Multi-point Codes from the GGS Curves
aa r X i v : . [ c s . I T ] F e b Manuscript submitted to doi:10.3934/xx.xx.xx.xxAIMS’ JournalsVolume X , Number , XX pp. X–XX
MULTI-POINT CODES FROM THE GGS CURVES C HUANGQIANG H UYau Mathematical Sciences Center, Tsinghua University,Peking, 100084, China S HUDI Y ANG ∗ School of Mathematical Sciences, Qufu Normal UniversityShandong, 273165, China (Communicated by ***) A BSTRACT . This paper is concerned with the construction of algebraic-geometric (AG)codes defined from GGS curves. It is of significant use to describe bases for the Riemann-Roch spaces associated with some rational places, which enables us to study multi-pointAG codes. Along this line, we characterize explicitly the Weierstrass semigroups and puregaps by an exhaustive computation for the basis of Riemann-Roch spaces from GGS curves.In addition, we determine the floor of a certain type of divisor and investigate the propertiesof AG codes. Multi-point codes with excellent parameters are found, among which, apresented code with parameters [ , , > ] over F yields a new record. Introduction.
In the early 1980s, Goppa [15] constructed algebraic geometric codes(AG codes for short) from algebraic curves. Since then, the study of AG codes becomesan important instrument in the theory of error-correcting codes. Roughly speaking, theparameters of an AG code are good when the underlying curve has many rational pointswith respect to its genus. For this reason maximal curves, that is curves attaining theHasse-Weil upper bound, have been widely investigated in the literature: for example theHermitian curve and its quotients, the Suzuki curve, the Klein quartic and the GK curve. Inthis work we will study multi-point AG codes on the GGS curves.In order to construct good AG codes we need to study Weierstrass semigroups and puregaps. Their use dates back to the theory of one-point codes. For example, the authors in[17, 30, 31, 32] examined one-point codes from Hermitian curves and developed efficientmethods to decode them. Korchm´aros and Nagy [22] computed the Weierstrass semigroupof a degree-three closed point of the Hermitian curve. Matthews [28] determined the Weier-strass semigroup of any r -tuple rational points on the quotient of a Hermitian curve. As isknown, Weierstrass pure gap is also a useful tool in coding theory. Garcia, Kim and Laximproved the Goppa bound using Weierstrass gaps at one place in [12, 13]. The concept ofpure gaps of a pair of points on a curve was initiated by Homma and Kim [18], and it had Mathematics Subject Classification.
Primary: 14H55, 11R58, 11T71.
Key words and phrases.
Algebraic geometric code, GGS curve, Weierstrass semigroup, pure Weierstrass gap.This work is partially supported by the NSFC (11701317, 11531007, 11571380, 11701320, 61472457) andTsinghua University startup fund. This work is also partially supported by China Postdoctoral Science Foun-dation Funded Project (2017M611801), Jiangsu Planned Projects for Postdoctoral Research Funds (1701104C),Guangzhou Science and Technology Program (201607010144) and the Natural Science Foundation of ShandongProvince of China (ZR2016AM04). ∗ Corresponding author: Shudi Yang. been pushed forward by Carvalho and Torres [7] to several points. Maharaj and Matthews[24] extended this construction by introducing the notion of the floor of a divisor and ob-tained improved bounds on the parameters of AG codes. The authors in [5, 26] countedthe gaps and pure gaps at two points on Hermitian curves and those from some specificKummer extensions, respectively. Recently, Yang and Hu [33, 34] extended their work byinvestigating Weierstrass semigroups and pure gaps at many points.We mention that Maharaj [23] showed that Riemann-Roch spaces of divisors from fiberproducts of Kummer covers of the projective line, can be decomposed as a direct sum ofRiemann-Roch spaces of divisors of the projective line. Maharaj, Matthews and Pirsic[25] determined explicit bases for large classes of Riemann-Roch spaces of the Hermitianfunction field. Along this research line, Hu and Yang [19] gave other explicit bases forRiemann-Roch spaces of divisors over Kummer extensions, which makes it convenient todetermine the pure gaps.In this work, we focus our attention on the GGS curves, which are maximal curvesconstructed by Garcia, G¨uneri and Stichtenoth [11] over F q n defined by the equations ( x q + x = y q + , y q − y = z m , where q is a prime power and m = ( q n + ) / ( q + ) with n > n =
3. Recall that Fanali and Giulietti [10] exhibited one-pointAG codes on the GK curves and obtained linear codes with better parameters with respectto those previously known. Two-point and multi-point AG codes on the GK maximalcurves were studied in [4] and [9], respectively. Bartoli, Montanucci and Zini [3] examinedone-point AG codes from the GGS curves. Inspired by the above work and [8, 19], here wewill investigate multi-point AG codes arising from GGS curves. To be precise, an explicitbasis for the corresponding Riemann-Roch space is determined by constructing a relatedset of lattice points. The properties of AG codes from GGS curves are also considered.Then the basis is utilized to characterize the Weierstrass semigroups and pure gaps withrespect to several rational places. In addition, we give an effective algorithm to computethe floor of divisors. Finally, our results will lead us to find new codes with better param-eters in comparison with the existing codes in MinT’s Tables [29]. A new record-giving [ , , > ] -code over F is presented as one of the examples.The remainder of the paper is organized as follows. Section 2 focuses on the con-struction of bases for the Riemann-Roch space from GGS curves. Section 3 studies theproperties of the related AG codes. In Section 4 we illustrate the Weierstrass semigroupsand the pure gaps. Section 5 devotes to the floor of divisors from GGS curves. Finally, inSection 6 we employ our results to construct multi-point codes with excellent parameters.2. Bases for Riemann-Roch spaces of the GGS curves.
Throughout this paper, we al-ways let q be a prime power and n > ( q , n ) over F q n is defined by the equations ( x q + x = y q + , y q − y = z m , (1)where m = ( q n + ) / ( q + ) . The genus of GGS ( q , n ) is ( q − )( q n + + q n − q ) / q n + − q n + + q n + + n =
3, it becomes the well-known maximal curve introduced by Giulietti and Korchm´aros[14], the so-called GK curve, which is not a subcover of the corresponding Hermitian curve.
ULTI-POINT CODES FROM THE GGS CURVES 3
Let α , β , γ ∈ F q n such that α q + α = β q + and β q − β = γ m . In particular, if γ = α , β ∈ F q . Denote by P α , β , γ the rational place of the function field F q n ( GGS ( q , n )) centered at the rational affine point with coordinates ( α , β , γ ) and P ∞ the place located atinfinity. They are exactly all the F q n -rational places. Take Q β : = ∑ α q + α = β q + P α , β , where β ∈ F q and we always view F q as a subfield of F q n . Notice that deg ( Q β ) = q . For lateruse, we write P µ : = P α µ , , where α µ ∈ F q with α qµ + α µ = µ < q . Particularlywe denote P : = P , , and Q : = P + P + · · · + P q − .The following proposition describes some principle divisors of the GGS curve. Proposition 1.
