MINIMUM DISTANCE DECODING OF GENERAL ALGEBRAIC GEOMETRY CODES VIA LISTS

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1 MINIMUM DISTANCE DECODING OF GENERAL ALGEBRAIC GEOMETRY CODES VIA LISTS NATHAN DRAKE AND GRETCHEN L MATTHEWS DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC USA GMATTHE@CLEMSONEDU Astract Algeraic geoetry codes are defined y divisors D and G on a curve over a finite field F Often, G is supported y a single F-rational point and the resulting code is called a one-point code Recently, there has een interest in allowing the divisor G to e ore general as this can result in superior codes In particular, one ay otain a code with etter paraeters y allowing G to e supported y distinct F-rational points, where > 1 In this paper, we deonstrate that a ultipoint algeraic geoetry code C ay e eedded in a one-point code C Exploiting this fact, we otain a iniu distance decoding algorith for the ultipoint code C This is accoplished via list decoding in the one-point code C 1 Introduction Algeraic geoetry codes AG codes are defined y divisors D and G on a curve over a finite field F Often, G is supported y a single F-rational point and the resulting code is called a one-point code Recently, there has een interest in allowing the divisor G to e ore general as this can result in superior codes [, 4, 5, 6, 35, 36] In particular, one ay otain a code with etter paraeters y allowing G to e supported y distinct F-rational points, where > 1 [6, 15, 16, 17, 3, 4, 5] We refer to such a code as a ultipoint code While ultipoint codes ay have etter paraeters than coparale one-point codes on the sae curve, ost decoding algoriths have een designed specifically for one-point codes; those that apply to general algeraic geoetry codes tend to decode up to a ound on the iniu distance such as the Goppa ound or the order ound rather than the iniu distance itself For exaple, Beelen s adaptation of ajority voting see [3] and references therein and the odification of the Berlekap-Massey-Sakata Algorith [9] due to Sakata and and Fujisawa [8], decode up to the generalized order ound, a lower ound on the iniu distance; however, the generalized order ound does not agree with the actual iniu distance in general though it does in certain cases, for exaple, for twopoint Heritian codes The inforative survey on decoding algeraic geoetry codes [3] includes a decoding schee which allows the possiility of correcting errors eyond the order ound In this paper, we provide a siple algorith for decoding Key words and phrases algeraic geoetry code, list decoding, iniu distance decoding, ultipoint code This project was supported y NSF DMS and NSA H

2 NATHAN DRAKE AND GRETCHEN L MATTHEWS a ultipoint code up to its iniu distance y eedding the ultipoint code in a one-point code and list decoding in a supercode This algorith also applies to general AG codes defined y a rational divisors G and D, provided not all rational points are in the support of D While list decoding dates ack to the late 1950 s [8, 9, 34], its power was not fully exploited for over 40 years This changed with Sudan s oservation that list decoding could e applied to Reed-Soloon codes to give a polynoial tie algorith which decodes eyond the iniu distance of the code eaning ore than d 1 errors [33] A ajor reakthrough in decoding AG codes cae with the generalizations of Sudan s algorith to aritrary AG codes of low rate y Shokrollahi and Wasseran [30] and to one-point AG codes y Guruswai and Sudan [11] While the results in [30] apply to general AG codes of restricted rate, nearly all susequent iproveents including, for exaple, [10, 19, 0, 1, 7] are restricted to the one-point case By eedding a ultipoint code into a one-point code, we can capitalize on these and provide a iniu distance decoder for a ultipoint code This paper is organized as follows This section concludes with a suary of notation to e used throughout the paper Section deonstrates how a general AG code in particular, a