Limits to List Decoding Reed-Solomon Codes

Transcription

1 Limis o Lis Decodig Reed-Solomo Codes Veaesa Guruswami Deparme of Compuer Sciece & Egieerig Uiversiy of Washigo Seale, WA vea@cswashigoedu Ari Rudra Deparme of Compuer Sciece & Egieerig Uiversiy of Washigo Seale, WA ari@cswashigoedu ASTRACT I his paper, we prove he followig wo resuls ha expose some combiaorial limiaios o lis decodig Reed- Solomo codes 1 Give disic elemes α 1,, α from a field F, ad subses S 1,, S of F each of size a mos l, he lis decodig algorihm of Guruswami ad Suda [7] ca i polyomial ime oupu all polyomials p of degree a mos which saisfy p(α i) S i for every i, as log as l < We show ha he performace of his algorihm is he bes possible i a srog sese; specifically, we show ha whe l =, he lis of oupu polyomials ca be super-polyomially large i Oe way o ierpre our resul is he followig The algorihm i [7] ca, whe give as ipu disic pairs (β i, γ i) F (he β i s eed o be disic), fid ad oupu all degree polyomials p such ha p(β i) = γ i for a leas values of i, provided > y our resul, a improveme o he Reed- Solomo lis decoder of [7] ha wors wih slighly smaller agreeme, say > /, ca oly be obaied by exploiig some propery of he β i s (for example, heir (ear) disicess) For Reed-Solomo codes of bloc legh ad dimesio where = δ for small eough δ, we exhibi a explici received word r wih a super-polyomial umber of Reed-Solomo codewords ha agree wih i o ( ε) locaios, for ay desired ε > 0 (we oe agreeme of is rivial o achieve) Such a boud was ow earlier oly for a o-explici ceer We remar ha fidig explici bad lis decodig cofiguraios is of sigifica ieres for example he bes ow rae vs disace rade-off is based o a bad lis decodig cofiguraio for algebraic-geomeric codes [14] which is uforuaely o explicily ow Research suppored i par by NSF Career Award CCF Permissio o mae digial or hard copies of all or par of his wor for persoal or classroom use is graed wihou fee provided ha copies are o made or disribued for profi or commercial advaage ad ha copies bear his oice ad he full ciaio o he firs page To copy oherwise, o republish, o pos o servers or o redisribue o liss, requires prior specific permissio ad/or a fee STOC 05, May -4, 005, alimore, Marylad, USA Copyrigh 005 ACM /05/0005 $500 Caegories ad Subjec Descripors E4 [Daa]: Codig ad Iformaio Theory; F1 [Theory of Compuaio]: Aalysis of Algorihms ad Problem Complexiy; Numerical Algorihms ad Problems Geeral Terms Algorihms, Theory Keywords Reed-Solomo Codes, Lis Decodig, CH Codes, Lis Recoverig, Johso boud 1 INTRODUCTION Reed-Solomo codes are a impora ad exesively sudied family of error-correcig codes The codewords of a Reed-Solomo code (heceforh, RS code) over a field F are obaied by evaluaig low degree polyomials a disic elemes of F If he degree of he polyomials is a mos, ad a polyomial p is ecoded as p(α 1), p(α ),, p(α ), his gives a [, +1, ] code, ie, a code of bloc legh, dimesio +1 ad disace (he disace propery follows from he fac ha wo disic degree polyomials ca agree o a mos pois) This is opimal i erms of disace o dimesio rade-off (mees he so-called Sigleo boud ), which alog wih he code s ice algebraic properies, give RS codes a promie place i codig heory As a resul he problem of decodig RS codes has received much aeio The bes ow polyomial ime algorihm oday (i erms of umber of errors correced) ca, give a received word y 1,, y F, fid ad oupu a lis of all degree polyomials p which saisfy p(α i) = y i for a leas values of i {1,, }, provided > [11, 7] Noe ha his is a lis decodig algorihm ha oupus a lis of all codewords wih he requisie agreeme The performace of he algorihm i [7] maches he so-called Johso boud (cf [8]) which gives a geeral lower boud o he umber of errors oe ca correc usig small liss i ay code, as a fucio of he disace of he code Our wor i his paper is moivaed by he quesio of wheher his resul is he bes possible (ie, wheher he Johso boud is igh for Reed-Solomo codes) y his we mea wheher aempig o decode wih a lower agreeme parameer migh lead o super-polyomially large liss as oupu, which of course will preclude a polyomial ime algorihm (a leas i he sadard model where he lis has o be produced

