Construction and Decoding of a Class of Algebraic Geometry Codes

JUSTESEN et d.: A CLASS OF ALGEBRAIC GEOMETRY CODES 813Now, take any s rational points Pi,, . . . , Pi,, and considerthe divisorG, = G - Pi, - 1 . . - Pi,.Since degree(G,)=mj-s=2g-1>2g-2, it followsfrom Riemann-Roth that the dimension of L(G,) is Z(G,,)= Z(G) - s = g: An element [ fi]/[ fo] E V, = L(G) belongsto L(G,) if and only if f,(Pj,)=O, j=l;..,s. Thisfollows by considering, from each point P,,, the twocases f,( Pi,) = 0 and fo( Pi,) f 0 and using the fact thatfo( P,,) = 0 iff ord Pi ( fo) = 1. Therefore, the solutions(&,*. -3 h k _ i) of the equationsX,fo( p;,) + . . : + X&J-1( P,) = 0, j=l;..,s(2.13)form a vector space of dimension g. This however, meansthat the rank of the matrix for the system (2.13) is k - g = s.Consequently, any s columns of G are linearly independent,and the minimum distance for the code H,(j) is atleast’ s + 1.We will briefly consider the question concerning theconditions under which the codes H,(j) are of the formG,( j’) for some j’, that is, the condition under which theduals of the codes we consider are of the same kind.The reason for this question is that the decoding methodworks for the codes H,(j), and it would be nice to havethe generator matrix of this code. In general, this seems tobe a difficult question [9], but based on formula (2.10) forthe dimensions it is fairly easy to derive a necessarycondition.Proposition I: Let j, j’> m -2, n > mj and n > mj’ IfH,(j) = G,( j’), then m divides n, and j + j’ = m - 3 +(n/m>-Proof: If H,(j) = G,( j’), then n - k(j) = k( j’) andtherefore,son-(mj-g+l)=mj’-g+ln=m(j’+ j)-2g+2=m(j’+ j)-m(m-3)from which the propositionfollows.We- shall return to this in connection with the codestreated in the next section. We conclude this section withthe following example.Example 1: Let C be the Klein quartic with the equationx’y + y3z + z3x = 0and let F be GF(8). This curve is considered in [7] and itis known that C is regular with genus g = 3. If n is thenumber of rational points on C, Serre’s improvement ofthe Weil bound gives n I 24.It is actually easy to find 24 points in GF(8) on thecurve, namely, the points (l,O,O), (O,l,O), (O,O,l) and the21 points of the form (1, p, y) where p is any nonzeroelement of GF(8), and y is any one of the three differentsolutions to z3 + p3z + /3 = 0. Putting z = p5w, the equa-tion becomes p(w3 + w + 1) = 0, and hence the givenequation has three different solutions.Using Theorem 1, we have thus constructed codes G,corresponding to j = 2,3,4,5, with parameters(24,6,16) (24,10,12) (24,14,8) (24X4).From (2.2) and Theorem 2 it follows that the codes H, forj = 2,3,4,5, have parameters(2% l&4) (24,14,8) (24,10,12) (24,6,16).Theorem 1 and Theorem 2 give lower bounds on theminimum distance, but for the codes considered here itturn out to be easy to find codewords of minimum weight;thus the theorem actually gives the true distance. It is not acoincidence that length and dimension for the two series ofcodes above are identical because it can be shown that forthese codes we haveG,(j) = %(7-j>, j= 2,3,4,5. (2.14)It is clear that the necessary condition in Proposition 1 issatisfied for these codes, and (2.14) then follows either bydirect calculation of the matrices or by arguing in the samemanner as in the Appendix for another class of codes.III.CLASSES OF CODESTo construct good codes of the type characterized inSection II, we need irreducible curves of low genus andwith a large number of rational points. We shall be interestedin obtaining infinite classes of good codes based onsuch curves.For any particular field F and 4 elements, the numberof rational points is, of