ultipoint code ay e eedded in a one-point code Section 3 presents a decoding algorith for such codes ased on a list decoding algorith for one-point codes In Section 4, this algorith is odified to handle ultipoint codes eedding in ultiple one-point supercodes Notation Let X e a projective curve of genus g over a finite field F Let FX denote the field of rational functions on X defined over F The divisor of a nonzero rational function f is denoted y f The coefficient of a point P on X in a divisor A on X is written as v P A, or v P f if A = f for soe rational function f Given a divisor A on X defined over F, let LA denote the set of rational functions f on X defined over F with divisor f A together with the zero function We often use the fact that given divisors A and B on X with A B, L A L B Let la denote the diension of LA as an F-vector space As is standard, given a plane curve X with defining equation fx, y = 0, P a denotes the affine point which is the coon zero of x a and y Let G e an F-rational divisor on X and D = n i=1 P i where P 1,, P n are pairwise distinct F-rational points on X, none of which are in the support of G An AG code defined y D and G is where C L D, G := {evf : f LG} evf := f P 1,, f P n We refer to such a code as an -point code if and only if the support of the divisor G consists of distinct F-rational points We do not assue that the divisor D is supported y all rational points other than those in the support of G as is soeties taken to e the case If the support of G is precisely Q 1,, Q, then we say that the code C L D, G is supported y Q 1,, Q If deg G < n, then C L D, G has length n, diension lg, and designed distance n deg G The iniu distance of the code C L D, G is at least its designed distance We use d C resp, d C to denote the iniu distance resp, designed distance of a code C A code of length n, diension k, and iniu distance d resp at least d is called an

3 MINIMUM DISTANCE DECODING OF GENERAL AG CODES VIA LISTS 3 [n, k, d] resp [n, k, d] code The Haing distance etween words w, w F n is d w, w := {i : w i w i } A iniu distance decoder for an [n, k, d] code C over F is a decoder that given a received word w F n returns the unique codeword c C with d w, c < dc if such a codeword exists and declares failure otherwise General references for algeraic geoetry codes are [14, 3] The set of positive integers is denoted Z + As usual, given v F n where n Z +, the i th coordinate of v is denoted y v i Eedding a general AG code in a one-point code In this section, we deonstrate that a general AG code C L D, G on a curve X over a finite field F ay e eedded in a one-point code provided that the divisor D is not supported y all F-rational points on X As a consequence, we see that each ultipoint code C L D, G eeds in a one-point code, as the divisor G contains rational points in its support Exaples are provided at the end of the section Lea 1 Let X e a nonsingular projective curve over a finite field F Suppose G is an F-rational divisor and D := P P n is supported y n distinct F- rational points, none of which are in the support of G If there exists an F-rational point P on X not in the support of D, then C L D, G is isoetric to a sucode of a one-point code C L D, αp on X for soe α Z Proof Write G = G + G where G +, G 0 Then deg G + deg G + P = 0 Since the field F is finite, the group of divisor classes of degree zero has finite order Hence, soe ultiple of the divisor G + deg G + P is a principal divisor Consequently, there exists a rational function f with divisor f = λ G + deg G + P for soe λ Z Multiplication y f induces an isoorphis of Rieann-Roch spaces L G L λ deg G + P λ 1 G + G h fh Since L λ deg G + P λ 1 G + G L λ deg G + P, L G is isoorphic to a suspace of L λ deg G + P Moreover, ultiplication y the function f induces a vector space isoorphis φ : F n q F n q v evf v where evf v := fp 1 v 1,, fp n v n Since f has no zeros aong P 1,, P n, the ap φ is weight-preserving, and hence, distance-preserving Thus, restriction of φ to C L D, G induces an isoetry φ of codes 1 C L D, G φ = CL D, λ deg G + P λ 1 G + G Therefore, C L D, λ deg G + P contains an isoetric copy of the code C L D, G