2 explicily) While we do quie show his o be he case i his paper, we give evidece i his direcio by demosraig ha i a somewha more geeral seig o which also he algorihm of Guruswami ad Suda [7] applies, is performace is ideed he bes possible The deails follow 11 Limiaios o lis recoverig The algorihm i [7] i fac solves he followig more geeral polyomial recosrucio problem i polyomial ime: Give disic pairs (β i, γ i) F (we sress ha he β i s eed o be disic), oupu a lis of all polyomials p of degree which saisfy p(β i) = γ i for more ha values of i {1,,, } I paricular, he algorihm ca solve he followig lis recoverig problem 1 for a [, +1, ] q Reed-Solomo code as log as he parameer l saisfies l < : Defiiio 1 (Lis Recoverig) For a q-ary code C of bloc legh, he lis recoverig problem is he followig We are give a se S i F q of possible symbols for he i h symbol for each posiio i, 1 i, ad he goal is o oupu all codewords c = c 1,, c such ha c i S i for every i Whe each S i has a mos l elemes, we refer o he problem as lis recoverig wih ipu liss of size l I oher words, give disic elemes α 1,, α from a field F, ad subses S 1,, S of F each of size a mos l, oe ca oupu all degree polyomials p which saisfy p(α i) S i for every i i polyomial ime I Secio, we demosrae ha his laer performace is he bes possible wih surprisig accuracy specifically, we show ha whe l =, here are seigs of parameers for which he lis of oupu polyomials eeds o be super-polyomially large i (Theorem 3) As a corollary, his rules ou a efficie soluio o he polyomial recosrucio algorihm ha wors eve uder he slighly weaer codiio > / I his respec, he square roo boud achieved by [7] is opimal, ad ay improveme o heir lis decodig algorihm which wors wih agreeme fracio / < r where r = / is he rae of he code, or i oher words which wors beyod he Johso boud, mus exploi he fac ha he evaluaio pois β i are disic (or almos disic ) We are emped o view his as evidece ha he r boud is he miimum agreeme uder which lis decodig is possible for he RS code, ad hope ha his wor paves he way owards eveual resoluio of his quesio While his par o ighess of Johso boud remais speculaory a his sage, for he problem of lis recoverig iself, our wor proves ha RS codes are ideed sub-opimal, as we describe below Guruswami ad Idy [6] prove ha here exiss a fixed R > 0 such ha for every ieger l here are codes of rae R which are lis recoverable give ipu liss of size l (he alphabe size ad oupu lis size will ecessarily grow wih l) O he oher had, by our wor Reed-Solomo codes for lis recoverig wih ipu liss of size l mus have rae a mos 1/l Thus, despie he fac ha mos of he iiial success i efficie lis decodig 1 The ermiology lis recoverig was coied i [6] hough he problem was cosidered i may guises before, icludig for example i [1] This i ur rules ou, for every ε > 0, a soluio o he polyomial recosrucio algorihm ha wors as log as p (1 ε) has bee for algebraic codes lie Reed-Solomo codes, oe eeds o loo for ew codes i oher domais i order o mae progress owards he cosrucio of ad algorihms for ear-opimal lis-decodable codes 1 Explici bad lis decodig cofiguraios The resul meioed above preses a explici bad lis recoverig cofiguraio, ie, a ipu isace o he lis recoverig problem wih a super-polyomial umber of soluios To prove resuls o limiaios of lis decodig, such as he ighess of he Johso boud, we eed o demosrae a received word (or ceer) r wih super-polyomially may codewords ha agree wih r a or more places A simple couig argume esablishes he exisece of such received words for cerai seigs of parameers,, [10, ] i paricular for = δ, oe ca ge = for ay δ δ > 0 I Secio 3, we demosrae a explici cosrucio of such a received word wih super-polyomial umber of codewords wih agreeme up o ( ε) for ay ε > 0 Noe ha such a cosrucio is rivial for = sice we ca ierpolae degree polyomials hrough ay se of pois I geeral, he ques for explici cosrucios of his sor (amely small Hammig balls wih several codewords) is well moivaed If achieved wih appropriae parameers hey will lead o a deradomizaio of he iapproximabiliy resul for compuig he miimum disace of a liear code [3] However, for his applicaio i is impora o ge Ω(1) codewords i a ball of radius ρ imes he disace of he code for some cosa ρ < 1 While we ge he former, we oly achieve ρ = 1 o(1) As aoher moivaio, we poi ou ha he curre bes rade-off bewee rae ad relaive disace is achieved by a o-liear code comprisig of precisely a bad lis decodig cofiguraio i cerai algebraic-geomeric codes [14] Uforuaely he associaed ceer is oly show o exis by a couig argume ad is explici specificaio will be required o ge explici codes wih hese parameers 13 Proof Approach We show our resul o lis recoverig Reed-Solomo codes by provig a super-polyomial (i = q m ) boud o he umber of polyomials over F q m of degree abou q m 1 ha ae values i F q a every poi i F q m, for ay prime power q Noe ha his implies ha here ca be a superpolyomial umber of soluios o lis recoverig whe ipu lis sizes are We esablish his boud o he umber of such polyomials by exploiig a follore coecio of such polyomials o a classic family of cyclic codes called CH codes, followed by a (exac) esimaio of he size of CH codes wih cerai parameers We also wrie dow a explici collecio of polyomials, obaied by aig F q-liear combiaios of raslaed orm fucios, all of which ae values oly i F q y he CH boud, we coclude ha his i fac is a precise descripio of he collecio of all such polyomials Our explici cosrucio of a ceer wih several RS codewords wih o-rivial agreeme wih i is obaied usig ideas from [] relaig o represeaios of elemes i a exesio fiie field by producs of disic liear facors 14 Relaed Wor Our wor, specifically he par ha deals wih precisely

