schdt, and M. Bossert, Decodng Punctured Reed-Soloon Codes up to the Sngleton Bound, n, 2008 7th Int. ITG Conf. on Source and Channel Codng (SCC), Jan 2008. [4] D. Blechenbacher, A. Kayas, and M. Yung, Decodng Interleaved Reed Soloon Codes over Nosy Channels, Theor. Coput. Sc., vol. 379, no. 3, pp. 348 360, Jul. 2007. [5] G. Schdt, V. R. Sdorenko, and M. Bossert, Collaboratve Decodng of Interleaved Reed Soloon Codes and Concatenated Code Desgns, IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 2991 3012, 2009. [6] V. Guruswa and C. Xng, Lst Decodng Reed Soloon, Algebrac-Geoetrc, and Gabduln Subcodes up to the Sngleton Bound, Electronc Colloq. Cop. Coplexty, vol. 19, no. 146, 2012. [7] L.-Z. Shen, F. we Fu, and X. Guang, Unque Decodng of Certan Reed-Soloon Codes, IEICE Trans. on Fund. of Electr., Co. and Coput. Scences, vol. E98-A, no. 12, pp. 2728 2732, Dec. 2015. [8] G. Schdt, V. R. Sdorenko, and M. Bossert, Syndroe Decodng of Reed-Soloon Codes Beyond Half the Mnu Dstance Based on Shft-Regster Synthess, IEEE Trans. Inf. Theory, vol. 56, no. 10, pp. 5245 5252, Oct. 2010. [9] R. E. Blahut, Theory and Practce of Error Control Codes, 1st ed. Addson-Wesley, 1983. [10] D. Slva, F. R. Kschschang, and R. Kötter, A Rank-Metrc Approach to Error Control n Rando Network Codng, IEEE Trans. Inf. Theory, vol. 54, no. 9, pp. 3951 3967, 2008. [11] V. R. Sdorenko, L. Jang, and M. Bossert, Skew-Feedback Shft-Regster Synthess and Decodng Interleaved Gabduln Codes, IEEE Trans. Inf. Theory, vol. 57, no. 2, pp. 621 632, Feb. 2011.