4 4 NATHAN DRAKE AND GRETCHEN L MATTHEWS Lea 1 is crucial for the decoding algorith presented in Section 3 The following exaples provide isoetries for coonly studied ultipoint codes Exaple In this exaple, we consider Heritian codes Recall that the Heritian curve X is defined y y q + y = x over F q Since the autoorphis group of this curve is douly-transitive, to study two-point codes, we ay restrict our attention to a code of the for C := C L D, ap + P 00 Suppose a, Z + Then ultiplication y f := y induces a vector space isoorophis L D, ap + P 00 = L a + q + 1 P + q + 1 P 00, q + 1 q + 1 ecause y = q + 1 P 00 P Hence, C = C L D, a + q + 1 P C L D, a + q + 1 P Now consider an -point Heritian code C := C L D, a 1 P + P αβi i= q + 1 P 00 suppported y collinear points P, P αβ,, P αβ where Z + for all i, 1 i and q + 1 Such codes were studied in [] where the authors show that if τ αβi := y β i α q x α then τ αβi = q + 1 P αβi P Thus we can take f = i= τ αβ i The ultiplication y f induces a vector space isoorphis L a 1 P + i= P αβi = L a 1 + q + 1 i= and an isoetry of codes C = C L D, a 1 + q + 1 i= since C L D, a 1 + q + 1 i= L a 1 + q + 1 i= L a 1 + q + 1 i= P + i= ai q + 1 P αβi P i= P P + i= P q + 1 P αβi q + 1 P αβi Exaple 3 In this exaple, let C := C L D, ap + P 00 e a two-point Suzuki code where a, Z + Recall that the Suzuki curve is defined over F q y the

5 MINIMUM DISTANCE DECODING OF GENERAL AG CODES VIA LISTS 5 equation y q y = x q0 x q x where = n, q = n+1, and n Z + Let w := y q q q x y q x q q q 0 +1 w = as shown in [1], ultiplication y Since q + q + 1 P 00 P f := w q+ q +1 gives rise to an isoorphis of Rieann-Roch spaces and consequently an isoetry of codes C = C L D, αp βp 00 C L D, αp where α = a + q + q q + q and β = q + q q + q Reark 4 Deterination of a suitale function f F X as in the proof of Lea 1 ay e found as follows Consider l λ G + deg G + P for increasing λ Z + until one is found with l λ G + deg G + P 0 Then copute a asis of L λ G + deg G + P to find such an f Hess [13] gives an algorith for effectively coputing ases of Rieann-Roch spaces For certain divisors on axial or optial function fields, ases of such spaces are known [] 3 A iniu distance decoder for general AG codes In this section, we outline the decoding algorith for general AG codes C L D, G, where not all rational points are in the support of D Note that this algorith applies to ultipoint codes and that the point of view taken here can e utilized with any list decoding algorith for one-point codes For clarity of exposition, we focus on the Guruswai-Sudan list decoding algorith as found in [11, Section IV B] Consider a nonzero AG code C := C L D, G on a nonsingular projective curve X of genus g over a field F where D := P P n, and assue that there is an F-rational point P on X not in the support of D For the purpose of decoding it is sufficient to consider the code φ C where φ is as in 1 To see this, suppose that w F n q is a received word using the code C and that E errors have occurred, where E dc 1 We ay identify w and evf w via the ap φ Then evf w is treated as a received word using φ C dφc 1 Since φ is distance preserving, E Hence, there is a unique nearest codeword evh φ C to evf w It follows that ev f 1 h is the unique codeword in C nearest w Consequently, throughout the reainder of this section, we assue that G = αp G where α Z +, P is an F-rational point not in the support of D, and G > 0 Algorith 31 Let C := C L P P n, αp G e a nonzero AGcode over the field F q Suppose that w F n q is a received word in which dc 1 or fewer errors have occurred Input: P 1,, P n, P, α, G, received word w F n q, agreeent paraeter t :=