course, bounded above by thenumber of F-points in P2,nIq2+q+l.Thus an infinite class of codes must use a sequence ofalphabets of increasing size.Example 2: The projective line is described by the equationx+y+z=o.It has degree m = 1 and genus g = 0, and n = q + 1 points:(0, 1, - 1) and (1, (Y,- 1 - a), (Y E F. Using (2.10) and Theorem2, we see that the code H,(j) has dimension n - j - 1and minimum distance j + 2, and thus the code is maximumdistance separable (MDS). Using the polynomialsxj,xj-ly,. . ., yj as a basis, we get the parity-check matrixof the doubly extended Reed-Solomon codes.For codes on a curve with genus g, we noted Serre’simprovement of Weil’s upper bound on the number ofrational points (2.8). Actually, for q = p”, the number ofrational points on an irreducible smooth curve is [14]2&Tn,=l+q- C c$. (3.1)i=lHere the (Y~ are pairs of complex conjugates with ]ai] = ~“1~.When n, is known for sufficiently many values of s, it ispossible to find the (Y~ and obtain the number of points in

JUSTESEN et d: A CLASS OF ALGEBRAIC GEOMETRY CODES 815and the curve Example 7: Let q = pzr, and consider the curvexp’P1zp*rP1 _ Zp’-lY p=r-l+ xp2r-1 p’pl= ()YIp’+la11aI2 . *. al,Xlwhich has ( p3’ - l)( p2’ - 1) rational points with all coordi-p’+la21a22 ...a2sx2 =().nates nonzero and the additional points (l,O, O),(O, l,O),. . . . . .(O,O, 1). Klein’s quartic may be considered as the firstxPr+la(,-2,l a(,-,), . * . a(,-,), II s Imember of this sequence for p = 2 and r = 1.The approach taken in Example 5 may be extended to The curve is regular if all (s - 2) X (s - 2) submatrices areother polynomials. However, we do not have families of nonsingular. For s < fi + 1, such a matrix may be obcurvesthat yield good codes over all fields. To get a good tained as the parity-check matrix of an MDS code withcode over a particular field, GF( p’), we take a polynomial alphabet GF( p’). The pr + 1st powers of the coordinatesu(w) with coefficients in GF( p) and roots in GF( p’). By then form codewords in the MDS code. From the weightsubstituting y”z for w, multiplying by a power of y, and distribution of an MDS code we could find the exacthomogenizing with x, we obtain an equation for a curve. number of nonzero coordinates. However, since mostUsually, the curve has ( pr - 1) deg( u) rational points with codewords in such a code have weight s, we may use thenonzero coordinates and a few more with one coordinate approximationequal to zero. It is necessary to check that the curve isirreducible. As discussed in Section II, it is sufficient thatthe curve be regular. If there are singular points, it is where p’ is the approximate number of codewords withnecessary to ensure that these are not intersections of first coordinate equal to one, and the remaining s - 1components of lower degree. Bezout’s theorem may be coordinates may assume pr + 1 values in GF( p”). Theused to predict the degree of such factors. The following genus of the curve is approximatelyexample is the result of a computer search for curves thatyield codes with low g/n over some fields that were notg = (l/2)@ -2)p”“P”.covered by previous constructions.Example 6: For the field GF(32) a complete search was Thus for fixed s, g/n vanishes for q + cc. In three-dimen-done for binary polynomials and substitutions to find thebest ratio g/n. Usingand w = yl’z, we getu(w) = w5+ w2+1x2y2z5 + x7z2 + y9 = 0with I57 rational points. The points (O,O, 1) and (I, 0,O) aresingular, but the curve is irreducible. Because of the twosingular points, the genus is reduced by two, relative to(2.9):g = (9-1)(9-2)/2-2 = 26.Thus g/n becomes 0.166. A similar search in GF(128)gave g/n = 0.085 for