Erasure List-Decodable Codes from Random and Algebraic Geometry Codes

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1 Erasure Lst-Decodable Codes from Random and Algebrac Geometry Codes Yang Dng, Lngfe Jn and Chaopng Xng arxv:40.276v [cs.it] 3 Jan 204 Abstract Erasure lst decodng was ntroduced to correct a larger number of erasures wth output of a lst of possble canddates. In the present paper, we consder both random lnear codes and algebrac geometry codes for lst decodng erasure errors. The contrbutons of ths paper are two-fold. Frstly, we show that, for arbtrary 0 < R < and ǫ > 0 R and ǫ are ndependent), wth hgh probablty a random lnear code s an erasure lst decodable code wth constant lst sze 2 O/ǫ) that can correct a fracton R ǫ of erasures,.e., a random lnear code acheves the nformaton-theoretc optmal trade-off between nformaton rate and fracton of erasure errors. Secondly, we show that algebrac geometry codes are good erasure lst-decodable codes. Precsely speakng, for any 0 < R < and ǫ > 0, a q-ary algebrac geometry code of rate R from the Garca-Stchtenoth tower can correct R + ǫ fracton of erasure errors wth lst sze O/ǫ). Ths mproves the Johnson bound appled to algebrac q geometry codes. Furthermore, lst decodng of these algebrac geometry codes can be mplemented n polynomal tme. Index Terms Erasure codes, Lst decodng, Algebrac geometry codes, Generalzed Hammng weghts. I. INTRODUCTION Erasure codes have receved great attentons for ther wde applcatons n recoverng packet losses n the nternet and storage systems. In the model of erasure channel, errors are descrbed as erasures, namely the recevers are supposed to know the postons where the erasures occurred. Compared wth other communcaton channels such as adversaral nose channel, erasure channel s much smpler. Thus, we can expect better parameters for erasure channel than adversaral nose channel. Instead of the unque decodng, the model of lst decodng for whch a decoder allows to output a lst of possble codewords was ndependently ntroduced by Elas and Wonzencraft [3], [8]. The decodng s consdered to be successful as long as the correct codeword s ncluded n the lst and the lst sze s not too bg. The problem of lst decodng for classcal adversaral nose channel has been extensvely studed see [3], [7], [8], [9], [6], [8], [9], for example). A fundamental problem n lst decodng s the tradeoff among the nformaton rate, decodng radus.e., fracton of errors that can be corrected) and the lst sze. In other words, f we fx one of these three parameters, then one s nterested n optmal tradeoff between the remanng two parameters. For nstance, f the lst sze s fxed to be constant or polynomal n the length of codes, the problem becomes a tradeoff between nformaton rate and decodng radus. Defnton.: τ,l)-erasure lst decodablty) Let Σ be a fnte alphabet of sze q, L > an nteger, and τ 0,). A code C Σ n s sad to be τ,l)-erasure lst-decodable, f for every r F τ)n q, and any subset T {,2,,n} of sze τ)n, one has {c C c T = r} L, where c T s the projecton of c onto the coordnates ndexed by T. In other words, gven any receved word wth at most τn erasures, there are at most L codewords that are consstent wth the unerased porton of the receved word. Known results It s known that, for an erasure channel where the codeword symbols are randomly and ndependently erased wth probablty τ, the capacty s τ see [4]). Although erasure lst decodng has been consdered prevously see [6], [0], [8], [9]), a lot of problems stll reman unsolved. Let us summarze some of prevous results on erasure lst decodng below. ) It was shown n [6] that, for any small ǫ > 0 and τ 0,), a τ,l)-erasure lst-decodable code of rate τ ǫ must satsfy L Ω ǫ ); and on the other hand, there exsts a τ,oexp ǫ )))-erasure lst-decodable code of rate τ ǫ. ) In [7, Proposton 0.], the Johnson bound for erasure decodng radus was derved. It says that, for any gven ǫ > 0, every q-ary code of relatve dstance δ < /q s δ+ δ ǫ,o/ǫ))-erasure lst-decodable. Ths means that, wth a constant lst sze, erasure decodng radus s enlarged by approxmaly δ compared wth unque erasure decodng whose decodng radus s only δ. On the other hand, t was shown further n [7, Proposton 0.2] that there exsts a q-ary code of length n and relatve dstance δ < /q that s not δ + δ δn) )-erasure lst-decodable for every small +ǫ,2ωǫ2 All authors are wth Dvson of Mathematcal Scences, School of Physcal and Mathematcal Scences, Nanyang Technologcal Unversty, Sngapore 63737, Republc of Sngapore emal: {dngyang,lfjn,xngcp}@ntu.edu.sg). The work s partally supported by the Sngapore A*STAR SERC under Research Grant