6 6 NATHAN DRAKE AND GRETCHEN L MATTHEWS n dc 1 Assuptions: t > αn 1 Initialization: Set r := gt + αn + gt + αn 4 g 1 t αn t + 1, αn rt 1 g s :=, α and Ω := Interpolation: Find a nonzero polynoial QT F q X [T ] satisfying: a Q f L rt 1 P for all f L αp, and v Pi Q f r for each i {1,, n} with f P i = w i 3 Factorization: Find all roots h L αp of the polynoial Q For each such h, if h P i = w i for at least t values of i, then add h to Ω In this way, we find all functions h L αp that possily give rise to the codewords in C := C L D, αp at distance at ost dc 1 fro w; that is, Ω := {h LαP : devh, w n t} is deterined 4 Check for zeros: Copute the order of h at Q i for each h Ω until the one is found with v Qi h v Qi G for all Q i in the support of G Output: ev h, the unique word in C with d ev h, w dc 1 Reark 3 1 Step of Algorith 31 produces a polynoial Q which fits the points P i, w i in the sense that Q w i P i = 0 for all i, 1 i n Hence, the eleents of Ω are aong the roots of Q See [11] for further discussion on this as well as the correctness of Steps 1-3 Steps 1-3 of Algorith 31 ay e replaced with those of any list decoding algorith for C L D, αp which yields Ω as its output 3 According to [7], the paraeters r and s ay e replaced with the following values in Step 1: αn t + tg + rt α r := t + 1 and s := g + 1 αn α where and := α n t 4gn + 4αgn t + g := t αn r + αt αn tg r + α 4 + g αg This ay result in a lower degree interpolating polynoial Q, as deg Q s In addition, it provides a etter ound on Ω and the nuer of functions that ust e checked in Step 4 4 The goal of Step 4 ay e achieved via parity checks Specifically, given a parity check atrix H for C, copute Hevh T for each h Ω until one is found with Hevh T = 0 Alternatively, one could deterine the additional parity checks v 1,, v r that words in C ust satisfy to e in the sucode

7 MINIMUM DISTANCE DECODING OF GENERAL AG CODES VIA LISTS 7 C Then, for each h found in Step 3, copute ev h v i, 1 i r, until an h is found satisfying all r checks 5 It is natural to copare Algorith 31 with ajority voting as in [3] However, ajority voting only corrects up to doc 1 errors, where d o C denotes the order ound on C Because the only step in Algorith 31 eyond list decoding is the check for zeros or parity checks as entioned in 4 aove, this algorith copares as favoraly to ajority voting as the underlying algorith for list decoding in the one-point code the choices for which see to e ever-iproving, for instance [1] and has the advantage of correcting up to dc 1 errors Due to the nature of Step 4 and Reark 3 4 aove, it is advantageous to iniize the difference in the diension of C and C Exaple 33 Let X denote the Heritian curve defined y y 8 + y = x 9 over F 64 Then X has genus g = 8 Consider the two-point code C := C L D, 344P 8P 00 where D := P P 511 is the su of all F 64 -rational points other than P and P 00 According to [17], C is a [511, 309, 175] code, and, thus can correct any 87 errors Suppose w F is a received word in which 87 errors have occured To decode w, eed C in C := C L D, 344P, a [511, 317, 167] code [15] Applying Steps 1-3 in Algorith 31 with the paraeter choices given in Reark 33 produces a list of at ost 1 functions h 1,, h 1 L 344P with d w, ev h i 87 There is a unique h i L 344P 8P 00 satisfying Equation To deterine this function, evaluate v P00 h i for 1 i 1 until h Ω is found so that v P00 h 8 Then decode w as ev h = h P 1, h P,, h P A iniu distance decoder for ultipoint codes using lists, ultiple eeddings, and gcd In this section, we discuss a odification of Algorith 31 in which a ultipoint code is eedded in ultiple one-point codes and the interpolating polynoial is otained as a greatest coon divisor This idea was inspired y [1] Consider a ultipoint code C := C L D, i=1 Q i where Z Given any function f whose divisor is supported only y points aong Q 1,, Q, ultiplication y f induces a vector space isoorphsi L Q i = L v Qi f Q i and an isoetry of codes i=1 C φ = CL D, i=1 v Qi f Q i i=1 Hence, for each such function f with v Qj f < a j for exactly one j, 1 j, C is isoetric to a sucode of the one-point code C L D, aj v Qj f Q j ; that is, φ C C L D, aj v Qj f Q j To ephasize that the eedding is induced y f L a j v Qj f Q j,