3 describig he collecio of polyomials ha ae values oly i F q, bears some similariy o [5] which also exhibied limis o lis recoverabiliy of codes Oe of he simple ye powerful ideas used i [5], ad also i he wor o exracor codes [1], is ha polyomials which are r h powers of a lower degree polyomial ae oly values i a muliplicaive subgroup cosisig of he r h powers i he field Specifically, he cosrucio i [1, 5] yields roughly l codewords for lis recoverig where l is he size of he S i s i Defiiio 1 Noe ha his gives super-polyomially may codewords oly whe he ipu liss are asympoically bigger ha / I our wor, we also use r h powers, bu he value of r is such ha he r h powers form a subfield of he field Therefore, oe ca also freely add polyomials which are r h powers ad he sum sill aes o values i he subfield This les us demosrae a much larger collecio of polyomials which ae o oly a small possible umber of values a every poi i he field Provig bouds o he size of his collecio of polyomials uses echiques ha are ew o his lie of sudy The echique behid our resuls i Secio 3 is closely relaed o ha of he rece resul of Cheg ad Wa [] o coecios bewee Reed-Solomo lis decodig ad he discree logarihm problem over fiie fields CH CODES AND LIST RECOVERING REED-SOLOMON CODES 1 Mai Resul We will wor wih polyomials over F q m of characerisic p where q is a power of p, ad m 1 We will deoe by F q m he se of ozero elemes i he field F q m Our goal i his secio is o prove he followig resul, ad i Secio we will use i o sae corollaries o limis o lis decodabiliy of Reed-Solomo codes (We will oly eed a lower boud o he umber of polyomials wih he saed propery bu he resul below i fac gives a exac esimaio, which i ur is used i Secio 3 o give a precise characerizaio of he cocered polyomials) Theorem 1 Le q be a prime power, ad m 1 be a ieger The, he umber of uivariae polyomials i F q m[z] of degree a mos qm 1 which ae values i F q whe evaluaed a every poi i F q m is exacly q m Tha is, {P (z) F q m[z] deg(p ) qm 1 ad α Fqm, q 1 P (α) F q} = q m I he res of his secio, we prove Theorem 1 The proof is based o a coecio of polyomials wih he saed propery o a family of cyclic codes called CH codes, followed by a esimaio of he size (or dimesio) of he associaed CH code We begi wih he defiiio of CH codes, (wha we defie are acually referred o more specifically as arrow-sese primiive CH codes, bu we will jus use he erm CH codes for hem) We poi he reader o [9], Ch 7, Sec 6, ad Ch 9, Secs 1-3, for deailed bacgroud iformaio o CH codes Defiiio Le α be a primiive eleme of F q m, ad le = q m 1 The CH code CH q,m,d,α of desiged disace d is a liear code of bloc legh over F q defied as: CH q,m,d,α = { c 0, c 1,, c F q c(α i ) = 0 for i = 1,,, d 1, where c(x) = c 0 + c 1x + + c x F q[x] } We will omi oe or more he subscrips i CH q,m,d,α for oaioal coveiece whe hey are clear from he coex Our ieres i CH codes is due o he followig follore resul, which preses a alerae view of CH codes as wha are called subfield subcodes (cf [9, Ch 7, Sec 7]) of Reed-Solomo codes (I is our opiio ha his alerae view, despie beig much easier o sae, does o ge he meio i deserves i he sadard codig exboos For sae of compleeess, ad sice we view our wor as a good opporuiy o do so, we prese a proof i Appedix A) Lemma 1 (CH codes are subfield subcodes of RS codes) Le q be a prime power ad m 1 a ieger Le = q m 1, d be a ieger i he rage 1 < d <, ad α be a primiive eleme of F q m The he codewords of CH q,m,d,α are i oe-oe correspodece wih elemes of he se { P (α 0 ), P (α 1 ),, P (α ) F q P F q m[z], deg(p ) d, ad P (γ) F q γ F q m} I ligh of he above lemma, i order o prove Theorem 1, we have o prove ha CH q,m,d,α = q m whe d = (q m 1)(1 1 ) We ur o his as ex We begi wih he followig boud o he size of CH codes [1, Ch 1], ad give a sech of is proof for he sae of compleeess Lemma (Dimesio of CH Codes) For ieger i,, defie i by he codiios i = i mod ad 0 i 1 The CH q,m,d,α = q