On Gröbner Basis for certain one-point AG codes
aa r X i v : . [ m a t h . AG ] A p r ON GR ¨OBNER BASIS FOR CERTAIN ONE-POINT AG CODES
F. FORNASIERO AND G. TIZZIOTTI
Abstract.
Heegard, Little and Saints worked out a Gr¨obner basis algorithm forHermitian codes and Farr´an, Munuera, Tizziotti and Torres extended such a resultfor codes on norm-trace curves. In this work we generalize such a result for codesarising from certain types of curves X over F q with plane model f ( y ) = g ( x ). Keywords: AG codes; Gr¨obner basisMSC codes: 11T71; 13P10 1.
Introduction
In the early 1980s, V.D. Goppa constructed error-correcting codes using algebraiccurves, the called algebraic geometric codes (AG codes), see [6] and [7]. The intro-duction of methods from algebraic geometry to construct good linear codes was oneof the major developments in the theory of error-correcting codes. From that momentmany studies and applications on this theory has emerged. In [10], Little, Saints andHeegard introduced an encoding algorithm for a class of AG codes via Gr¨obner basis,similar to the usual one for cyclic codes. This encoding method is efficient and alsointeresting from a theoretical point of view. It is known that the main drawback ofGr¨obner basis is the high computational cost required for its calculation. Indeed, it iswell known that the complexity of computing a Gr¨obner basis is doubly exponential ingeneral. But, in [11], using an appropriate automorphism of the Hermitian curve, Littleet al. introduced the concept of root diagram that allows to construct an algorithm forcomputing a Gr¨obner basis with a lower complexity for one-point Hermitian codes. In[4], the results of [11] were extended to codes arising from the norm-trace curve, whichis a generalization of the Hermitian curve. In both works, the one-point AG codes aris-ing from curves over finite fields F q with q elements and the construction of the rootdiagram is made by using automorphisms whose order is equal to q −
1. In this work,we will construct the root diagram, and consequently an algorithm for computing aGr¨obner basis, for codes arising from certain curves over F q with automorphisms whoseorder divides q −
1, thus we get results more general than those achieved previously.As examples, we have codes over the curves y q + y = x q r +1 and y q + y = x m , and codesover Kummer extensions, which have been applied in coding theory, see [9] and [12],and [2], respectively.This paper is organized as follows. In Section 2 we recall some background onGr¨obner basis for modules, AG codes and root diagram. In Section 3 we present a wayto construct the root diagram for one-point AG codes C arising from certain types of curves X over F q with plane model f ( y ) = g ( x ). In addition, we present the way toobtain the Gr¨obner basis for such C . Finally, in Section 4 we present examples of thosecurves and the necessary informations to construct the root diagram and the Gr¨obnerbasis studied in the previous section.2. Background
Gr¨obner basis for F q [ t ] -modules. We introduce some notations about Gr¨obner ba-sis for F q [ t ]-modules that we shall needed later. For a complete treatment see [1] and[3]. A monomial m in the free F q [ t ]-module F q [ t ] r is an element of the form m = t i e j ,where i ≥ e , . . . , e n is the standard basis of F q [ t ] r . Fixed a monomial ordering,for all element f ∈ F q [ t ] r , with f = 0, we may write f = a m + · · · + a ℓ m ℓ , where, for1 ≤ i ≤ ℓ , 0 = a i ∈ F q and m i is a monomial in F q [ t ] r satisfying m > m > . . . > m ℓ .The term a m is called leading term of f and denoted by LT ( f ), the coefficient a andthe monomial m are called leading coefficient , LC ( f ), and leading monomial , LM ( f ),respectively. A Gr¨obner basis for a submodule M ⊆ F q [ t ] r is a set G = { g , . . . , g s } such that { LT ( g ) , . . . , LT ( g s ) } generates the submodule LT ( M ) formed by the lead-ing terms of all elements in M . The monomials in LT ( M ) are called nonstandard while those in the complement of LT ( M ) are the standard monomials for M . Werecall that every submodule M ⊆ F q [ t ] n has a Gr¨obner basis G , which induces aa division algorithm: given f ∈ F q [ t ] r there exist a , . . . , a s , R G ∈ F q [ t ] r such that f = a g + . . . + a s g s + R G ([1] Algorithm 1.5.1, or [3] Theorem 3).In this work we will use the POT (position over term) ordering over F q [ t ] r whichis defined as follows.Let { e , . . . , e r } be the standard basis in F q [ t ] r , with e > . . . > e r . The POTordering on F q [ t ] r is defined by t i e j > t k e ℓ if j < ℓ , or j = ℓ and i > k .We say that f ∈ F q [ t ] r is reduced with respect to a set P = { p , . . . , p l } of non-zeroelements in F q [ t ] r if f = or no monomial in f is divisible by a LM ( p i ), i = 1 , . . . , l .A Gr¨obner basis G = { g , . . . , g s } is reduced if g i is reduced with respect to G − { g i } and LC ( g i ) = 1 for all i , and non-reduced otherwise. Every submodule of F q [ t ] r has aunique reduced Gr¨obner basis (see [1], Theorem 3.5.22).2.2. Linking AG codes and F q [ t ] -modules. Let X be a projective, non-singular, geomet-rically irreducible, algebraic curve of genus g > F q . Let P , . . . , P n , Q , . . . , Q ℓ be n + ℓ distinct rational points on X and m , . . . , m ℓ be integers. Consider the divisors D = P + · · · + P n , G = m Q + · · · + m ℓ Q ℓ . The algebraic geometry code (AG code) C X ( D, G ) associated to the curve X , is defined as(2.1) C X ( D, G ) := { ( f ( P ) , . . . , f ( P n )) ∈ F nq : f ∈ L ( G ) } , N GR ¨OBNER BASIS FOR CERTAIN ONE-POINT AG CODES 3 where L ( G ) is the space of rational functions f on X such that f = 0 or div( f )+ G ≥ f ) denote the (principal) divisor of the function f ∈ L ( G ). The number n = | Supp ( D ) | is the length of C X ( D, G ), where
Supp ( D ) denotes the support of thedivisor D , and the dimension of C X ( D, G ) is its dimension as an F q -vector space, whichis generally denoted by k . The elements in C X ( D, G ) are called codewords . If G = aP ,for some rational point P on X , and D is the sum of the all others rational points on X the AG code C X ( D, λP ) is called one-point AG code . For more details about AGcodes, see e.g. [8].Let S n be the symmetric group. S n acts on F nq via τ ( a , . . . , a n ) = ( a τ (1) , . . . , a τ ( n ) ),where τ ∈ S n . The automorphism group of a code C is defined asAut( C ) := { σ ∈ S n : σ ( C ) = C } . In [7], Goppa already observed that the underlying algebraic curve induces auto-morphism of the associated AG codes as follows.