8 8 NATHAN DRAKE AND GRETCHEN L MATTHEWS we soeties write φ j instead of φ The following algorith exploits these ultiple eeddings Algorith 41 Let C := C L D, i=1 Q i e an -point code over the finite field F q where D := P P n Suppose that w F n q is a received word in which dc 1 or fewer errors have occurred Input: n, a 1,, a, received word w F n q, agreeent paraeter t := n dc 1 1 Eedding: Choose a nonepty suset J {1,, } For each j J, find a one-point code C j := C L D, a j jj Q j such that C φj = C L D, a j jj Q j ij Q i C j 1 i i j is the eedding induced y a rational function f j whose divisor is supported y no points other than Q 1,, Q with v Qi f j = ij for all 1 i, ij for all i j, jj < a j and t > a j jj n Set Ω := Interpolation: For each j J, apply Steps 1 and of Algorith 31 to C j with received word φ j w to yield nonzero polynoials H j T F q X [T ] satisfying Conditions a and of Step Set Q T := gcd {H j f j T : j J} 3 Factorization: Find the roots of QT as in the standard factorization step If a root h of QT satisfies h P i = w i for at least t values of i, 1 i, then add h to Ω In this way, we otain Ω = h L a j Q j + j J 1 i i / J d C 1 i Q i : d ev h, w dc 1, that is, we find all functions that possily give rise to the codewords in C at distance d 1 fro w, where i = in { ij : j / J} 4 Check for zeros: Copute the order of h at Q i for each h found in Step 4 until the one is found with v Qi h for all i / J Output: ev h, the unique word in C with d ev h, w Theore 4 Given a ultipoint code C := C L D, i=1 Q i as aove, Algorith 41 provides a ininu distance decoder for C Proof Suppose that w is a received word in which at ost dc 1 errors have occured Then there exists a unique codeword ev h that is the transitted word resulting in the received word w We ust show that the output of Algorith 41 is ev h Assue h L i=1 Q i and d ev h, w dc 1 We clai that h is a root of Q T To see this, we show that h is a root of H j f j T for all j J Note

9 MINIMUM DISTANCE DECODING OF GENERAL AG CODES VIA LISTS 9 that aong the roots of H j T are eleents of { } d C 1 Ω j := f L a j jj Q j : d ev f, φ j w We prove that f j h Ω j for all j J Let j J It is iediate that f j L a j jj P j y definition of f j Since ev f j h = φ j ev h and φ j is distance preserving, d C 1 d ev f j h, φ j w = d ev h, w Hence, h is a root of Q T and so will e found in Step 4 of Algorith 41 As seen in Algorith 41 and Theore 4, ultiple eeddings and greatest coon divisor ay e coined to yield an interpolating polynoial Q of saller degree While these ultiple eeddings ay e advantageous in the root-finding step of list decoding, the cost of calcuating ultiple interpolating polynoials to produce Q ay outweigh the enefits 5 Conclusion In this paper, we present a iniu distance decoding algorith for a general AG code C L D, G on a curve X over a finite field F, provided there is an F- rational point on X not in the support of D This decoding algorith applies to all ultipoint codes It relies on eedding the code C L D, G into a one-point code C L D, αp and applying list decoding to the one-point code A ethod for utilizing ultiple eeddings and greatest coon divisor is also presented Acknowledgeents The authors thank the referees for suggestions that iproved the content and presentation of this work References [1] A Barg, E Krouk, H C van Tilorg, On the coplexity of iniu distance decoding of long linear codes IEEE Trans Infor Theory , no 5, [] P Beelen, The order ound for general AG codes, Finite Fields Appl , no 3, [3] P Beelen and T Hoholdt, The decoding of algeraic geoetry codes, in Advances in Algeraic Geoetry Codes, Series on Coding Theory and Cryptology World Scientific 5, E Martinez- Moro, C Munuera, and D Ruano, eds, [4] C Carvalho, C Munuera, E da Silva, and F Torres, Near orders and codes, IEEE Trans Infor Theory , no 