I(q,m,d) where I(q, m, d) = {i 0 i 1, iq j d for all j, 0 j m 1} (1) for = q m 1 (Noe ha for his value of, if i = i 0 + i 1q + i m 1q m 1, he iq = i m 1 + i 0q + i 1q + + i m q m 1, ad so iq is obaied by a simple cyclic shif of he q-ary represeaio of i) Proof I follows from Defiiio ha he CH codewords are simply polyomials c(x) over F q of degree a mos ( 1) which vaish a α i for 1 i < d Noe ha if c(x), c (x) are wo such polyomials, he so is c(x) + c (x) Moreover, sice α = 1, xc(x) mod (x 1) also vaishes a each desigaed α i I follows ha if c(x) is a codeword, he so is r(x)c(x) mod (x 1) for every polyomial r(x) F q[x] I oher words CH q,m,d is a ideal i he quoie rig R = F q[x]/(x 1) I is well ow ha R is a pricipal ideal rig, ie, a rig i which every ideal is geeraed by

4 oe eleme Therefore here is a uique moic polyomial g(x) F q[x] such ha CH q,m,d,α = {g(x)h(x) h(x) F q[x]; deg(h) deg(g)} I follows ha CH q,m,d,α = q deg(g), ad so i remais o prove ha deg(g) = I(q, m, d) where I(q, m, d) is defied as i (1) I is easily argued ha he polyomial g(x) is he moic polyomial of lowes degree over F q ha has α i for every i, 1 i < d, as roos Now comes he simple bu crucial propery: sice g(x) has coefficies i F q, ad q is a power of he characerisic of he field, g(x q ) = g(x) q ideically as polyomials, ad i paricular for every γ F q m g(γ) = 0 if ad oly if g(γ q ) = 0 Recallig he way i was defied ad ha γ = 1 for all γ F qm, he above implies ha for every i, 0 i 1, ad every j, 0 j m 1, g(α i ) = 0 if ad oly if g(α iqj ) = 0 () Usig he above we claim ha for 0 i 1, if i / I(q, m, d), he g(α i ) = 0 This immediaely gives us he lower boud deg(g) I(q, m, d), as g has a leas I(q, m, d) disic roos Ideed, suppose i / I(q, m, d) The here mus exis some j, 0 j m 1 such ha iq j > d, or equivalely i = iq j d 1 Sice g(x) belogs o he CH code, g(α i ) = g(α iqj ) = 0, which by () implies g(α i ) = 0 For he oher direcio, defie he polyomial h(x) F q m[x] as Y h(x) = (x α i ) i / I(q,m,d) 0 i y defiiio of h ad I(q, m, d), i is easily see ha γ is a roo of h if ad oly if γ q is a roo of h y a wellow algebra fac (cf [13, Thm 11]), his implies ha he coefficies of h lie i F q, ie, h(x) F q[x] Also oe ha if d < i 1, he clearly i / I(q, m, d), ad herefore h(α i ) = 0 for d < i 1, or equivalely h(α i ) = 0 for 1 i < d Thus h(x) belogs o he CH code ad herefore mus be divisible by g(x) Hece deg(g) deg(h) = I(q, m, d), which combied wih our earlier lower boud gives deg(g) = I(q, m, d) Le s ow use he above o compue he size of CH q,m,d,α where d = (q m 1) qm 1 We eed o compue he quaiy I(q, m, d), ie, he umber of i, 0 i < q m 1 such ha iq j q m 1 qm 1 = 1 + q + + q m 1 for each j = 0, 1,, m 1 This codiio is equivale o sayig ha if i = i 0 + i 1q + + i m 1q m 1 is he q-ary expasio of i, he all he m iegers whose q-ary represeaios are cyclic shifs of (i 0, i 1,, i m 1) are a mos 1+q+ +q m 1 Clearly, his codiio is saisfied if ad oly if ha for each j = 0, 1,, m 1, i j {0, 1} There are m choices for i wih his propery, ad hece we coclude I(q, m, d) = m whe d = (q m 1) qm 1 Togeher wih Lemma 1, we coclude ha he umber of polyomials of degree a mos qm 1 over F q m which ae o values oly i F q a every poi i F q m is precisely q m This is exacly he claim of Theorem 1 efore movig o o sae implicaios of above resul for Reed-Solomo lis decodig, we sae he followig geeralizaio of Theorem 1 which ca proved i he same maer (he resul of Theorem 1 is he case whe parameer = m): Theorem Le q be a prime power, ad m 1 be a ieger The, for each, 1 m, he umber of uivariae polyomials i F q m[z] of degree a mos P j=1 qm j which ae values i F q whe evaluaed a every poi i F q m is exacly qp ( m j ) Ad he umber of such polyomials of degree sricly less ha q m 1 is exacly q (amely jus he cosa polyomials, so here are o polyomials wih his propery for degrees bewee 1 ad q m 1 1) Implicaios for Reed-Solomo Lis Decodig I he resul of Theorem 1, if we imagie eepig q 3 fixed ad le m grow, he for he choice = q m ad = (q m 1)/(q 1) (so ha = q), Theorem 1 immediaely gives us he followig egaive resul o polyomial recosrucio algorihms ad Reed-Solomo lis decodig 3 Theorem 3 For every prime power q 3, here exis ifiiely may pairs of iegers, such ha = q for which here are Reed-Solomo codes of dimesio ( + 1) ad bloc legh, such ha lis recoverig hem wih ipu liss of size requires super-polyomial (i fac q1/ lg q ) oupu lis size The above resul is exacly igh i he followig sese I is easy o argue combiaorially (via he Johso ype bouds, cf [8]) ha whe l <, he umber of codewords is polyomially bouded Moreover [7] preses a polyomial