2 2 ǫ > 0. Ths mples that the best bound on erasure lst decodng radus of a q-ary code of relatve mnmum dstance δ s δ + δ. ) In [6], Guruswam showed that, for any small ǫ > 0, wth hgh probablty a random lnear code of rater = Ωǫ/log/ǫ)) s σ, O/σ))-erasure lst-decodable for every σ satsfyng ǫ σ. Furthermore, by the concatenaton method Guruswam showed n [6] that, for any small ǫ > 0, one can construct a famly of concatenated bnary) ǫ,o/ǫ))- erasure lst-decodable codes of rate Ωǫ 2 /log/ǫ)) n polynomal tme. A slghtly better rate was obtaned for nonlnear codes over larger alphabet sze n [9]. Our results and comparson Our contrbutons of ths paper are two-fold. ) Frstly, we show that, for arbtrary 0 < R < and ǫ > 0 R and ǫ are ndependent), wth hgh probablty a random lnear code s R ǫ,2 O/ǫ) )-erasure lst-decodable,.e., a random lnear code acheves the nformaton-theoretc optmal tradeoff between nformaton rate and fracton of erasure errors that can be corrected. Whle Theorem 2 n [6] whch was derved from [] only shows exstence of R ǫ,2 O/ǫ) )-erasure lst-decodable codes for arbtrary 0 < R < and ǫ > 0. ) Secondly, we show that algebrac geometry codes are good erasure lst-decodable codes. Precsely speakng, for any 0 < τ < andǫ > 0, aq-ary algebrac geometry code from the Garca-Stchtenoth tower has rate at least τ + q ǫ and sτ, O/ǫ))-erasure lst-decodable. Furthermore, lst decodng of these algebrac geometry codes can be mplemented n polynomal tme. On the other hand, f we apply the Johnson bound gven n [7, Proposton 0.] to general algebrac geometry codes, we can only clam that a q-ary algebrac geometry code from the Garca-Stchtenoth tower has rate τ + τ q ǫ and s τ,o/ǫ))-erasure lst-decodable. Ths rate s always smaller than our rate for any τ 0,). Ths mples that the Johnson bound could be mproved for some specal class of codes although t s optmal n general. Open problems For adversaral error channel, t has been shown that, gven decodng radus 0 < τ <, the optmal rate for lst decodng s R = H q τ), where H q x) = xlog q q ) xlog q x x)log q x) s the q-ary entropy functon. More precsely speakng, for any small ǫ > 0 and τ wth 0 < τ < /q, wth hgh probablty a random code s H q τ) ǫ,o ǫ ))-lst decodable. Furthermore, every q-ary H q τ) ǫ,l)-lst-decodable code has lst sze at least Ωlog/ǫ). It s stll an open problem to determne f there exsts a q-ary H q τ) ǫ,l)-lst decodable code wth lst sze L smaller than O ǫ ). Under the stuaton of erasure lst decodng, the optmal rate R that one can acheve s R = τ. If we denote L τ,q ǫ) to be the smallest nteger L for whch there are q-ary τ,l)-erasure lst-decodable codes of rate at least τ ǫ for nfntely many lengths n, then t follows from our result and [6] that Ω ǫ ) L τ,qǫ) 2 O/ǫ). Now the frst open problem s Open Problem : Determne L τ,q ǫ). In the lterature, there are not many results on constructve bounds on erasure lst decodng except for suffcently large q or small rate [6], [0]. The second open problem would be Open Problem 2: Narrow the rate gap between τ + q and τ by constructng erasure lst-decodable such that codes explctly,.e., construct a q-ary τ,l)-erasure lst-decodable codes of rate R > τ + q the lst sze L s ether a constant or a polynomal n length. Organzaton The paper s organzed as follows. In Secton 2, we ntroduce some necessary nataton and defntons and known results as well. Secton 3 s devoted to random codes. In the last secton, we show that algebrac geometry codes are good erasure lst-decodable codes. II. PRELIMINARIES In ths paper, we only focus on lnear codes. Recall that a q-ary [n,k] q lnear code s an F q -lnear subspace of F n q wth dmenson k, where F q s a fnte feld wth q elements and q