the curvex3yz10 + xl32 + y14 = 0which has 891 rational points. This curve was constructedfrom the polynomialand the substitutionu(w)=wlO+w+lw = yn3z.sional space s = 4, there are codes of length n = (q - 1)’and genus g = fi( q + fi - 1). In GF(64) these codes haveparameters (3969,81/c, 81(42 - k)) for 112 k I 38.IV. DECODING PROCEDURESIn Section II we constructed, for a given irreduciblecurve C, two kinds of codes, G,(j) and H,(j). For thecodes H,( j)-or shortened versions thereof-it followsfrom Theorem 2 that the numberD=mj-2g+2is a lower bound on the minimum distance. This number Dis called the designed distance for the code in question. Inthe following, we shall describe a decoding procedurefor these codes which turns out to be a generalizationof the well-known Peterson algorithm for decodingReed-Solomon codes [16, sec. 9.41. In the general setup,this procedure does not correct errors up to half thedesigned distance but only up to a smaller number. For allthe examples we have considered it has been possible toextend the general method in such a way that any numberof errors up to half the designed distance can be corrected.However, we have not yet been able to describe and provesuch extensions for the general case.As noted earlier, a good code based on a plane curve Let us consider an absolutely irreducible plane curvecannot have length much greater than q312. To obtain C(x, y, z). Among the rational points on this curve welonger codes, we must consider curves in spaces of higher only take points of the form (1, y;, zi). These points aredimension. If we restrict the curves considered to be inter- denoted Q,; . ., Q,.sections of hypersurfaces, it is possible to generalize our Let H,‘(j) be the code whose parity-check matrix isapproach to such curves. We briefly discuss a particular obtained from the parity-check matrix of H,(j) by deletclassof codes in the following example. ing the columns corresponding to the points (0, y,, zi).

816 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 35, NO. 4, JULY 1989More precisely, if nient to look at exactly t positions in e. Let I denote theset of these indices, and let e, denote the value at positionG:(j) = ((.f(Q,>,. . . J(Q,>) If E q) (4.1)theni E I. If S = He and S = (So, S,, . . . , S,_ i), we then haveH,‘(j) =GJ( j)‘. (4.4 s I ,,,, =S,= C e;yjzp, 7(11,12) =I, f=O;.*,s-1.1EI(4.10)As before, m is the degree of C and g the genus of C, andin accordance with (2.10) we define the number k(j),(j2m-2) ask(j) =mj-g+l. (4.3)From Theorem 1 and its proof, and Theorem 2, we havethe following.Theorem 3: Suppose that n > mj and j 2 m - 2. Thenfor the code H,‘(j) with length n, the dimension is n -k(j), and the minimum distance satisfiesd,,>mj-2g+2=k(j)-g+l. (4.4)We now consider a code H,‘(j) for which the assumptionsin Theorem 3 are satisfied. Let fo; * . , f,-i be thej+2polynomials x “y ‘*z ‘3, j, + j, + j, = j, where s =i 2i .ThenH= {f,(Q,)}, i=O ;+*,s-1, l=l;*.,n (4.5)is a parity-check matrix for H,‘(j). Since we only insertpoints Q, with first coordinate equal to 1, it is sufficientinstead of the fi to consider the polynomials in twovariables‘Pity? ‘1 = cPi,,i,(Y, z, = Yilzi2t i, + i, I j. (4.6)The notation in (4.6) implicitly involves an arrangementof the pairs (iI, i2), i, + i, 4 j, in a sequence. We shall usethe ordering known from Cauchy multiplication of series,that is, the pairs are arranged in the sequence (O,O),(1,0>,(0,1),(2,0>,(1,1)..., etc. By r we denote the mapfor which T(il, i2) = i in (4.6). It can be seen that 7(il, i2)= (1/2)(i, + i, + l)(il + i,)+ i,, and to find T-‘(i), firstsolve (i) Sic (jil). Then i,fi,= j-l and i,=i- ji 2 i .If we putQ; = (Y,, z,), Z=l;..,n (4.7)we can write the matrix (4.5) asH= { v;(Q,‘)}, i=O;e.,s-1, I=l;.