ime algorihm o recover all he soluio codewords i his case The algorihm i [7] solves he more geeral problem of fidig all polyomials of degree a mos which agree wih a leas ou of disic pairs (β i, γ i) wheever > The followig corollary saes ha, i ligh of Theorem 3, his is esseially he bes possible rade-off oe ca hope for from such a geeral algorihm We view his as providig he message ha a lis decodig algorihm for Reed-Solomo codes ha wors wih fracioal agreeme / ha is less ha r where r is he rae, mus exploi he fac ha he evaluaio pois β i are disic or almos disic (by which we mea ha o β i is repeaed oo may imes) Noe ha for small values of r (close o 0), our resul covers eve a improveme of he ecessary fracioal agreeme by O(r) which is subsaially smaller ha r Corollary 1 Suppose A is a algorihm ha aes as ipu disic pairs (β i, γ i) F for a arbirary field F ad oupus a lis of all polyomials p of degree a mos for which p(β i) = γ i for more ha pairs The, here exis ipus uder which A mus oupu a lis of superpolyomial size Proof Noe ha i he lis recoverig seig of Theorem 3, he oal umber of pairs = l = < ( +1), ad he agreeme parameer = The r < + 1 r = We remar ha we used he oaio = q m 1 i he previous subsecio, bu for his Subsecio we will ae = q m

5 1 + = = Therefore here ca be super-polyomially may cadidae polyomials o oupu eve whe he agreeme parameer saisfies > / 3 A precise descripio of polyomials wih values i base field We proved i Secio 1, for Q = qm 1, here are exacly q m polyomials over F q m of degree Q or less ha evaluae o a value i F q a every poi i F q m The proof of his obais he coefficies of such polyomials usig a Fourier rasform of codewords of a associaed CH code, ad as such gives lile isigh io he srucure of hese polyomials Oe of he aural quesios o as is: Ca we say somehig more cocree abou he srucure of hese q m polyomials? I his secio, we aswer his quesio by givig a exac descripio of he se of all hese q m polyomials We begi wih he followig well-ow fac which simply saes ha he Norm fucio of F q m over F q aes oly values i F q Lemma 3 For all x F q m, x qm 1 F q Theorem 4 Le q be a prime power, ad le m 1 Le α be a primiive eleme of F q m The, here are exacly q m uivariae polyomials i F q m[z] of degree a mos Q = q m 1 which ae values i F q whe evaluaed a every poi i F q m, ad hese are precisely he polyomials i he se m 1 N = { β i(z + α i ) Q β 0, β 1,, β m 1 F q} Proof y Lemma 3, clearly every polyomial P i he se N saisfies P (γ) F q for all γ F q m The claim ha here are exacly q m polyomials over F q m of degree Q or less ha ae values oly i F q was already esablished i Theorem 1 So he claimed resul ha N precisely describes he se of all hese polyomials follows if we show ha N = q m Noe ha by defiiio, N q m To show ha N q m, i clearly suffices o show (by lieariy) ha if m 1 β i(z + α i ) Q = 0 (3) as polyomials i F q m[z], he β 0 = β 1 = = β m 1 = 0 We will prove his by seig up a full ra homogeeous liear sysem of equaios ha he β i s mus saisfy For his we eed Lucas heorem, saed below Lemma 4 (Lucas Theorem, cf [4]) Le p be a prime Le a ad b be posiive iegers wih p-ary expasios a 0 + a 1p + + a rp r ad b 0 + b 1p + + b rp r respecively The `a b = `a0 b 0 `a1 b 1 `ar b r mod p which gives us `a b 0 mod p if ad oly if a j b j for all j {0, 1,, r} Defie he se T = { j S q j S {0,, m 1} } Applyig Lemma 4 wih p beig he characerisic of he field F q, we oe ha whe operaig i he field F q m, he biomial coefficie of z j i he expasio of (z + α i ) Q is 1 if j T ad 0 oherwise I follows ha (3) holds if ad oly if P m 1 (α i ) Q j β i = 0 for all j T, which by he defiiio of T ad he fac ha Q = 1 + q + q + + q m 1 is equivale o (α j ) i β i = 0 for all j T (4) m 1 Le us label he m elemes {α j j T } as α 0, α 1,, α m 1 (oe ha hese are disic elemes of F q m sice α is primiive i F q m) The coefficie marix of he homogeeous sysem of equaios (4) wih uows β 0,, β m 1 is he he Vadermode marix 1 α 0 α 0 α m α 1 α1 α m α m 1 α m 1 α m 1 m 1 which has full ra Therefore, he oly soluio o he sysem (4) is β 0 = β 1 = = β m 1 = 0, as desired 1 C A 4 Some furher facs o he CH code ad lis recoverig associaed RS code The resuls i he previous subsecios show ha a large umber q m of polyomials over F q m ae o values i F q a every evaluaio poi, ad his proved he ighess of he square-roo boud for agreeme = = q m ad oal umber of pois = q (recall Corollary 1) I is a aural quesio wheher similarly large lis size ca be show a oher pois (, ), specifically for slighly smaller ad For example, wha if = (q 1) ad we cosider lis recoverig from liss of size q 1 I paricular, how