s a prme power. n s called the length of the code and k s the dmenson of the code. The nformaton rate of the code C s defned as R = k/n whch represents the effcency of the code. Another mportant parameter of the code s the dstance whch represents the error correctng capablty. The dstance of a lnear code C s defned to be the mnmum Hammng weght of nonzero codewords of C, denoted by d = dc). The relatve dstance δ = δc) s defned to be the quotent d/n. From Defnton., one knows that, n a τ,l)-erasure lst decodable code C of length n, for every r F q τ)n and T {,2,...,n} wth T = τ)n the number of the codewords n the output lst that are consstent wth r at the coordnates ndexed by T s at most L. Thus, f C s lnear, t s equvalent to sayng that the number of the codewords

3 3 that are 0 at the coordnates ndexed by T s at most L,.e., {c C c T = 0} L. Hence, an [n,k,d] q -lnear code s d )/n, )-erasure lst-decodable, but not d/n, )-erasure lst-decodable. Defnton 2.: Erasure lst decodng radus ELDR)) ) For an nteger L and a lnear code C of length n, we denote Rad L C) := max{s Z >0 : C s s/n,l)-erasure lst-decodable}. ) For an nfnte famly C = {C } of q-ary lnear codes wth length tendng to and an nteger L, we denote { } RadL C ) ELDR L C) := lmnf, where n s the length of C. Defnton 2.2: For an nteger L and 0 τ, the maxmum rate for lnear τ,l)-erasure lst-decodable code famles s defned to be R L τ) := sup C: ELDR LC) τ RC). The notaton of erasure lst decodng for lnear codes actually had already been studed n the form of generalzed Hammng weght, see [7]. However, the explct relatonshp between erasure lst decodng and generalzed Hammng weght had not been made clear untl the work n [6]. The concept of generalzed Hammng weght was ntally ntroduced n [7] and later receved great attenton due to applcatons n cryptography, desgn of codes, t-reslent functons and so on []. Defnton 2.3: Generalzed Hammng Weght) The r-th generalzed Hammng weght of a code C, denoted by d r C), s defned to be the sze of the smallest support of an r-dmensonal subcode of C,.e., d r C) = mn{ SuppD) : D s a subspace of C of dmenson r}, where SuppD) = { : c,...,c n ) D,c 0}. Note that d C) s exactly the mnmum dstance d of C. The characterzaton of erasure lst decodablty through generalzed Hammng weght s gven below. Lemma 2.4: see [6]) A lnear code C of length n s s/n,l)-erasure lst-decodable f and only f d r C) > s, where r = log q L +. The lnk stated n Lemma 2.4 establshes a two-way brdge. Results for erasure lst decdng can be derved drectly from the exstng results on generalzed Hammng weght, and thus the applcatons of generalzed Hammng weght are nherted. In the meanwhle, some new propertes for generalzed Hammng wght can be obtaned as well f one can develop some fresh deas on erasure lst decodng. In [6], Guruswam made use of the connecton between generalzed Hammng weght and erasure lst decodng to establsh some bounds for rate R L τ) through the exstng bounds on generalzed Hammng weght. Lemma 2.5: see [6]) One has ) For every nteger L and every τ, 0 τ, R L τ) τ r log q n q r q H qτ) r where r = log q L +. In partcular, for any small ǫ > 0 and τ 0,), there exsts a τ,oexp ǫ )))-erasure lstdecodable code of rate τ ǫ. ) For small ǫ > 0 and τ wth 0 < τ <, a τ,l)-erasure lst-decodable code of rate τ ǫ must satsfy L Ω ǫ ). III. RANDOM LIST DECODABLE ERASURE CODES Random ǫ, O/ǫ))-erasure lst-decodable codes of rate R = Ωǫ/ log/ǫ)) was dscussed n [6] by usng a characterzaton of generator matrces of erasure lst-decodable codes. However, the rate s qute small and actually s dependent on ǫ. In ths secton, we are gong to show that for any 0 R R s ndependent of ǫ), wth probablty q Ωn) a random lnear code C of length n and rate R s R ǫ,2 O/ǫ) )-erasure lst-decodable. Our approach s through a characterzaton of party-check matrces of erasure lst-decodable codes. Proposton 3.: If k/n R > 0 when n tends to, then for a random matrx H over F q of sze n k) n, the probablty that H s full-rank s approachng when n tends to. Proof: On one hand, t s easy to compute that the total number of random matrx H over F q of sze n k) n wth full rank s q n )q n q) q n q n k ).