5, [5] C Carvalho and F Torres, On Goppa codes and Weierstrass gaps at several points Des Codes Cryptogr , no, 11 5 [6] I Duursa and R Kirov, An extension of the order ound for AG codes, preprint [7] N Drake and G L Matthews, Paraeter choices and a etter ound on the list size in the Guruswai-Sudan algorith for algeraic geoetry codes, Des Codes Cryptogr, to appear [8] P Elias, List decoding for noisy channels, Tech Rep 335, Res La Electron, MIT, Caridge, MA, 1957 [9] P Elias, Error-correcting codes for list decoding, IEEE Trans Infor Theory , 5 1 [10] V Guruswai and M Sudan, On representations of algeraic-geoetry codes, IEEE Trans Infor Theory , no 4, [11] V Guruswai and M Sudan, Iproved decoding of Reed-Soloon and algeraic-geoetric codes, IEEE Trans Infor Theory , [1] J Hansen and H Stichtenoth, Group codes on certain algeraic curves with any rational points, Appl Algera Engrg Co Coput , no 1, 67 77

10 10 NATHAN DRAKE AND GRETCHEN L MATTHEWS [13] F Hess, Coputing Rieann-Roch spaces in algeraic function fields and related topics, J Syolic Coput 33 00, no 4, [14] T Høholdt, J H van Lint, and R Pellikaan, Algeraic geoetry codes, in Handook of Coding Theory, V Pless, W C Huffan, and R A Brualdi, Eds, 1, Elsevier, Asterda 1998, [15] M Hoa and S J Ki, Toward the deterination of the iniu distance of two-point codes on a Heritian curve, Des Codes Cryptogr , no 1, [16] M Hoa and S J Ki, The two-point codes on a Heritian curve with designed iniu distance, Des Codes Cryptogr , no 1, [17] M Hoa and S J Ki, The two-point codes with the designed iniu distance on a Heritian curve in even characteristic, Des Codes Cryptogr , no 3, [18] M Hoa and S J Ki, The coplete deterination of the iniu distance of two-point codes on a Heritian curve, Des Codes Cryptogr , no 1, 5 4 [19] H O Keeffe and P Fitzpatrick, Gröner asis solutions of constrained interpolation proles, Linear Algera Appl 351/35 00, [0] R Koetter and A Vardy, Algeraic soft-decision decoding of Reed-Soloon codes IEEE Trans Infor Theory , no 11, [1] K Lee and M E O Sullivan, List decoding of Heritian codes using Gröner ases, J Syolic Coput, to appear [] H Maharaj, G Matthews, and G Pirsic, Rieann-Roch spaces of the Heritian function field with applications to algeraic geoetry codes and low-discrepancy sequences, J Pure Appl Algera, , no 3, [3] G L Matthews, Codes fro the Suzuki function field, IEEE Trans Infor Theory, , no 1, [4] G L Matthews, Weierstrass pairs and iniu distance of Goppa codes, Des Codes Cryptogr 001, [5] G L Matthews, Weierstrass seigroups and codes fro a quotient of the Heritian curve, Des Codes Cryptogr , no 3, [6] G L Matthews and T W Michel, One-point codes using places of higher degree, IEEE Trans Infor Theory , no 4, [7] S Sakata, On fast interpolation ethod for Guruswai-Sudan list decoding of one-point algeraic-geoetry codes Applied algera, algeraic algoriths and error-correcting codes Melourne, 001, , Lecture Notes in Coput Sci 7, Springer, Berlin, 001 [8] S Sakata and M Fujisawa, Fast decoding of two-point AG codes, preprint [9] S Sakata, J Justesen, Y Madelung, H E Jensen, and T Høholdt, Fast decoding of AGcodes up to the designed iniu distance, IEEE Trans on Infor Theory, IT , [30] M A Shokrollahi and H Wasseran, List decoding of algeraic-geoetric codes, IEEE Trans Infor Theory , [31] J H Silveran, The arithetic of elliptic curves Graduate Texts in Matheatics, 106 Springer-Verlag, New York, 1986 [3] H Stichtenoth, Algeraic Function Fields and Codes, Springer-Verlag, 1993 [33] M Sudan, Decoding of Reed-Soloon codes eyond the error correction ound, J Copl 13, , 1997 [34] J M Wozencraft, List decoding, Quarterly progress report, Research Laoratory of Electronics, MIT, 48:90-95, 1958 [35] C P Xing and H Chen, Iproveents on paraeters of one-point AG codes fro Heritian codes, IEEE Trans Infor Theory 48 no 00, [36] L Xu, Iproveent on paraeters of Goppa geoetry codes fro axial curves using the Vlădut-Xing ethod, IEEE Trans Infor Theory 51 no 6 005, 07 10