*,n. (4.8)Recall that the designed distance is defined asIn the decoding situation, we know the vector S and,assuming that at most t errors have occurred, we knowthat equation (4.10) has a unique solution. Some of thevalues e, for this solution may be zero, namely, when theactual number of errors is less than t. The decodingproblem is then to determine from (4.10) the error positionsI and the error values e,.The key idea in the decoding algorithm we shall describeis the following: we first determine a locator polynomiala( y, z) which among its roots has the points Q!, i E I. Byintersecting the curve given by a( y, z) = 0 and the givencurve (where x = l), we find possible error positions. Insertingthese points in an equation corresponding to (4.10),but possibly with more than t points, we obtain a systemof equations from which the values e, are calculated.Before describing the details, we illustrate the idea in thefollowing example.Example 8: We consider the code obtained from Example1 by shortening by two positions for which x = 0. It isa (22,12,8)-code. In this example we shall demonstratehow at most three errors may be corrected.The ten syndrome equations (4.10) may be written1 eiy$z+ = S,1,z,i=l1, + I, 23.We now assume that we have received a word such that thevalues of the syndromes are given by S, = (Ye, S,, = (Y,s,, = d, s,, = d, s,, = a, so, = a5, s3, = a5, s,, = 0, s,,=CY 6, and S,, = (Y’. If the errors were located on a straightline, we could find a nonzero error locator a, + a,,y + a,,,~which would be a linear recursion among the syndromes;namely, sincewe gethl+ UlOY + ~o14&I,I,= 0,%osI,,,+~ol~,l+lI +~ols,l,I,+l=o-3 2However, the syndromes do not satisfy such a relation. Byadding a second-order term, we can always find a locatorpolynomialD=mj-2gt2. (4.9) u( y, z) = y2+ qoy + u,,z + a,.In the following analysis, t denotes an (unknown) number, By the same reasoning as before, the coefficients may bet I (D - 1)/2, and we suppose that the actual number of determined by requiring that u be a recursion among theerrors is less than or equal to t. known syndromes:We then develop a decoding algorithm, and to make thisalgorithm work, we have to put restrictions on t. The finalvalue of t is implicit in the inequalities in Theorem 4.Now let us consider a vector 0, for which 0 - e E H,‘(j),1; 1; ~,)(zJ={~~]where the error vector e is nonzero for at most t positions.We then calculate the vector Hv = He. It will be conve- which implies a( y, z) = y2 + ti’y + (Y~z, where (Y is a root

. .JUSTESEN et d.: A CLASS OF ALGEBRAIC GEOMETRY CODES817of x3+x +l. Intersecting C and u, we obtain the fivepoints (l,O,O), (l,l, a), (1, (~,l), (1, (Ye, a3), and (1, (Ye, 1). Ifwe expand the syndrome equations to include these fivepoints, we get a unique solution: eil =l, ej2= (Ye, e13 = 0,ej4 = a, ei5 = 0.The important term in the decoding is a certain matrixg, whose construction we will now describe. First, however,we assume the existence of a number h such thatClearly, (4.11) implies thatIf h’ = j - h, we havet+lsk(h)sD-g-t. (4.11)tsD-g-lk(h’)=k(j-h)=mj-2g+2-k(h)2 .(4.12)provided that h 2 m - 2 and j - h 2 m - 2. In this case, itfollows from (4.11) thatk( h’) 2 t + g. (4.13)To summarize, we assume the existence of a number hsuch that m - 2 I h I j - m + 2 and (4.11) is satisfied.Putting h, = (l&h + l)( h + 2) and hi = (l/2)( h’ + l)( h’+ 2), we define S bywheres= {%,A l=O;..,h;-1, r=O;..,h,-1Lemmaa/,,= S 1, + ‘l,l, + f-2& 12) = 1 7(r,, r2) = r. (4.14)2: For any 1 and r in (4.14), we haveI, + r1 + I, + r, I j.Proof: From (4.14) and the definition of r, we haveI, + I, I h’ and r1 + r2 I h.This means that all terms S,,, is s are known. Let usdenote the indices in I in (4.10) by i,, i,, . . . , i,, andconsider the following three matrices:F= uJ,,.L l=O;**, hi-l, u=l;.