may polyomials of degree Q = (q m 1)/(q 1) ae o values i F q \ {0} a pois i F q m I is easily see ha whe = = q m, here are precisely (q 1) such polyomials, amely he cosa polyomials which equal a eleme of F q Ideed, by he Johso boud, sice > Q for he choice = ad = (q 1), we should o expec a large lis size However, eve for he slighly smaller amou of agreeme = = Q, here are oly abou a liear i umber of codewords, as Lemma 5 below shows Hece obaiig super-polyomial umber of codewords a oher pois o he square-roo boud whe he agreeme is less ha he bloc legh remais a ieresig quesio which perhaps he CH code coecio jus by iself cao resolve Lemma 5 Le q be a prime power ad le m > 1 For ay polyomial P (z) over F q m[z], le i s Hammig weigh be defied as {β F q m P (β) 0} The, here are exacly (q 1)q m uivariae polyomials i F q m[z] of degree a mos Q = (qm 1) which ae values i F q whe evaluaed a every poi i F q m ad which have Hammig weigh (q m 1) Furhermore, hese are precisely he polyomials i he se W = {λ(z + β) Q β F q m, λ F q} Proof I is obvious ha all he polyomials i W saisfy he required propery ad are disic polyomials We ex show ha ay polyomial of degree a mos Q which saisfies he required properies belogs o W compleig he proof Le P (z) be a polyomial of degree a mos Q which saisfies he required properies We mus show ha P (z) W

6 Le γ F q m be such ha P (γ) = 0 Clearly, for each β (F q m {γ}), P (β)/(β γ) Q F q y a pigeohole argume, here mus exis some λ F q such ha P (β) = λ(β γ) Q for a leas qm 1 = Q values of β i F q m {γ} Sice P (γ) = 0, we have ha he degree Q polyomials P (z) ad λ(z γ) Q agree o a leas Q + 1 field elemes, which meas ha hey mus be equal o each oher Thus he polyomial P (z) belogs o W ad he proof is complee 3 EPLICIT HAMMING ALLS WITH SEV- ERAL REED-SOLOMON CODEWORDS Throughou his secio, we will be cocered wih a [q, + 1] Reed-Solomo code RS[q, ] over F q We will be ieresed i a ceer r F q q such ha a super-polyomial umber of codewords of RS[q, ] agree wih r o or more posiios, ad he aim would be o prove such a resul for o-rivially larger ha I is easy o prove he exisece of such a r wih a leas `q /q codewords wih agreeme a leas wih r Oe way o see his is ha his quaiy is he expeced umber of such codewords for a received word ha is he evaluaio of a radom polyomial of degree [10] 4 A relaed way is suggesed i [] based o a eleme β i F q h = F q(α), for some posiive ieger h, ha ca be wrie as a produc Q `q a T (α+a) for a leas /q h subses T F q wih T = he exisece of such a β agai follows by a rivial couig argume Here we use he fac ha for cerai seigs of parameers ad fields such a β ca be explicily specified wih oly a sligh loss i he umber of subses T (see Theorem 5 below), ad hereby ge a explici ceer r wih several close-by codewords from RS[q, ] Theorem 5 Le ε > 0 be arbirary Le q be a prime power, h be a posiive ieger ad α be such ha F q(α) = F q h For ay β F q, le N (β) deoe he umber of h Q -uples a 1, a,, a of disic a i F q such ha β = i=1 (α + ai) If ( 4 + )(h + 1) ɛ ad5 q max(, (h 1) (+ɛ) (+ɛ) ), he for all β F q, N (β) > ( 1)q h 1 h Proof From he proof of Theorem 4 i [], we obai N (β) E 1 E, where E 1 = q ( )q 1 ad E q h 1 = (1 + ` )(h 1) q Observe ha from he choice of q, ` = q We firs give a lower boud o E 1 Ideed, usig ` q ad q h 1 < q h, we have E 1 > q (q )q 1 = q h + q h 1 Noe ha from our choice of, we have > ( 4 + )h, ha ɛ is, h > ( 4+ɛ ) Furher, from our choice of q, (h 4+ɛ 1) q +ɛ 1 We ow boud E from above From our bouds ad (h 1), we have E (1 + q 4+ɛ )q( 4+ɛ ) 1 < o ` (1 + q )q h 1 = q h ( 1)q h 1, where he secod iequaliy comes from our boud o h Combiig he bouds o E 1 ad E proves he heorem 4 The boud ca be improved slighly o `q /q 1 by usig a radom moic polyomial 5 We also eed ɛ <, bu his will be saisfied sice we will hi of ɛ as a fixed cosa ad le q, h ad grow q h We ow sae our mai resul of his secio cocerig Reed-Solomo codes: Theorem 6 Le ɛ > 0 be arbirary, q a prime power, ad h ay posiive ieger If ( 4 + )(h + 1) ad q ɛ max(, (h 1) (+ɛ) (+ɛ) ) he for every i he rage h 1, here exiss a explici received word r F q q such q!