Let the curve
GGS ( q , n ) be given in (1) and assume that α µ ∈ F q with µ < q are the solutions of x q + x = . Then we obtain ( ) div ( x − α µ ) = m ( q + ) P µ − m ( q + ) P ∞ , ( ) div ( y − β ) = mQ β − mqP ∞ for β ∈ F q , ( ) div ( z ) = ∑ β ∈ F q Q β − q P ∞ . For convenience, we use Q ν (0 ν q −
1) to represent the divisors Q β ( β ∈ F q ) . Inparticular, the symbol Q performs well in both cases v = ∈ Z and β = ∈ F q .For a function field F , the Riemann-Roch vector space with respect to a divisor G isdefined by L ( G ) = n f ∈ F (cid:12)(cid:12)(cid:12) div ( f ) + G > o ∪ { } . Let ℓ ( G ) be the dimension of L ( G ) . From the famous Riemann-Roch Theorem [30, Theo-rem 1.5.15], we know that ℓ ( G ) − ℓ ( W − G ) = − g + deg ( G ) , where W is a canonical divisor and g is the genus of the associated curve.In this section, we consider divisors supported at all rational places of the form G : = ∑ q − µ = r µ P µ + ∑ q − ν = s ν Q ν + tP ∞ , where r µ ’s, s ν ’s and t are integers. We wish to show that theRiemann-Roch space L ( G ) is generated by some elements, say E i , j , k for some i , j , k , andthe number of such elements equals ℓ ( G ) . For this purpose, we proceed as follows.Let j = ( j , j , · · · , j q − ) and k = ( k , k , · · · , k q − ) . For ( i , j , k ) ∈ Z q + q − , we define E i , j , k : = z i q − ∏ µ = ( x − α µ ) j µ q − ∏ ν = ( y − β ν ) k ν . (2)Here and thereafter, we denote | v | to be the sum of all the coordinates of a given vector v .Then | j | = ∑ q − µ = j µ and | k | = ∑ q − ν = k ν . By Proposition 1, one can compute the divisor of E i , j , k : div ( E i , j , k ) = iP + q − ∑ µ = ( i + m ( q + ) j µ ) P µ + q − ∑ ν = ( i + mk ν ) Q ν − (cid:0) q i + m ( q + ) | j | + mq | k | (cid:1) P ∞ . (3)For later use, we denote by N : = N ∪ { } the set of nonnegative integers and denoteby ⌈ x ⌉ the smallest integer not less than x . It is easy to show that j = ⌈ a / b ⌉ if and only if0 b j − a < b , where b ∈ N and a ∈ Z . CHUANGQIANG HU, SHUDI YANG
Put r = ( r , r , · · · , r q − ) and s = ( s , s , · · · , s q − ) . Let us define a set of lattice pointsfor ( r , s , t ) ∈ Z q + q , Ω r , s , t : = n ( i , j , k ) (cid:12)(cid:12)(cid:12) i + r > , i + m ( q + ) j µ + r µ < m ( q + ) for µ = , · · · , q − , i + mk ν + s ν < m for ν = , · · · , q − , q i + m ( q + ) | j | + mq | k | t o , or equivalently, Ω r , s , t : = n ( i , j , k ) (cid:12)(cid:12)(cid:12) i + r > , j µ = (cid:24) − i − r µ m ( q + ) (cid:25) for µ = , · · · , q − , k ν = (cid:24) − i − s ν m (cid:25) for ν = , · · · , q − , q i + m ( q + ) | j | + mq | k | t o . (4)Our key result depends on the following lemma. However, its proof is technical and itwill be completed later. Lemma 2.1.
Assume that t > g − − q w, where w is the minimum element of all thecoordinates of r and s , then the cardinality of Ω r , s , t is Ω r , s , t = − g + t + | r | + q | s | . Now we can easily prove the main result of this section.
Theorem 2.2.
Let G : = ∑ q − µ = r µ P µ + ∑ q − ν = s ν Q ν + tP ∞ . The elements E i , j , k with ( i , j , k ) ∈ Ω r , s , t constitute a basis for the Riemann-Roch space L ( G ) . In particular ℓ ( G ) = Ω r , s , t .Proof. Let ( i , j , k ) ∈ Ω r , s , t . It follows from the definition that E i , j , k ∈ L ( G ) , where G = ∑ q − µ = r µ P µ + ∑ q − ν = s ν Q ν + tP ∞ . From Equation (3), we have v P ( E i , j , k ) = i , which indicatesthat the valuation of E i , j , k at the rational place P uniquely depends on i . Since latticepoints in Ω r , s , t provide distinct values of i , the elements E i , j , k are linearly independent ofeach other, with ( i , j , k ) ∈ Ω r , s , t . In order to indicate that they constitute a basis for L ( G ) ,the only thing left is to prove that ℓ ( G ) = Ω r , s , t . For the case of r sufficiently large, it follows from the Riemann-Roch Theorem andLemma 2.1 that ℓ ( G ) = − g + deg ( G )= − g + | r | + q | s | + t = Ω r , s , t . This implies that L ( G ) is spanned by elements E i , j , k with ( i , j , k ) in the set Ω r , s , t .For the general case, we choose r ′ > r large enough and set G ′ : = r ′ P + ∑ q − µ = r µ P µ + ∑ q − ν = s ν Q ν + tP ∞ and r ′ = ( r ′ , r , · · · , r q − ) . From above argument, we know that theelements E i , j , k with ( i , j , k ) ∈ Ω r ′ ,s , t span the whole space of L ( G ′ ) . Remember that L ( G ) is a linear subspace of L ( G ′ ) , which can be written as L ( G ) = n f ∈ L ( G ′ ) (cid:12)(cid:12)(cid:12) v P ( f ) > − r o . ULTI-POINT CODES FROM THE GGS CURVES 5
Thus, we choose f ∈ L ( G ) and suppose that f = ∑ ( i , j , k ) ∈ Ω r ′ , s , t a i E i , j , k , since f ∈ L ( G ′ ) by definition. The valuation of f at P is v P ( f ) = min a i = { i } . Then theinequality v P ( f ) > − r gives that, if a i =
0, then i > − r . Equivalently, if i < − r , then a i =
0. From the definition of Ω r , s , t and Ω r ′ ,s , t , we get that f = ∑ ( i , j , k ) ∈ Ω r , s , t a i E i , j , k . Then the theorem follows.We now turn to prove Lemma 2.1 which requires a series of results including telescopicsemigroups listed as follows.
Definition 2.3 ([21], Definition 6.1) . Let ( a , · · · , a k ) be a sequence of positive integerssuch that the greatest common divisor is 1. Define d i = gcd ( a , · · · , a i ) and A i = { a / d i , · · · , a i / d i } for i = , · · · , k . Let d =
0. Let S i be the semigroup generated by A i . If a i / d i ∈ S i − for i = , · · · , k , we call the sequence ( a , · · · , a k ) telescopic. A semigroup is called telescopicif it is generated by a telescopic sequence. Lemma 2.4 ([21], Lemma 6.4) . If ( a , · · · , a k ) is telescopic and M ∈ S k , then there existuniquely determined non-negative integers x i < d i − / d i for i = , · · · , k, such thatM = k ∑ i = x i a i . We call this representation the normal representation of M.