*,tb,, u = y;;z;z +, 12) = 1E= {c,,“}, u=l ,-.*,t, u=l,sCu.u = eiu c u u = 0 u#u,t(4.15)(4.16)producton the right side of (4.18) isfc Y,y;e,,,Y(y:p= bL,L u=l;*.,t, r = 0, . . . >A,-1For any polynomial u( y, z) in (4.20) we put(7(x, y, z) = Cua,hXh-a-byaZb,d 0, r = Y~Zi~ T( r,, r2) = r.(4.17)T(a,b) E {O;*.,h,-1).IWe then have the following lemma.Lemma 3: The following decomposition holds:We shall use the phrase “the curve C(x, y, z) is not adivisor of a( y, z)” to mean that C(x, y, z) is not a divisori?= FEP. (4.18) of the above polynomial u(x, y, z).--Lemma 6: The system of equations (4.19) always has aProof: The rth column in EP is the vector (ej‘,yzz,7), solution u such that the corresponding polynomial a( y, z)u=l,. . . , t. Hence the element at position (I, r) in the does not have the curve C as a divisor.0 =lfLemma 4: The matrix F has rank t.Proof: Consider the code H,( h’). By (4.15) F consistsof t columns of a parity-check matrix for this code, namely,the t columns corresponding to the points (1, yi,, zi,,),(,) =I,. . . , t. The minimum distance of H,(h’) is by Theorem2 greater than or equal to mh’- 2g + 2. However, by(4.13) we haveand the lemma follows.t

818 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 35, NO. 4, JULY 1989Proof The system (4.19) has h, unknowns and byLemma 3 the rank of $ is at most t. Hence for any fixedh, -(t + 1) positions, there is a proper solution to (4.19)which is zero at these positions.Consider a fixed term ~“-~-~y~z~ in the equation forthe curve C for which a + b is maximal. Suppose thatC(x, y, z) is a divisor of u(x, y, z). Then a(x, y, z) containsterms of the form P(x, y, ~)x”-~-~y~z~, where thedegree of the polynomial P(x, y, z) is h - m. These termscorrespond to at most(4.22)terms in a( y, z). Therefore, if the coefficients for theseterms are all zero, then a( y, z) cannot have C as a divisor.The lemma now follows ifi”-7”) 2 h&+1).This, however, is the same asthat is, k(h)of h.(“;2j-(h-;+2)rt+l;2 t + 1, which is satisfied by the definitionWe now describe the decoding algorithm.1. For the received vector 0, calculate the syndromevector S = Hv.2. Consider the matrix ,!? defined in (4.14). Find thesolution u to the system (4.19) for which the correspondingpolynomial a( y, z) has minimal degreeand does not have the curve C as divisor.3. Find, among the points Q;, . . . , Q;, those points Q,,for which u(Q,) = 0. Denote these points byQ;‘, Q;l, . . . , Q;:.4. Insert the coordinates for Q;‘; . ., Q;/ in the systemof equations (4.10), and calculate the values e,.Theorem 4: The algorithm just described corrects t errorsif there exists a number h such thatwhere m-2mh+t,andtherefore,mh

JUSTESEN et d.: A CLASS OF ALGEBRAIC GEOMETRY CODES 819equations and must determine u of degree I 3 with tencoefficients. In this case, all terms of degree 3 are usedsince b < m. In what follows, we assume that we havecalculated the syndromes S, h and that they are given inthe following matrices; we write the integer i for the fieldelement (Y’, where (Y is a primitive element in GF(16); *indicates the zero element:10 * 1 8 0 * 9 2 0* 8 0 9 2 0 1 14 91 0 * 2 0 4 14 9 48 9 2 1 14 9 10 14 80 2 0 14 9 4 14 8 1* 0 4 9 4 12 8 1 59 1 14 10 14 8 2 3 22 14 9 14 8 1 3 2 140 9 4 8 1 5 2 14 74 4 12 1 5 * 14 7 21 10 14 2 3 2 3 11 214 14 8 3 2 14 11 2 99 8 1 2 14 7 2 9 *4 1 5 14 7 2 9 * 1312 5 * 7 7 9 * 13 12Consider the followingexample:10 0 5 5 0 3 10 10 2 40 5 0 10 10 2 0 5 6 105 0 3 10 2 4 5 6 10 95 10 10 0 5 6 5 0 3 20 10 2 5 6 10 0 3 2 133 2 4 6 10 9 3 2 13 010 0 5 5 0 3 10 10 2 410 5 6 0 3 2 10 2 4 102 6 10 3 2 13 2 4 10 54 10 9 2 13 0 4 10 5 00 5 0 10 10 2 0 5 6 105 0 3 10 2 4 5 6 10 96 3 2 2 4 10 6 10 9 10i 109 132 130 10 4 105 50 109 10 9 10102%I(JlO001%O011002030021(712= 0.We may solve this system to obtain the unique solution0 = 1 + a”y + a52 + d3y2 + cx’zy + a5z2 + d2y3=O.