( +h ha here are a leas codewords of RS[q, ] which ) agree wih r i a leas posiios We will prove he above heorem a he ed of his secio As ɛ 0, ad q, g, h i he above, we ca ge super-polyomially may codewords wih agreeme (1+δ) for some δ = δ(ε) > 0 for a Reed-Solomo code of dimesio edig o q 1/ As ɛ, we ca ge superpolyomially may codewords wih agreeme edig o wih dimesio sill beig q Ω(1) We record hese as wo corollaries below We oe ha he o-explici boud `q /q gives a super-polyomial umber of codewords for agreeme /δ for dimesio abou = q δ o(1), where as our explici cosrucio ca give agreeme a mos (or dimesio a mos q) Corollary For all 0 < γ < 1, here exiss δ > 0 such ha for all large eough prime powers q, here exiss a explici r F q q such ha he Reed-Solomo code RS[q, = q δ ] coais a super-polyomial (i q) umber of codewords wih agreeme a leas ( γ) wih r Corollary 3 For all 0 < γ < 1, here exiss δ > 0, such ha for all large eough prime powers q, here is a explici r F q q such ha he Reed-Solomo code RS[q, = q 1/ γ ] coais a super-polyomial (i q) umber of codewords wih agreeme a leas (1 + δ) wih r Proof of Theorem 6 I wha follows, we fix H(x) o be a polyomial of degree h ha is irreducible over F q For he res of his proof we will deoe F q[x]/(h(x)) by F q h Also oe ha for ay roo α of H, F q(α) = F q h Pic ay l where 0 l h 1 ad oe ha q ad saisfy he codiios of Theorem 5 For ay = (b 0, b 1,, b l ), where b i F q wih a leas oe o zero b j; defie L (x) def = P l bixi Fix r(x) o be a arbirary o-zero polyomial of degree a mos h 1 y heir defiiios, r(α) ad L (α) are elemes of F q h We will se he ceer r o be r(a) H(a) a F q Noe ha sice H(x) is a irreducible polyomial, H(a) 0 for all a F q, ad r is a well-defied eleme of F q q We ow proceed o boud from below he umber of polyomials of degree def = + l h which agree wih r o posiios For each o-zero uple F l+1 q, defie Q (x) = r(x) Clearly, Q(α) L (x) F q For oaioal h coveiece we will use N o deoe N (Q (α)) The, for j = 1,, N here exis A (,j) where A (,j) F q ad A (,j) = such ha P (j) def (α) = Q a A (,j) (α + a) = Q (α) y Theorem 5, we have N ( 1)q h 1 for every le us deoe by N his laer quaiy Recallig he defiiio of Q, we have ha for ay (, j), r(α) (j) (j) L = P (α) (α), or equivalely r(α)+p (α)l(α) = 0 Sice H is he irreducible polyomial of α over F q, his implies ha H(x) divides P (j) (x)l(x) + r(x) i Fq[x]

7 Fially we defie T (j) (x) o be a polyomial of degree = + l h such ha T (j) (j) (x)h(x) = P (x)l(x) + r(x) (5) Clearly T (j) ( a) equals r( a)/h( a) for each a A (,j) ad hus he polyomial T (j) agrees wih r o a leas posiios To complee he proof we will give a lower boud o he umber of disic polyomials i he collecio {T (j) } For a fixed, ou of he N choices for P (j),! choices of j would lead o he same6 polyomial of degree Sice N N, here are a leas (ql+1 1)N! choices of pairs (, j) Clearly for j 1 j he polyomials P (j 1) (x) ad P (j ) (x) are disic, however we could have P (j 1) 1 (x)l 1 (x) = P (j ) (x)l (x) (boh are equal o say S(x)) leadig o T (j 1) 1 (x) = T (j ) (x) However he degree of S is a mos + l = + h, ad hece S ca have a mos + h roos, ad herefore a mos `+h facors of he form Q a T (x + a) wih T = I follows ha o sigle degree polyomial is coued more ha `+h imes i he collecio {T (j) }, ad hece here mus be a leas (q l+1 1)N q!`+h!`+h disic polyomials amog hem, where we used N = ( 1)q h 1 ad (q l+1 1)( 1) q l+1 = q +h+1 sice = + l h Compariso wih he Cheg-Wa paper [] Our resuls i his subsecio build upo he resuls i [] however, our aim is slighly differe compared o heirs i ha we wa o ge a large collecio of codewords close by o a received word I paricular i Theorem 5, we ge a esimae o N (β) while Cheg ad Wa oly require N (β) > 0 Also Cheg ad Wa cosider equaio (5) oly wih he choice L (x) = 1 4 CONCLUSIONS AND OPEN QUESTIONS Our wor exposes limiaios o he id of rade-offs for lis recoverig achievable usig Reed-Solomo codes, ad i paricular demosraes ha RS codes are quie far from he bes possible i his regard Specifically, hey ca have rae a mos 1/l for lis recoverig wih ipu liss of size l whe he bes, albei o-explici, codes ca achieve cosa rae I is ieresig ha he resul is exacly igh: lis recoverig for ipu liss of size l whe l < is possible i polyomial ime, while a l = we migh have o cofro super-polyomial umber of soluio codewords Our wor raises several ieresig quesios for fuure wor; we lis some of hem below: We have show ha RS codes of rae 1/l cao be lis recovered wih ipu liss of size l i polyomial ime whe l is a prime power Ca oe show a similar resul for oher values of l? Usig he desiy of primes ad our wor, we ca boud he rae by O(1/l), bu 6 If a 1,, a is a soluio of he equaio β = Q i=1 (α + a i) he so is a σ(1),, a σ() for ay permuaio σ o {1,, } if i is rue i will be ice o show i is a mos 1/l for every l We have show ha he boud for polyomial recosrucio is he bes possible give geeral pairs (β i, γ i) F as ipu I remais a big challege o deermie wheher his is he case also whe he β i s are all disic, or equivalely wheher he Johso boud is he rue lis decodig radius of RS codes We