Lemma 2.5 ([21], Lemma 6.5) . For the semigroup generated by the telescopic sequence ( a , · · · , a k ) we have l g ( S k ) = k ∑ i = ( d i − / d i − ) a i , g ( S k ) = ( l g ( S k ) + ) / , where l g ( S k ) and g ( S k ) denote the largest gap and the number of gaps of S k , respectively. Lemma 2.6.
Let m = ( q n + ) / ( q + ) , g = ( q − )( q n + + q n − q ) / for an odd integern > . Let t ∈ Z . Consider the lattice point set Ψ ( t ) defined by n ( a , b , c ) (cid:12)(cid:12)(cid:12) a < m , b q , c > , q a + mqb + m ( q + ) c t o , If t > g − , then Ψ ( t ) has cardinality Ψ ( t ) = − g + t . Proof.
Let a = q , a = mq , a = m ( q + ) . It is easily verified that the sequence ( a , a , a ) is telescopic. By Lemma 2.4 every element M in S has a unique representation M = a a + a b + a c , where S is the semigroup generated by ( a , a , a ) . One obtains fromLemma 2.5 that l g ( S ) = ( q − )( q n + )( q + ) − q , g ( S ) = ( l g ( S ) + ) = ( q − )( q n + + q n − q ) = g . CHUANGQIANG HU, SHUDI YANG
It follows that the set Ψ ( t ) has cardinality 1 − g + t provided that t > g − = l g ( S ) , whichfinishes the proof.From Lemma 2.6, we get the number of lattice points in Ω , , t . Lemma 2.7.
If t > g − , then the cardinality of Ω , , t is Ω , , t = − g + t . Proof.
Note that Ω , , t : = n ( i , j , k ) (cid:12)(cid:12)(cid:12) i > , j µ = (cid:24) − im ( q + ) (cid:25) for µ = , · · · , q − , k ν = (cid:24) − im (cid:25) for ν = , · · · , q − , q i + m ( q + ) | j | + mq | k | t o . (5)Set i : = a + m ( b + ( q + ) c ) with 0 a < m , 0 b q and c >
0. Then Equation (5) givesthat Ω , , t ∼ = n ( a , b , c ) (cid:12)(cid:12)(cid:12) a < m , b q , c > , q a + mqb + ( q n + ) c t o . Here and thereafter, the notation A ∼ = B means that two lattice point sets A and B arebijective. In the last formula, the bijection comes from the mappings ( i , j , k ) ( a + m ( b + ( q + ) c ) , − c , − b − ( q + ) c ) and ( a , b , c ) ( i + mk , ( q + ) j − k , − j ) by denot-ing j = j = · · · = j q − and k = k = · · · = k q − . Thus the assertion Ω , , t = − g + t isderived from Lemma 2.6.The next three lemmas state some elementary properties of Ω r , s , t . Lemma 2.8.
The lattice point set Ω r , s , t as defined above is symmetric with respect tor , r , · · · , r q − and s , s , · · · , s q − , respectively. In other words, we have Ω r , s , t = Ω r ′ , s ′ , t by denoting r ′ = ( r ′ , r ′ , · · · , r ′ q − ) and s ′ = ( s ′ , s ′ , · · · , s ′ q − ) , where the sequences (cid:0) r i (cid:1) q − i = and (cid:0) s i (cid:1) q − i = are equal to (cid:0) r ′ i (cid:1) q − i = and (cid:0) s ′ i (cid:1) q − i = up to permutation, respectively.Proof. Recall that Ω r , s , t is defined by Ω r , s , t = n ( i ′ , j ′ , k ′ ) (cid:12)(cid:12)(cid:12) i ′ + r > , j ′ µ = (cid:24) − i ′ − r µ m ( q + ) (cid:25) for µ = , · · · , q − , k ′ ν = (cid:24) − i ′ − s ν m (cid:25) for ν = , · · · , q − , q i ′ + m ( q + ) | j ′ | + mq | k ′ | t o , where j ′ = ( j ′ , · · · , j ′ q − ) and k ′ = ( k ′ , · · · , k ′ q − ) . It is important to write i ′ = i + m ( q + ) l with 0 i < m ( q + ) . Let j ′ µ = j µ − l for µ > k ′ ν = k ν − ( q + ) l for ν >
1. Then Ω r , s , t ∼ = n ( i , l , j , k ) (cid:12)(cid:12)(cid:12) i + m ( q + ) l > − r , i < m ( q + ) , j µ = (cid:24) − i − r µ m ( q + ) (cid:25) for µ = , · · · , q − , ULTI-POINT CODES FROM THE GGS CURVES 7 k ν = (cid:24) − i − s ν m (cid:25) for ν = , · · · , q − , q i + m ( q + ) (cid:0) l + | j | (cid:1) + mq | k | t o , where j = ( j , · · · , j q − ) and k = ( k , · · · , k q − ) . The first inequality in Ω r , s , t gives that l > j : = (cid:24) − i − r m ( q + ) (cid:25) . So we write l = j + ι with ι >
0. Then Ω r , s , t ∼ = n ( i , ι , j , j , k ) (cid:12)(cid:12)(cid:12) i < m ( q + ) , ι > , j µ = (cid:24) − i − r µ m ( q + ) (cid:25) for µ = , , · · · , q − , k ν = (cid:24) − i − s ν m (cid:25) for ν = , · · · , q − , q i + m ( q + ) (cid:0) j + ι + | j | (cid:1) + mq | k | t o . The right hand side means that the number of the lattice points does not depend on theorder of r µ , 0 µ q −
1, and the order of s ν , 1 ν q −
1, which concludes the desiredassertion.
Lemma 2.9.
Let r = ( r , r , · · · , r q − ) ∈ N q and s = ( s , s , · · · , s q − ) ∈ N q − . If t > g − , then Ω r , s , t = Ω , s , t + | r | . Proof.
Let us take the sets Ω r , s , t and Ω r ′ , s , t for consideration, where r ′ = ( , r , · · · , r q − ) .It follows from the definition that the complement set ∆ : = Ω r , s , t \ Ω r ′ , s , t is given by n ( i , j , k ) (cid:12)(cid:12)(cid:12) − r i < , j µ = (cid:24) − i − r µ m ( q + ) (cid:25) for µ = , · · · , q − , k ν = (cid:24) − i − s ν m (cid:25) for ν = , · · · , q − , q i + m ( q + ) | j | + mq | k | t o . Clearly ∆ = ∅ if r =
0. To determine the cardinality of ∆ with r >
0, we denote i : = a + m ( b + ( q + ) c ) with integers a , b , c satisfying 0 a < m , 0 b q and c −
1. Then j µ − c for µ > k ν − b − ( q + ) c for ν >
1. A straightforward computation shows q i + m ( q + ) | j | + mq | k | q a + mqb + m ( q + ) c q ( m − ) + mq − m ( q + ) = g − . So the last inequality in ∆ always holds for all t > g −
1, which means that the cardinalityof ∆ is determined by the first inequality, that is ∆ = r . Then we must have Ω r , s , t = Ω r ′ , s , t + r , whenever r >
0. Repeating the above argument and using Lemma 2.8, we get Ω r , s , t = Ω , s , t + | r | , where r = ( r , r , · · · , r q − ) . CHUANGQIANG HU, SHUDI YANG
Lemma 2.10.