+ a3y2z + axyz2 + z3.Since u has degree 3, there may be 15 intersections withC. In this case 14 of the points have x # 0. We list (y, z):(o,~),(o,~3),(o,~12),(~,o),(Q,,~14),(~2,~10),(,*,(Y7),((Y*,Ly13),((Y5,(Y10),((Y10,Ly14),(~7,~‘),(a”,o),((Y”,(Y2),(cy12,0).We now solve the 30 independent syndrome equationswith these points inserted to find the error values, of whichat most nine are assumed to be nonzero. Taken in the sameorder as the locations, the ej areSince a curve of degree 2 may intersect C in 10 points, itmay occur that there is an error locator of degree 2.This system has rank only 7, but we get a unique solutionof degree 2u=l+y+y?There are 10 points on the curve with y = (Y~ or y = &O.(a52 a> (a5, a”> (a5, a’> ( a5, do) ( (Y5, d3)(cP,a2) (.lO,c?) (alO,2) (alO,(Yll) (cP,d4).The error values are found as before:a5 a8 (Y13,l,(Y3,(Y1O,(YX,(Y2,O,(Y5.? >As mentioned in the beginning of Section IV, we havefound it possible to decode all errors of weight t < d/2 inthe cases we have investigated. If j = 6, then the codeparameters are (60,35,20). Again we shall correct nineerrors, and consider the first 25 syndromes in the twonumerical cases treated earlier. In the first case we get tenequations and find the same unique solution for a u ofdegree 3. In the second case the ten equations have lowrank, and we do not immediately get a solution. However,if we assume that u has degree 2, the last five equationsinvolve only the known syndromes, and we again find thesame solution as before. If u has degree 3, but some of theleading coefficients are zero, it is always possible to obtainadditional equations. However, we have not found a proofthat the system has sufficiently high rank, and we shall notdiscuss these cases in further detail.For practical applications, the codes over GF(256) areparticularly interesting. We shall discuss such codes in thefollowing example and concentrate on a code with rateL003

820 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 35, NO. 4, JULY 1989close to 7/8, which is the rate used in some systems using Hansen and Anders B. Sorensen are also gratefully acconcatenatedcodes. knowledged.Example 10: In Example 4 we found the parameters ofcodes over GF(256) derived from Hermitian curves. If weAPPENDIXexclude points with x = 0, the length becomes n = 4080. We will prove that for the Hermitian curves given by theTheorem 4 is satisfied by 102 codes with 15 5 h 5116. For equation (Example 4) xl+’ + y’+* + z”+l = 0 over F= GF(q),this sequence of codes the dimension decreases and the where q = r2, we haveminimum distance increases in steps of 34 fromH,‘(j) =G:(q-2- j) h-21 jiq-fi.(4080,3570,391) to (4080,136,3825). The first of thesecodes will correct 135 errors if the decoding algorithm of The curve has q’/* - q’/’ = r3 - r points with x coordinateSection IV is used. As indicated in the beginning of this equal to 1, and hence gives codes H,‘(j) of length n = r3 - r.section, the matrix s is reduced to 255 X 136, and we solveSince r +l divides r3 - r (with quotient r2 - r), the necessarycondition in Proposition 1 is satisfied.for a locator polynomial with 136 coefficients. If there areTo prove the statement we have to show that if F(y, z) is any135 errors, we have 8 I degree(u) 215. For a locatorpolynomial