cojecure his o be he case Oe approach ha migh give a leas parial resuls would be o use some of our ideas (i paricular hose usig he orm fucio, possibly exeded o oher symmeric fucios of he auomorphisms of F q m over F q) ogeher wih ideas i he wor of Jusese ad Høhold [10] who used he Trace fucio o demosrae ha a liear umber of codewords could occur a he Johso boud Ca oe show a aalog of Theorem 5 o producs of liear facors for he case whe is liear i he field size q (he currely ow resuls wor oly for up o q 1/ )? This is a ieresig field heory quesio i iself, ad furhermore migh help owards showig he exisece of super-polyomial umber of Reed- Solomo codewords wih agreeme (1 + ε) for some ε > 0 for cosa rae (ie whe is liear i )? I is impora for he laer, however, ha we show ha N (β) is very large for some special field eleme β i a exesio field, sice by a rivial couig argume i follows ha here exis β F q for which h N (β) `q /(q h 1) 5 REFERENCES [1] Elwy erleamp Algebraic Codig Theory McGraw-Hill Series i Sysems Sciece, 1968 [] Qi Cheg ad Daqig Wa O he lis ad bouded disace decodabiliy of Reed-Solomo codes I Proceedigs of he 45h Aual Symposium o Foudaios of Compuer Sciece (FOCS), pages , Ocober 004 [3] Ilya Dumer, Daiele Micciacio, ad Madhu Suda Hardess of approximaig he miimum disace of a code IEEE Trasacios o Iformaio Theory, 49(1): 37, Jauary 003 [4] Adrew Graville The arihmeic properies of biomial coefficies I hp://wwwcecmsfuca/orgaics/papers/graville/, 1996 [5] Veaesa Guruswami, Joha Hȧsad, Madhu Suda, ad David Zucerma Combiaorial bouds for lis decodig IEEE Trasacios o Iformaio Theory, 48: , May 00 [6] Veaesa Guruswami ad Pior Idy Expader-based cosrucios of efficiely decodable codes I Proceedigs of he 4d Aual Symposium o Foudaios of Compuer Sciece (FOCS), pages , 001 [7] Veaesa Guruswami ad Madhu Suda Improved decodig of Reed-Solomo ad algebraic-geomeric codes IEEE Trasacios o Iformaio Theory, 45: , 1999

8 [8] Veaesa Guruswami ad Madhu Suda Exesios o he Johso oud Mauscrip; available from hp://heorylcsmiedu/ madhu/papershml, 000 [9] F J MacWilliams ad Neil J A Sloae The Theory of Error-Correcig Codes Elsevier/Norh-Hollad, Amserdam, 1981 [10] Jør Jusese ad Tom Høhold ouds o lis decodig of MDS codes IEEE Trasacios o Iformaio Theory, 47(4): , May 001 [11] Madhu Suda Decodig of Reed-Solomo codes beyod he error-correcio boud Joural of Complexiy, 13(1): , 1997 [1] Amo Ta-Shma ad David Zucerma Exracor Codes IEEE Trasacios o Iformaio Theory, 50(1): , 004 [13] J H va Li Iroducio o Codig Theory Graduae Texs i Mahemaics 86, (Third Ediio) Spriger-Verlag, erli, 1999 [14] Chaopig ig Noliear codes from algebraic curves improvig he Tsfasma-Vladu-Zi boud IEEE Trasacios o Iformaio Theory, 49(7): , 003 APPENDI A CH CODES ARE SUFIELD SUCODES Lemma Le q be a prime power ad m 1 a ieger Le = q m 1, d be a ieger i he rage 1 < d <, ad α be a primiive eleme of F q m The he codewords of CH q,m,d,α are i oe-oe correspodece wih elemes of he se { P (α 0 ), P (α 1 ),, P (α ) P F q m[z], deg(p ) d, ad P (γ) F q γ F q m} We ex proceed o show he iclusio S 1 S Suppose c 0, c 1,, c S 1 For 0 j 1, defie (his is he iverse Fourier rasform ) a j = 1 c iα ji, where by 1, we mea he muliplicaive iverse of 1 i he field F q m Noe ha a j = 1 c(α j ) = 1 c(α j ) where c(x) = P cixi So, by he defiiio of S 1, i follows ha a j = 0 for j > d Therefore he polyomial P (z) F q m defied by d P (z) = a jz j = a jz j has degree a mos ( d) We ow claim ha for P (α s ) = c s for 0 s 1 Ideed, P (α s ) = = a jα sj = c i 1 (α s i ) j = c s, c iα ji «α sj where i he las sep we used he fac ha P (αs i ) j = 0 wheever i s, ad equals whe i = s Therefore, c 0, c 1,, c = P (α 0 ),, P (α ) We are prey much doe, excep ha we have o chec also ha P (0) F q (sice we waed P (γ) F q for all γ F q m, icludig γ = 0) Noe ha P (0) = a 0 = 1 P ci Sice = qm 1, we have + 1 = 0 i F q m ad so 1 = 1 Fq This ogeher wih he fac ha c i F q for every i implies ha P (0) F q as well, compleig he proof Proof Our goal is o prove ha he wo ses S 1 = { c 0, c 1,, c c(α i ) = 0 for i = 1,,, d 1, where c(x) = c 0 + c 1x + + c x F q[x] }, S = { P (α 0 ), P (α 1 ),, P (α ) P F q m[z], deg(p ) d, ad P (γ) F q γ F q m}, are ideical We will do so by showig boh he iclusios S S 1 ad S 1 S We begi wih showig S S 1 Le P (z) = P d ajzj F q m[z] be a polyomial of degree a mos ( d) ha aes values i F q The, for r = 1,,, d 1, we have d P (α i )(α r ) i = a j α ij d α ri = a j (α r+j ) i = 0, where i he las sep we use ha P γi = 0 for every γ F q m \ {1} ad α r+j 1 sice 1 r + j 1 ad α is primiive Therefore, P (α 0 ), P (α 1 ),, P (α ) S 1