Let s = ( s , s , · · · , s q − ) ∈ N q − . If t > g − , then the following identityholds: Ω , s , t = Ω , , t + q | s | . Proof.
For convenience, let us denote r : = ( s , s , · · · , s ) to be the q -tuple with all entriesequal s , where s >
0, and write Ω r , s , t as Γ s , ( s , ··· , s q − ) , t . To get the desired conclusion,we first claim that Γ s , ( s , ··· , s q − ) , t = Γ s ′ , ( s ′ , ··· , s ′ q − ) , t , (6)where the sequence (cid:0) s i (cid:1) q − i = is equal to (cid:0) s ′ i (cid:1) q − i = up to permutation.Note that Γ s , ( s , ··· , s q − ) , t is equivalent to n ( i ′ , j ′ , k ′ , · · · , k ′ q − ) (cid:12)(cid:12)(cid:12) i ′ + s > , i ′ + m ( q + ) j ′ + s < m ( q + ) , i ′ + mk ′ ν + s ν < m for ν = , · · · , q − , q i ′ + m ( q + )( q − ) j ′ + mq | k ′ | t o , where | k ′ | = ∑ q − ν = k ′ ν . By setting i ′ : = i + m κ where 0 i < m and k ′ ν : = k ν − κ for ν > Γ s , ( s , ··· , s q − ) , t ∼ = n ( i , κ , j ′ , k , · · · , k q − ) (cid:12)(cid:12)(cid:12) i < m , i + m κ + s > , i + m κ + m ( q + ) j ′ + s < m ( q + ) , i + mk ν + s ν < m for ν = , · · · , q − , q i + m ( q + )( q − ) j ′ + mq ( κ + | k | ) t o . Put κ : = k + ε where k = (cid:24) − i − s m (cid:25) and ε = − ( q + ) j + η with 0 η < q +
1. Onegets that 0 i + mk + s < m , which leads to 0 i + m κ + m ( q + ) j − m η + s < m .So the inequality 0 i + m κ + m ( q + ) j + s < m + m η m ( q + ) holds because ε = − ( q + ) j + η . Thus we must have j = j ′
0. Therefore Γ s , ( s , ··· , s q − ) , t ∼ = n ( i , j , η , k , k , · · · , k q − ) (cid:12)(cid:12)(cid:12) i < m , j , η < q + , k ν = (cid:24) − i − s ν m (cid:25) for ν = , , · · · , q − , q i − m ( q + ) j + mq ( k + η + | k | ) t o . The right hand side means that the lattice points do not depend on the order of s ν with0 ν q −
1, by observing that k ν is determined by s ν . In other words, we have shownthat the number of lattice points in Γ s , ( s , ··· , s q − ) , t does not depend on the order of s ν with0 ν q −
1, concluding the claim we presented by (6). So it follows from (6) andLemma 2.9 that Γ , ( s , s , ··· , s q − ) , t = Γ s , ( , s , ··· , s q − ) , t = Γ , ( , s , ··· , s q − ) , t + qs . ULTI-POINT CODES FROM THE GGS CURVES 9
By repeatedly using Lemma 2.9, we get Γ , ( s , s , ··· , s q − ) , t = Γ , ( , , ··· , ) , t + q ( s + s + · · · + s q − ) , concluding the desired formula Ω , s , t = Ω , , t + q | s | .We are now in a position to give the proof of Lemma 2.1. Proof. [ Proof of Lemma 2.1]
By taking w : = min µ q − ν q − n r µ , s ν o , we obtain from thedefinition that Ω r , s , t is equivalent to Ω r ′ , s ′ , t ′ , where r ′ = ( r − w , · · · , r q − − w ) , s ′ =( s − w , · · · , s q − − w ) and t ′ = t + q w . Hence Ω r , s , t = Ω r ′ , s ′ , t ′ . On the other hand,by observing that r µ − w > s ν − w > t ′ > g −
1, we establish from Lemmas 2.7,2.9 and 2.10 that Ω r ′ , s ′ , t ′ = Ω , s ′ , t ′ + | r ′ | = Ω , , t ′ + q | s ′ | + | r ′ | = − g + t ′ + q | s ′ | + | r ′ | = − g + t + q | s | + | r | . It then follows that Ω r , s , t = − g + t + q | s | + | r | , completing the proof of Lemma 2.1.We finish this section with a result that allows us to give a new form of the basis forour Riemann-Roch space L ( G ) with G = ∑ q − µ = r µ P µ + ∑ q − ν = s ν Q ν + tP ∞ . Denote λ : =( λ , · · · , λ q − ) and γ : = ( γ , · · · , γ q − ) . For ( u , λ , γ ) ∈ Z q + q − , we define Λ u , λ , γ : = τ u q − ∏ µ = f λ µ µ q − ∏ µ = h γ ν ν , where τ : = z q n − x − α , f µ : = x − α µ x − α for µ >
1, and h ν : = y − β ν y − β for ν >
1. We have fromProposition 1 thatdiv ( Λ u , λ , γ ) = q − ∑ µ = (cid:16) q n − u + m ( q + ) λ µ − m | γ | (cid:17) P µ + q − ∑ ν = ( q n − u + m γ ν ) Q ν − (cid:16) ( m ( q + ) − q n − ) u + m ( q + ) | λ | + m | γ | (cid:17) P + uP ∞ . There is a close relationship between the elements Λ u , λ , γ and E i , j , k explored as follows. Corollary 2.1.
Let G : = ∑ q − µ = r µ P µ + ∑ q − ν = s ν Q ν + tP ∞ . Then the elements Λ u , λ , γ with ( u , λ , γ ) ∈ Θ r , s , t form a basis for the Riemann-Roch space L ( G ) , where the set Θ r , s , t isgiven by n ( u , λ , γ ) (cid:12)(cid:12)(cid:12) u > − t , q n − u + m γ ν + s ν < m for ν = , · · · , q − q n − u + m ( q + ) λ µ − m | γ | + r µ < m ( q + ) for µ = , · · · , q − , ( m ( q + ) − q n − ) u + m ( q + ) | λ | + m | γ | r o . (7) In addition we have Θ r , s , t = Ω r , s , t .Proof. It suffices to prove that the set n Λ u , λ , γ (cid:12)(cid:12)(cid:12) ( u , λ , γ ) ∈ Θ r , s , t o equals the set n E i , j , k (cid:12)(cid:12)(cid:12) ( i , j , k ) ∈ Ω r , s , t o . In fact, for fixed ( u , λ , γ ) ∈ Z q + q − , we obtain Λ u , λ , γ equals E i , j , k with i = − ( m ( q + ) − q n − ) u − m ( q + ) | λ | − m | γ | , j µ = u + | λ | + λ µ for µ = , · · · , q − , k ν = ( q + )( u + | λ | ) + | γ | + γ ν for ν = , · · · , q − . On the contrary, if we set u = − q i − m ( q + ) | j | − mq | k | , λ µ = q i + q n − | j | + m | k | + j µ for µ = , · · · , q − , γ ν = ( q + )( i + q n − | j | ) + q n − | k | + k ν for ν = , · · · , q − , then E i , j , k is exactly the element Λ u , λ , γ . Therefore, if we restrict ( i , j , k ) in Ω r , s , t , then wemust have ( u , λ , γ ) is in Θ r , s , t and vice versa. This completes the proof of this corollary.In the following, we will demonstrate an interesting property of Ω r , s , t for GK curveswith a specific vector s . Corollary 2.2.