4 4 On the other hand, the total number of matrces H over F q of sze n k) n s q nn k). Let E denote the event that an n k) n random matrx H over F q s full-rank, then PrE) = qn )q n q) q n q n k ) q nn k). To show lm n PrE) =, t suffces to show that lm n lnpre) 0. When n tends to, we have 0 ln qn )q n q) q n q n k ) q nn k) = n =k+ ln q ) n =k+ 2q ) 2n q k 0. Ths completes the proof. Lemma 3.2: Let s be a postve nteger, then an [n,k] q code C s s/n,l)-erasure-lst-decodable f and only f any submatrx H n k) s of the party check matrx H n k) n of C has rank at least s log q L. Proof: By Defnton. and the fact that C s a lnear code, C s s/n,l)-erasure-lst-decodable f and only f {c C c T = 0} L for T {,2,...,n} wth sze n s. Ths mples that C s s/n,l)-erasure-lst-decodable f and only f for any submatrx H n k) s of H n k) n, {x F s q H n k) s x = 0} L,.e., the soluton space of H n k) s has dmenson at most log ql. Therefore, H n k) s has rank at least s log ql. Theorem 3.3: For every small ǫ > 0, a real 0 < R < and suffcently large n, wth probablty at least q Ωn), a random lnear code over F q of length n and rate R s R ǫ,2 O ǫ ) )-erasure lst-decodable. Proof: Put l = ǫ 2 R)log q2+) and L = q l. Thus, L = 2 O ǫ ). We randomly pck a matrx H n k) n. Then wth probablty approachng, H n k) n s full rank from Proposton 3.. Let such a full rank matrc H n k) n be the party check matrx of our lnear code C. Then we are gong to prove that wth probablty at most q Ωn), C s not s,l)-erasure lst-decodable for s = n k ǫn. By Lemma 3.2, ths happens only f some n k) s submatrx of H has rank less than s log q L. Denote n k by K. Let A denote the number of full-rank matrces H n k) n n whch there exsts s = n k ǫn columns wth rank ) at most s l. Note that the total number of matrces of sze K s over F q wth rank at most s l s equal to q s ) q s q )q K ). Thus, we have s l K A n s s l < 2 n ) K q s ) q s q )q K ) ) K q s+k 2 )+Kn s) s l ) K )q Kn s) s l 2 n q s+k)s l) s l)2 +Kn s) 2 n+k q s+k)s l) s l)2 +Kn s) q n+k)log q 2 q 2K ǫn)k ǫn l) K ǫn l)2 +Kn K+ǫn) < q n2 R)log2 ǫl)+kn. Substtutng the value of l to the above equaton, we have lm sup n A q n )q n q) q n q n k ) lmsup n A lm q n k)n n q n k)n q n )q n q) q n q n k ) lm n q n = 0. Ths mples that wth probablty at most q n, a random matrx H n k) n has full rank and a submatrx of sze n k) s of rank at most s l. The clamed result follows from settng of our parameters.