of degree less than or equal to r* - 2 = s, thenpolynomial of degree 15, there may be 255 intersectionswith C. From the 510 syndrome equations, we obtain theerror values, of which at most 135 are assumed to benonzero.It would be interesting to extend the decoding algorithmin several directions. We lack a general proof that allerrors of weight < d/2 can be corrected. Further, it wouldbe interesting to decode the high-rate codes which areexcluded by the assumptions in Theorem 4, and it wouldbe of interest to include the points with x = 0. It appearsto be a straightforward generalization to apply the samealgorithm to the codes in higher dimensional space describedin Example 7.As mentioned in earlier sections, the decoding algorithmmay be seen as a generalization of Peterson’s algorithm forBCH codes. For long codes the algorithm involves verytime-consuming solutions of systems of linear equations. Itwould be a maJor improvement if more efficient algorithmssimilar to those available for BCH codes could befound.VI.DISCUSSIONWe have treated algebraic geometry codes in termswhich should be familiar to readers with a background instandard coding theory. Many details in the constructionof generator and parity-check matrices, and in the decodingalgorithm, are closely related to methods with a longtradition in coding theory.The use of projective geometries in the construction oferror-correcting codes is not new, either. The essentiallynew aspect is the restriction of the space to the points on acurve and the results directly related to the properties ofthe curve.The new codes are better than any previously knownwhen the alphabet is large enough. This is true even forcodes of moderate length. Unfortunately, our knowledgeof good curves over finite fields is still rather limited, andmore extensive work in this direction is needed.ACKNOWLEDGMENTThe authors wish to thank the referees for their remarksand suggestions. Several useful discussions with Johan P.We write F(y,z) asc F(Y,z) =O. (1)(l,Y,Z)ECF(y,z)= i a,y’+ i b,z’+ c a,, y’zj. (4, j=O j=l i>O,J>Oi-t jssThe points (1, y, z) of C have the form (1, a,O), where (Y is one ofthe r + 1 elements of GF(r2), where a’+l = - 1, or the form(l,cll$) where a is an element of GF(r2), where (Y’+’ # -1 and/?isanyoneofther+lsolutionstoz’+’=-l-a”+’.The term P,=oaiyJ then contributes na, +C~=oa,XnsGFcr2jaj= 0 to (1) since r divides n and j I r2 -2 [15, p. 3211. Theterm CT=, b,z’ contributes (for a fixed cu, where 01”+l# - 1)C-J=,b,C@i = 0 to (1) since the inner sum is the sum of the rootsof the equation zr+’ = (- 1 - ar+‘)J.The last termc a,,yldi>O,j>OI+jsscontributesC a,, C a’ C P’i>O,j>O a BIt j

JUSTESEN et a/.: A CLASS OF ALGEBRAIC GEOMETRY CODES821[9] H. .I. Tiersma, “Remarks on codes from Hermitian curves,” IEEETruns. fnform. Theory, vol. IT-33, no. 4, pp. 60%609,1987.[lo] A. N. Skorobogatov and S. G. Vladut, “On the decoding ofalgebraic geometric codes,” Inst. Problems of Information TransmissionMoscow, preprint 1988.[ll] W. Fulton, Algebruic Curues. Reading, MA: Benjamin Cummings,1969.[12] J. P. Serre, “Sur le nombre des points rationnels dune courbealgebraique sur un corps fini,” C. R. Acad. Sci., vol. 296, pp.397-402, 1983.[13] M. Deuring, Lectures on the Theory of Algebraic Functions of OneVuriable. Berlin: Springer Lecture Notes in Mathematics, vol.314,1973.. f141 _ R. D. Hartshorne, Algebraic Geometry. New York: Springer-Verlag, Graduate Texts in Mathematics,-vol. 52, 1977.ll51L- 1.I. W. P. Hirschfeld. Projective Geometries ouer Finite Fields. OXford,UK: Oxford University, 1979.[16] W. W. Peterson and E. .I. Weldon, Jr., Error-Correcting Codes.Cambridge, MA: MIT Press, 1972.