Let n = and the vectors r, s be given by r : = ( r , r , · · · , r q − ) , s : = ( s ′ , s ′ , · · · , s ′ q − )= ( s , s , · · · , s | {z } q + , s , s , · · · , s | {z } q + , · · · , s q − , s q − , · · · , s q − | {z } q + ) . Then the lattice point set Ω r , s , t is symmetric with respect to r , r , · · · , r q − , t. In otherwords, we have Ω r , s , t = Ω r ′ , s , t ′ , where the sequence (cid:0) r i (cid:1) qi = is equal to (cid:0) r ′ i (cid:1) qi = up topermutation by putting r q : = t and r ′ q : = t ′ .Proof. Denote r : = ( r , ˙ r ) = ( r , r , · · · , r q − ) and Ω ( r , ˙ r ) , s , t : = Ω r , s , t . By Lemma 2.8, itsuffices to prove that Ω ( r , ˙ r ) , s , t = Θ ( r , ˙ r ) , s , t = Ω ( t , ˙ r ) , s , r . The first identity follows directly from Corollary 2.1. Applying Corollary 2.1 againgives the set Θ ( r , ˙ r ) , s , t as n ( u , λ , γ ) (cid:12)(cid:12)(cid:12) u + t > , u + m γ ν + s ′ ν < m for ν = , · · · , q − , u + m ( q + ) λ µ − m | γ | + r µ < m ( q + ) for µ = , · · · , q − , q u + m ( q + ) | λ | + m | γ | r o . ULTI-POINT CODES FROM THE GGS CURVES 11
From our assumption, it is obvious that | γ | is divisible by q +
1. So if we take i : = u , j µ : = λ µ − | γ | q + µ > k ν : = γ ν for ν >
1, then Θ ( r , ˙ r ) , s , t is equivalent to n ( i , j , k ) (cid:12)(cid:12)(cid:12) i + t > , i + mk ν + s ′ ν < m for ν = , · · · , q − , i + m ( q + ) j µ + r µ < m ( q + ) for µ = , · · · , q − , q i + m ( q + ) | j | + mq | k | r o . The last set is exactly Ω ( t , ˙ r ) , s , r by definition. Hence the second identity is just shown,completing the whole proof.3. The AG codes from GGS curves.
This section settles the properties of AG codes fromGGS curves. Generally speaking, there are two classical ways of constructing AG codesassociated with divisors D and G , where G is a divisor of arbitrary function field F and D : = Q + · · · + Q N is another divisor of F such that Q , · · · , Q N are pairwise distinct rationalplaces, each not belonging to the support of G . One construction is based on the Riemann-Roch space L ( G ) , C L ( D , G ) : = n ( f ( Q ) , · · · , f ( Q N )) (cid:12)(cid:12)(cid:12) f ∈ L ( G ) o ⊆ F Nq . The other one depends on the space of differentials Ω ( G − D ) , C Ω ( D , G ) : = n ( res Q ( η ) , · · · , res Q N ( η )) (cid:12)(cid:12)(cid:12) η ∈ Ω ( G − D ) o . It is well-known the codes C L ( D , G ) and C Ω ( D , G ) are dual to each other. Further C Ω ( D , G ) has parameters [ N , k Ω , d Ω ] with k Ω = N − k and d Ω > deg ( G ) − ( g − ) , where k = ℓ ( G ) − ℓ ( G − D ) is the dimension of C L ( D , G ) . If moreover 2 g − < deg ( G ) < N then k Ω = N + g − − deg ( G ) . The reader is referred to [30] for more information.In this section, we follow the notation given in Section 2. Let D be the direct sum ofall F q n -rational places except the F q -rational places of the function field F q n ( GGS ( q , n )) ,namely, D : = ∑ α , β , γγ = P α , β , γ , where α , β , γ are elements in F q n satisfying α q + α = β q + and β q − β = γ m . Now we will study the AG code C L ( D , G ) with G : = ∑ q − µ = r µ P µ + ∑ q − ν = s ν Q ν + tP ∞ . The length of C L ( D , G ) is N : = deg ( D ) = q n + ( q n − q + ) − q . It is well known that the dimension of C L ( D , G ) is given bydim C L ( D , G ) = ℓ ( G ) − ℓ ( G − D ) . (8)Set R : = N + g −
2. If deg ( G ) > R , we deduce from the Riemann-Roch Theorem and (8)that dim C L ( D , G ) = ( − g + deg ( G )) − ( − g + deg ( G − D ))= deg ( G ) − deg ( G − D ) = N , which implies that C L ( D , G ) is trivial. So we only consider the case 0 deg ( G ) R .Now, we use the following lemmas to calculate the dual of C L ( D , G ) . Lemma 3.1 ([30], Proposition 2.2.10) . Let τ be an element of a function field such thatv P i ( τ ) = for all rational places P i contained in the divisor D. Then the dual of C L ( D , G ) is C L ( D , G ) ⊥ = C L ( D , D − G + div ( d τ ) − div ( τ )) . Lemma 3.2 ([30], Proposition 2.2.14) . Suppose that G and G are divisors with G = G + div ( ρ ) for some ρ ∈ F \{ } and supp G ∩ supp D = supp G ∩ supp D = ∅ . Let N : = deg ( D ) and ̺ : = ( ρ ( P ) , · · · , ρ ( P N )) with P i ∈ D. Then the codes C L ( D , G ) and C L ( D , G ) are equivalent and C L ( D , G ) = ̺ · C L ( D , G ) . The dimension and the dual code of C L ( D , G ) can be described below. Theorem 3.3.
Let A : = ( q n + )( q − ) − , B : = mq ( q n − q ) + ( q n + )( q − ) − and ρ : = + ∑ n − i = z ( q n + )( q − ) ∑ ij = q j . Then the dual code of C L ( D , G ) is given as follows. ( ) The dual of C L ( D , G ) is represented asC L ( D , G ) ⊥ = ̺ · C L ( D , q − ∑ µ = ( A − r µ ) P µ + q − ∑ ν = ( A − s ν ) Q ν + ( B − t ) P ∞ ) , where ̺ : = ( ρ ( P α , β , γ ) , · · · , ρ ( P α N , β N , γ N )) with P α i , β i , γ i ∈ D. ( ) In particular, for n = , we have ρ = andC L ( D , G ) ⊥ = C L ( D , q − ∑ µ = ( A − r µ ) P µ + q − ∑ ν = ( A − s ν ) Q ν + ( B − t ) P ∞ ) . Proof.