5 5 IV. ALGEBRAIC GEOMETRY CODES ARE GOOD ERASURE LIST-DECODABLE CODES In the prevous secton, we proved that random codes are good erasure lst-decodable codes. There s stll a lack of constructve results on erasure lst decodng. Though Guruswam [6] presented a constructve result from concatenated codes, the rate s extremely small. In ths secton, we show that algebrac geometry AG for short) codes are good erasure lst-decodable codes and furthermore they can be lst decoded n polynomal-tme. As a preparaton, we recall some basc results on AG codes frst. Readers may refer to [5] for more detals. Let X be a smooth, projectve, absolutely rreducble curve of genus gx) we wll use g nstead of gx) f there s no confuson n the context) defned over F q. We denote by F q X) the functon feld of X. Denote by NX) the number of ratonal ponts of X. Let P = {P,...,P n } be a set of n dstnct ratonal ponts over F q. Let G be a dvsor such that SuppG) {P,...,P n } =. Defne LG) as the Remann-Roch space assocated to G and denote dmlg) = lg). The algebrac geometry code CG,P) s defned as the mage of LG) n F n q under the followng evaluaton map C : LG) F n q, f fp ),...,fp n )). If n > degg, then CG,P) s an [n, degg g +, n degg] q -AG code. Throughout ths secton, we always assume that n s bgger than degg). The gonalty of a curve X was ntroduced n [3]. It s defned to be the smallest degree of a nonconstant map from X to the projectve lne. We denote the gonalty of X by tx). More specfcally, f X s defned over a feld F q and F q X) s the functon feld of X, then tx) s the mnmum degree of the feld extensons of F q X) over a ratonal functon feld. It s easy to see that f gx) = 0, then tx) =. If gx) = or 2, then tx) = 2. However, for general g, the gonalty s no longer determned by genus. In general, we have the followng lower bound for tx). Lemma 4.: [3]) Let X be a curve defned over F q of genus g wth N ratonal ponts. Then tx) N/q +). By usng the lower bound on tx), one has the followng proposton. n ) ) Proposton 4.2: CG, P) s n degg)+ n q+,q -erasure lst-decodable. Proof: Let s be a postve nteger wth s n degg)+ n q+. For any subset T {,2,,n} of sze n s, we clam that {c CG,P) c T = 0} q. Ths s equvalent to provng that dml G ) P. Suppose dml G P ) 2, the one can choose a nonzero functon f L G P ), then f)+g P 0. Let H = f)+g P 0. Then t s clear that degh = deg G ) n P = degg)+s n and q + dmlh) = dml G ) P 2. Choose a functon z LH)\F q, then [F q X) : F q z)] s at most degh) n q+ < N q+. Ths contradcts Lemma 4.. Our desred result follows from Defnton.. Proposton 4.2 can be extended by the Grsmer bound through the followng lemma. Lemma 4.3: If a dvsor G satsfes lg) t and degg < N, then degg N qt q t, where N stands for the number of ratonal ponts on X. Proof: Suppose P,...,P N are N dstnct ratonal ponts on X. By the strong approxmaton theorem, there exsts x F q X) such that Suppx)+G) {P,...,P N } =. Then lx)+g) = lg) and degx) +G) = degg). Thus, we can obtan an algebrac geometry code Cx)+G,{P,...,P N }) wth parameters [N,lG),d N degg] q. By the Grsmer bound [2], we have lg) d t d t N q q N degg) q. Thus, the desred result follows from the above nequalty. Theorem 4.4: If G satsfes lg) t and degg < n, then CG,P) s erasure lst-decodable. ) n n degg)+ qt q t n,q )- t