Define H : = n z ∈ F ∗ q n (cid:12)(cid:12)(cid:12) ∃ y ∈ F q n with y q − y = z m o . Consider the element τ : = ∏ γ ∈ H ( z − γ ) . Then τ is a prime element for all places P α , β , γ in D and its divisor isdiv ( τ ) = ∑ γ ∈ H div ( z − γ ) = D − deg ( D ) P ∞ , where D = ∑ α , β , γγ = P α , β , γ and N = deg ( D ) = q n + ( q n − q + ) − q . Moreover by a samediscussion as in the proof of Lemma 2 in [1], we have τ = + k − ∑ i = w ∑ ij = q j + ∑ k − j = q j + + k − ∑ i = w ∑ ij = q j + , where n = k + n > w = z ( q n + )( q − ) . Then a straightforwardcomputation shows d τ = w ∑ k − j = q j + (cid:16) + k − ∑ i = w ∑ ij = q j (cid:17) dw = w qn − qq − (cid:16) + k − ∑ i = w ∑ ij = q j (cid:17) dw , dw = − z ( q n + )( q − ) − dz . ULTI-POINT CODES FROM THE GGS CURVES 13
Let ρ : = + ∑ k − i = w ∑ ij = q j and denote its divisor by div ( ρ ) . Set A : = m ( q n − q ) + ( q n + )( q − ) − , S : = q A − g + . Since div ( dz ) = ( g − ) P ∞ (see Lemma 3.8 of [16]), it follows from Proposition 1 thatdiv ( d τ ) = A · div ( z ) + div ( dz ) + div ( ρ )= A ∑ β ∈ F q Q β − (cid:16) q A − g + (cid:17) P ∞ + div ( ρ )= A q − ∑ µ = P µ + A q − ∑ ν = Q ν − SP ∞ + div ( ρ ) . Let η : = d τ / τ be a Weil differential. The divisor of η isdiv ( η ) = div ( d τ ) − div ( τ )= A q − ∑ µ = P µ + A q − ∑ ν = Q ν − D + (cid:16) deg ( D ) − S (cid:17) P ∞ + div ( ρ ) . By writing B : = deg ( D ) − S = mq ( q n − q ) + ( q n + )( q − ) −
1, we establish fromLemma 3.1 that the dual of C L ( D , G ) is C L ( D , G ) ⊥ = C L ( D , D − G + div ( η ))= C L ( D , q − ∑ µ = ( A − r µ ) P µ + q − ∑ ν = ( A − s ν ) Q ν + ( B − t ) P ∞ + div ( ρ )) . Denote ̺ : = ( ρ ( P α , β , γ ) , · · · , ρ ( P α N , β N , γ N )) with P α i , β i , γ i ∈ D . Then we deduce the firststatement from Lemma 3.2. The second statement then follows immediately. Theorem 3.4.
Suppose that deg ( G ) R. Then the dimension of C L ( D , G ) is given by dim C L ( D , G ) = (cid:26) Ω r , s , t if 0 deg ( G ) < N , N − Ω ⊥ r , s , t if N deg ( G ) R , where Ω ⊥ r , s , t : = Ω r ′ , s ′ , B − t with r ′ = ( A − r , · · · , A − r q − ) and s ′ = ( A − s , · · · , A − s q − ) .Proof. For 0 deg ( G ) < N , we have by Theorem 2.2 and Equation (8) thatdim C L ( D , G ) = ℓ ( G ) = Ω r , s , t . For N deg ( G ) R , Theorem 3.3 yields thatdim C L ( D , G ) = N − dim C L ( D , G ) ⊥ = N − Ω ⊥ r , s , t . So the proof is completed.Goppa bound provides an estimate for the minimum distance of C L ( D , G ) . Techniquesfor improving the Goppa bounds will be dealt with in the next two sections. Weierstrass semigroups and pure gaps.
In this section, we will characterize the Weier-strass semigroups and pure gaps on GGS curves, which will enables us to obtain improvedbounds on the parameters of AG codes.We need some preliminary notation and results before we begin. For an arbitrary func-tion field F , let Q , · · · , Q l be distinct rational places of F , then the Weierstrass semigroup H ( Q , · · · , Q l ) is defined by n ( s , · · · , s l ) ∈ N l (cid:12)(cid:12)(cid:12) ∃ f ∈ F with ( f ) ∞ = l ∑ i = s i Q i o , and the Weierstrass gap set G ( Q , · · · , Q l ) is defined by N l \ H ( Q , · · · , Q l ) , where N de-notes the set of nonnegative integers. The details are found in [27].Homma and Kim [18] introduced the concept of pure gap set with respect to a pair ofrational places. This was generalized by Carvalho and Torres [7] to several rational places,denoted by G ( Q , · · · , Q l ) , which is given by n ( s , · · · , s l ) ∈ N l (cid:12)(cid:12)(cid:12) ℓ ( G ) = ℓ ( G − Q j ) for all 1 j l , where G = l ∑ i = s i Q i o . In addition, they showed in Lemma 2.5 of [7] that ( s , · · · , s l ) is a pure gap at ( Q , · · · , Q l ) if and only if ℓ ( s Q + · · · + s l Q l ) = ℓ (( s − ) Q + · · · + ( s l − ) Q l ) . A useful way to calculate the Weierstrass semigroups is given as follows, which can beregarded as an easy generalization of Lemma 2.1 due to Kim [20].
Lemma 4.1 ([7], Lemma 2.2) . For rational places Q , · · · , Q l with l r, the setH ( Q , · · · , Q l ) is given by n ( s , · · · , s l ) ∈ N l (cid:12)(cid:12)(cid:12) ℓ ( G ) = ℓ ( G − Q j ) for all j l , where G = l ∑ i = s i Q i o . In the rest of this section, we will restrict our study to the divisor G : = ∑ q − µ = r µ P µ + tP ∞ and denote r = ( r , r , · · · , r q − ) . Our main task is to determine the Weierstrass semigroupsand the pure gaps at distinct rational places P , P , · · · , P q − , P ∞ . Before we proceed, someauxiliary results are presented in the following. Denote Ω r , , t by Ω ( r , r , ··· , r q − ) , t for theclarity of description. Lemma 4.2.
For the lattice point set Ω ( r , r , ··· , r q − ) , t , we have the following assertions. ( ) Ω ( r , r , ··· , r q − ) , t = Ω ( r − , r , ··· , r q − ) , t + if and only if q − ∑ µ = (cid:24) r − r µ m ( q + ) (cid:25) + q ( q − ) l r m m t + q r m ( q + ) . ( ) Ω ( r , r , ··· , r q − ) , t = Ω ( r , r , ··· , r q − ) , t − + if and only if q − ∑ µ = (cid:24) q n − t − r µ m ( q + ) (cid:25) + q ( q − ) (cid:24) q n − tm (cid:25) t + r − q n − tm ( q + ) . Proof.