6 6 Proof: Let s be an nteger satsfyng s n degg+ qt q t n. For any T {,2,,n} of sze n s, we have deg G ) q t P = degg T = degg n+s q t n < N qt q t. By Lemma 4.3, we have l G ) P t. Our desred result follows from Defnton.. Remark 4.5: When t =, Theorem 4.4 shows that CG,P) s n n degg) ),) -erasure lst-decodable. For t = 2, we obtan the result of Proposton 4.2. Combng Lemma 2.4 and Theorem 4.4, we mmedately obtan the followng lower bound on generalzed Hammng weght of algebrac geometry codes. Corollary 4.6: For t degg) g +, the t-th generalzed Hammng weght of CG,P) satsfes q t d t CG,P)) n degg)+ q t n. Now we come to the man result of ths secton. Theorem 4.7: Let q be a square. For any small ǫ > 0 and τ wth 0 < τ < + q ǫ, there exsts a famly {CG,P)} of algebrac geometry code wth length tendng to such that CG,P) have rate at least τ + q ǫ and are τ,o ǫ ))-erasure lst-decodable. Furthermore, t can be lst decoded n Onlog q n)3 ) tme, where n s the length of the code. Proof: Choose a curvex/f q n the Garca-Stchtenoth tower [5]. Then NX)/gX). Let P = {P,P 2,...,P n } wth n = NX). Choose the last ratonal pont P of X such that P P. Put q t m := n τn + q t n and G = mp. By Theorem 4.4, CG,P) s ) n n m+ qt q t n,q )-erasure t lst-decodable for any constant t. Hence, CG,P) s τ,q t )-erasure lst-decodable. Pck ǫ = q qt q t = qq t ), then qt = O ǫ ). Moreover, the rate of CG,P) s at least n m g +) τ + qt q q t = τ + q q ǫ. Ths proves the frst statement of the theorem. Fnally by [4], we know that a bass of LG) can be found n Onlog q n) 3 ) tme, where n s the length of the code. Assume that we have already found a bass f,...,f k of LG). Suppose that c = c,...,c n ) was transmtted and c T was receved wth T {,2,...,n} and T τ)n. A functon f LG) s n the lst f and only f fp ) = c for all T. Let f = k j= λ jf j wth λ j beng unknowns. Then one has k j= λ jf j P ) = c for all T. Ths s a system of lnear equatons wth T equatons and k unknowns. It can be solved n On 3 ) tme. Ths completes the proof. REFERENCES [] A. Ashkhmn, A. Barg and S. Ltsyn, New upper bounds on generalzed weghts, IEEE. Trans. Inform. Theory, 45, pp , 999. [2] G. D. Cohen, S. N. Ltsyn and G. Zémor, Upper bounds on generalzed dstances, IEEE. Trans. Inform. Theory, 40, pp , 994. [3] P. Elas, Lst-decodng for nosy channels, MIT, Res. Lab. Electron., Cambrdge, MA, Tech. Rep. 335, 957. [4] P. Elsa, Codng for two nosy channels, Informaton Theory, Thrd London Symposum, pp. 6-76, 995. [5] A. Garca, H. Stchenoth, A tower of Artn-Schreer extensons of functon felds attanng the Drnfeld-Vlădut bound, Inventones Mathematcae, 2, pp.2 222, 995. [6] V. Guruswam, Lst decodng from erasure: Bounds and code constructons, IEEE. Trans. Inform. Theory, 49, pp , [7] V. Guruswam, Lst decodng of error correctng codes, Number 3282 n Lecture Notes n Computer Scence. Sprnger, [8] V. Guruswam and P. Indyk, Lnear-tme st decodng n error-free settngs, Lecture Notes n Computer Scence. 342, PP , [9] V. Guruswam and P. Indyk, Near-optmal lnear tme codes for unque decodng and new lst-decodable codes over small alphabets, In Proceedngs of the 34nd Annual ACM Symposum on Theory of Computng, pp.82-82, [0] V. Guruswam and M. Sudan, Lst decodng algorthms for certan cancatenated codes, In Proceedngs of the 32nd Annual ACM Symposum on Theory of Computng, pp.8-90, [] T. Helleseth, T. Kløve, V. I. Levenshten and Ø. Ytrehus, Bounds on mnmum support weghts, IEEE. Trans. Inform. Theory, 4, pp , 995. [2] S. Lng and C. P. Xng, Codng Theory A Frst Course, Cambrdge Unversty Press, [3] R. Pellkaan, On the gonalty of curves, abundant codes and decodng, Lecture notes n Math., 58, 32-44, Sprnger, Berln, 992. [4] K. W. Shum, I. Aleshnkov, P. V. Kummer, H. Stchtenoth and V. Deolalkar, A low-complexty algorthm for the constructon of algebrac-geometry codes better than the Glbert-Varshamov bound, IEEE. Trans. Inform. Theory, 47, pp , 200. [5] H. Stchtenoth, Algebrac Functon Felds and Codes, Sprnger Verlag, 993. [6] M. Sudan, Lst decodng: Algorthms and applcatons, SIGACT news, 3, pp.6-27, [7] V. We, Generalzed Hammng weght for lnear codes, IEEE. Trans. Inform. Theory, 37, pp.42-48, 99. [8] J. M. Wozencraft, Lst decodng, Quarterly Progress Report MIT, Res. Lab. Electron., Cambrdge, MA, 48, 958. [9] V. V. Zyablov and M. S. Pnsker, Lst cascade decodng n Russan), Probl. Inf. Transm., 7, pp.29-34, 98.