Consider two lattice point sets Ω ( r , r , ··· , r q − ) , t and Ω ( r − , r , ··· , r q − ) , t , which aregiven in Equation (4). Clearly, the latter one is a subset of the former one, and the comple-ment set Φ of Ω ( r − , r , ··· , r q − ) , t in Ω ( r , r , ··· , r q − ) , t is given by Φ : = n ( i , j , k ) (cid:12)(cid:12)(cid:12) i + r = , ULTI-POINT CODES FROM THE GGS CURVES 15 j µ = (cid:24) − i − r µ m ( q + ) (cid:25) for µ = , · · · , q − , k ν = (cid:24) − im (cid:25) for ν = , · · · , q − , q i + m ( q + ) | j | + mq | k | t o , where j = ( j , j , · · · , j q − ) and k = ( k , k , · · · , k q − ) . It follows immediately that the set Φ is not empty if and only if − q r + m ( q + ) q − ∑ µ = (cid:24) r − r µ m ( q + ) (cid:25) + mq ( q − ) l r m m t , which concludes the first assertion.For the second assertion, we obtain from Corollary 2.1 that the difference between Ω ( r , r , ··· , r q − ) , t and Ω ( r , r , ··· , r q − ) , t − is exactly the same as the one between Θ r , , t and Θ r , , t − . In an analogous way, we define Ψ as the complementary set of Θ r , , t − in Θ r , , t , namely Ψ : = n ( u , λ , γ ) (cid:12)(cid:12)(cid:12) u = − t , q n − u + m γ ν < m for ν = , · · · , q − , q n − u + m ( q + ) λ µ − m | γ | + r µ < m ( q + ) for µ = , · · · , q − , − (cid:16) ( m ( q + ) − q n − ) u + m ( q + ) | λ | + m | γ | (cid:17) + r > o . The set Ψ is not empty if and only if m ( q + ) q − ∑ µ = (cid:24) q n − t − r µ m ( q + ) (cid:25) + mq ( q − ) (cid:24) q n − tm (cid:25) r + ( q n + − q n − ) t , completing the proof of the second assertion.We are now ready for the main results of this section dealing with Weierstrass semi-groups and pure gap sets, which play an interesting role in finding codes with good param-eters. For simplicity, we write r l = ( r , r , · · · , r l ) and define W j ( r l , t , l ) : = l ∑ i = i = j (cid:24) r j − r i m ( q + ) (cid:25) + ( q − − l ) (cid:24) r j m ( q + ) (cid:25) + q ( q − ) l r j m m − t + q r j m ( q + ) , for j = , · · · , l , and W ∞ ( r l , t , l ) : = l ∑ i = (cid:24) q n − t − r i m ( q + ) (cid:25) + ( q − − l ) (cid:24) q n − tm ( q + ) (cid:25) + q ( q − ) (cid:24) q n − tm (cid:25) − t − r − q n − tm ( q + ) . Theorem 4.3.
Let P , P , · · · , P l be rational places as defined previously. For l < q, thefollowing assertions hold. ( ) The Weierstrass semigroup H ( P , P , · · · , P l ) is given by n ( r , r , · · · , r l ) ∈ N l + (cid:12)(cid:12)(cid:12) W j ( r l , , l ) for all j l o . ( ) The Weierstrass semigroup H ( P , P , · · · , P l , P ∞ ) is given by n ( r , r , · · · , r l , t ) ∈ N l + (cid:12)(cid:12)(cid:12) W j ( r l , t , l ) for all j l and W ∞ ( r l , t , l ) o . ( ) The pure gap set G ( P , P , · · · , P l ) is given by n ( r , r , · · · , r l ) ∈ N l + (cid:12)(cid:12)(cid:12) W j ( r l , , l ) > for all j l o . ( ) The pure gap set G ( P , P , · · · , P l , P ∞ ) is given by n ( r , r , · · · , r l , t ) ∈ N l + (cid:12)(cid:12)(cid:12) W j ( r l , t , l ) > for all j l and W ∞ ( r l , t , l ) > o . Proof.
The desired conclusions follow from Theorem 2.2, Lemmas 2.8, 4.1 and 4.2.In the literature, Beelen and Montanucci [6, Theorem 1.1] studied the Weierstrass semi-groups H ( P ) at any point P on the GK curve. For the GGS curve, H ( P ) was determinedin [3, Proposition 4.3] and G ( P ) was described independently in [2, Corollary 18] for P = P ∞ , while H ( P ∞ ) and G ( P ∞ ) were given in [16, Corollary 3.5 and Theorem 3.7], re-spectively. Here we give another characterization and an alternative proof of H ( P ) and G ( P ) using our main theorem. Corollary 4.1.
With notation as before, we have the following statements. ( ) H ( P ) = n k ∈ N (cid:12)(cid:12)(cid:12) ( q − ) (cid:24) km ( q + ) (cid:25) + q ( q − ) (cid:24) km (cid:25) q km ( q + ) o . ( ) Let a , b , c ∈ Z . Then a + m ( b + ( q + ) c ) ∈ N is a gap at P if and only if exactly one ofthe following two conditions is satisfied: ( i ) a = , < b q − and c q − − b; ( ii ) < a < m, b q, c q − − b + (cid:22) bq + − q am ( q + ) (cid:23) andb − (cid:22) bq + − q am ( q + ) (cid:23) q − .Proof. The first statement is an immediate consequence of Theorem 4.3 ( ) .We now focus on the second statement. It follows from Theorem 4.3 ( ) that the Weier-strass gap set at P is G ( P ) = n k ∈ N (cid:12)(cid:12)(cid:12) ( q − ) (cid:24) km ( q + ) (cid:25) + q ( q − ) (cid:24) km (cid:25) > q km ( q + ) o . Let k ∈ G ( P ) and write k = a + m ( b + ( q + ) c ) , where 0 a < m , 0 b q and c > a + mb = k = m ( q + ) c and ( q − ) (cid:24) km ( q + ) (cid:25) + q ( q − ) (cid:24) km (cid:25) − q km ( q + )= ( q − ) c + q ( q − )( q + ) c − q c = − c , which contradicts to the fact k ∈ G ( P ) . So a + mb =
0. There are two possibilities. ( i ) If a =
0, then 0 < b q . In this case, k = mb + m ( q + ) c is a gap at P if and only if ( q − ) (cid:24) km ( q + ) (cid:25) + q ( q − ) (cid:24) km (cid:25) > q km ( q + ) , ULTI-POINT CODES FROM THE GGS CURVES 17 or equivalently, ( q − )( c + ) + q ( q − )( b + ( q + ) c ) > q c + q bq + , leading to the first condition 0 c q − − b and 0 < b q − ( ii ) If 0 < a < m , then 0 b q . In this case, we have similarly that k = a + mb + m ( q + ) c is a gap at P if and only if ( q − )( c + ) + q ( q − )( + b + ( q + ) c ) > q c + q am ( q + ) + q bq + , which gives the second condition 0 c q − − b + (cid:22) bq + − q am ( q + ) (cid:23) . Note that q − − b + (cid:22) bq + − q am ( q + ) (cid:23) > q − − q − > . The proof is finished.5.
The floor of divisors.
In this section, we investigate the floor of divisors from GGScurves. The significance of this concept is that it provides a useful tool for evaluatingparameters of AG codes. We begin with general function fields.
Definition 5.1 ([25], Definition 2.2) . Given a divisor G of a function field F / F q with ℓ ( G ) >
0, the floor of G is the unique divisor G ′ of F of minimum degree such that L ( G ) = L ( G ′ ) . The floor of G will be denoted by ⌊ G ⌋ .The floor of a divisor can be used to characterize Weierstrass semigroups and pure gapsets. Let G = s Q + · · · + s l Q l . It is not hard to see that ( s , · · · , s l ) ∈ H ( Q , · · · , Q l ) if andonly if ⌊ G ⌋ = G . Moreover, ( s , · · · , s l ) is a pure gap at ( Q , · · · , Q l ) if and only if ⌊ G ⌋ = ⌊ ( s − ) Q + · · · + ( s l − ) Q l ⌋ . Maharaj, Matthews and Pirsic in [25] defined the floor of a divisor and characterized itby the basis of the Riemann-Roch space.
Theorem 5.2 ([25], Theorem 2.6) . Let G be a divisor of a function field F / F q and letb , · · · , b t ∈ L ( G ) be a spanning set for L ( G ) . Then ⌊ G ⌋ = − gcd n div ( b i ) (cid:12)(cid:12)(cid:12) i = , · · · , t o . The next theorem extends Theorem 3.4 of [7] by determining the lower bound of mini-mum distance in a more general situation.
Theorem 5.3 ([25], Theorem 2.10) . Assume that F / F q is a function field with genus g.Let D : = Q + · · · + Q N where Q , · · · , Q N are distinct rational places of F, and let G : = H + ⌊ H ⌋ be a divisor of F such that H is an effective divisor whose support does not containany of the places Q , · · · , Q N . Then the minimum distance of C Ω ( D , G ) satisfiesd Ω > ( H ) − ( g − ) . The following theorem provides a characterization of the floor over GGS curves, whichcan be viewed as a generalization of Theorem 3.9 in [25] related to Hermitian functionfields.
Theorem 5.4.
Let H : = ∑ q − µ = r µ P µ + ∑ q − ν = s ν Q ν + tP ∞ be a divisor of the GGS curve givenby (1) . Then the floor of H is given by ⌊ H ⌋ = q − ∑ µ = r ′ µ P µ + q − ∑ ν = s ′ ν Q ν + t ′ P ∞ , where r ′ = max n − i (cid:12)(cid:12)(cid:12) ( i , j , k ) ∈ Ω r , s , t o , r ′ µ = max n − i − m ( q + ) j µ (cid:12)(cid:12)(cid:12) ( i , j , k ) ∈ Ω r , s , t o for µ = , · · · , q − , s ′ ν = max n − i − mk ν (cid:12)(cid:12)(cid:12) ( i , j , k ) ∈ Ω r , s , t o for ν = , · · · , q − , t ′ = max n q i + m ( q + ) | j | + mq | k | (cid:12)(cid:12)(cid:12) ( i , j , k ) ∈ Ω r , s , t o . Proof.
Let H = ∑ q − µ = r µ P µ + ∑ q − ν = s ν Q ν + tP ∞ . It follows from Theorem 2.2 that the el-ements E i , j , k of Equation (2) with ( i , j , k ) ∈ Ω r , s , t form a basis for the Riemann-Rochspace L ( H ) . Note that the divisor of E i , j , k is iP + q − ∑ µ = ( i + m ( q + ) j µ ) P µ + q − ∑ ν = ( i + mk ν ) Q ν − (cid:0) q i + m ( q + ) | j | + mq | k | (cid:1) P ∞ . By Theorem 5.2, we get that ⌊ H ⌋ = − gcd n div ( E i , j , k ) (cid:12)(cid:12)(cid:12) ( i , j , k ) ∈ Ω r , s , t o . The desired conclusion then follows.6.
Examples of codes on GGS curves.
In this section we treat several examples of codesto illustrate our results. The codes in the next example will give new records of betterparameters than the corresponding ones in MinT’s tables [29].
Example 1.
Now we study codes arising from GK curves, that is, we let q = n = F -rational points and itsgenus is g =
10. Here we will study multi-point AG codes from this curve by employingour previous results.Let us take H = P + P + P ∞ for example. Then it can be computed from Equation(4) that the elements (cid:0) − i , − i − m ( q + ) j , − i − mk , − i − mk , − i − mk , q i + m ( q + ) j + mq ( k + k + k ) (cid:1) , with ( i , j , k , k , k ) ∈ Ω , , , , , , are as follows: ( , , , , , − ) , ( , , − , − , − , ) , ( , , − , − , − , ) , ( , , , , , ) , ( − , − , − , − , − , ) , ( − , − , , , , ) , ( − , , , , , ) , ( − , , − , − , − , ) , ( − , , , , , ) . ULTI-POINT CODES FROM THE GGS CURVES 19
So we obtain from Theorem 5.4 that ⌊ H ⌋ = P + P + P ∞ . Let D be a divisor consist-ing of N =
216 rational places away from the places P , P , Q , Q , Q and P ∞ . Accord-ing to Theorem 5.3, if we let G = H + ⌊ H ⌋ = P + P + P ∞ , then the code C Ω ( D , G ) has minimum distance at least 2 deg ( H ) − ( g − ) =
18. Since 2 g − < deg ( G ) < N ,the dimension of C Ω ( D , G ) is k Ω = N + g − − deg ( G ) = C Ω ( D , G ) has parameters [ , , > ] . One can verify that our resulting code improvethe minimum distance with respect to MinT’s Tables. Moreover C Ω ( D , G ) is equivalent to C L ( D , G ′ ) , where G ′ = P + P + Q + Q + Q + P ∞ , and its generating matrix canbe determined by Theorem 3.3.Additionally, we remark that more AG codes with excellent parameters can be found bytaking H = aP + bP + P ∞ , where a , b ∈ { , , } and 9 a + b
12. The floor of such H is computed to be ⌊ H ⌋ = aP + bP + P ∞ . Let D be as before. If we take G = H + ⌊ H ⌋ = aP + bP + P ∞ , then we can produce AG codes C Ω ( D , G ) with parameters [ , − a − b , > a + b − ] . All of these codes improve the records of the corresponding onesfound in MinT’s Tables. Example 2.
Consider the curve GGS ( q , n ) of (1) with q = n =
5. This curve has3969 F -rational points and its genus is g =
46. It follows from Theorem 4.3 that n ( , j , ) (cid:12)(cid:12)(cid:12) j o ⊆ G ( P , P , P ∞ ) . Let D be a divisor consisting of N = P , P , Q , Q , Q and P ∞ .Applying Theorem 3.4 of [7] (see also Theorem 1, [19]), if we take G = P + P + P ∞ ,then the three-point code C Ω ( D , G ) has length N = N + g − − deg ( G ) = [ , , > ] . Unfortunately, this F -code cannot be compared with the one inMinT’s Tables because the alphabet size given is at most 256. Acknowledgements.
The authors are very grateful to the editor and the anonymous re-viewers for their valuable comments and suggestions that improved the quality of this pa-per.
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