Proposition 2.1.
Let Aut ( X ) be the automorphism group of X over F q and considerthe subgroup Aut
D,G ( X ) = { σ ∈ Aut ( X ) : σ ( D ) = D and σ ( G ) = G } . Each σ ∈ Aut
D,G ( X ) induces an automorphism of C X ( D, G ) by b σ ( f ( P ) , . . . , f ( P n )) = ( f ( σ ( P )) , . . . , f ( σ ( P n ))) . Assume that X has a nontrivial automorphism σ ∈ Aut
D,G ( X ) and let H be thecyclic subgroup of Aut ( X ) generated by σ . Let Supp ( D ) = O ∪ . . . ∪ O r be thedecomposition of the support of D into disjoint orbits under the action of σ . Then, byProposition 2.1, the entries of codewords in C X ( D, G ) corresponding to the points ineach O i are permuted cyclically by σ . We will denote σ = Id , where Id is the identityautomorphism, and, for a positive integer j , σ j = σ ◦ σ ◦ . . . ◦ σ | {z } j . In this way, for each i = 1 , . . . , r , by choosing any one point P i, ∈ O i , we can enumerate the other pointson O i as P i,j = σ j ( P i, ), where j runs from 0 to | O i | −
1. Using this fact, we get thefollowing result.
Lemma 2.2.
Let C X ( D, G ) be an AG code arising from X over F q . Suppose that X has a nontrivial automorphism σ ∈ Aut
D,G ( X ) . If Supp ( D ) = O ∪ . . . ∪ O r isthe decomposition of the support of D into disjoint orbits under the action of σ , thenthere is an one-to-one correspondence between C X ( D, G ) and a submodule C of the freemodule F q [ t ] r .Proof. Suppose that
Supp ( D ) = O ∪ . . . ∪ O r is the decomposition of the supportof D into disjoint orbits under the action of σ . For each i = 1 , . . . , r , let O i = { P i, , . . . , P i, | O i |− } , where for each P i,j ∈ O i we have that P i,j = σ j ( P i, ) be as above,and let h i ( t ) = P | O i |− j =0 f ( P i,j ) t j . F. FORNASIERO AND G. TIZZIOTTI
The r -tuples ( h ( t ) , . . . , h r ( t )) can be seen also as an element of the F q [ t ]-module A = L ri =1 F q [ t ] / h t | O i | − i . So, the collection ˜ C of r -tuples obtained from all f ∈ L ( G )is closed under sum and multiplication by t . Define C := π − ( ˜ C ), where π is thenatural projection from F q [ t ] r onto L ri =1 F q [ t ] / h t | O i | − i . Thus, we get an one-to-onecorrespondence between C X ( D, G ) and C ≤ F q [ t ] r . (cid:3) (cid:3) By the previous lemma, an AG code C X ( D, G ) can be identified to a submodule C ≤ F q [ t ] r and the standard theory of Gr¨obner basis for modules may be applied.Suppose that C X ( D, G ) has length n and dimension k . A Gr¨obner basis G = { g (1) , . . . , g ( r ) } for C ≤ F q [ t ] r with exactly r elements allows us to obtain a systematicencoding of C . Since { LT ( g (1) ) , . . . , LT ( g ( r ) ) } generates LT ( C ), then the nonstandardmonomials appearing in the r -uples ( h ( t ) , . . . , h r ( t )) can be obtained from the g ( i ) ’s.By ordering these monomials in decreasing order we obtain the so-called informationpositions of ( h ( t ) , . . . , h r ( t )), which are the first k monomials m l = t i l e j l , l = 1 , . . . , k .Let V C ( h ( t ) , . . . , h r ( t )) be the vector of coefficients of the terms of ( h ( t ) , . . . , h r ( t ))listed in the POT order. We have the following systematic encoding algorithm: Algorithm 2.3.Input:
A Gr¨obner basis G , monomials { m , . . . , m k } and w = ( w , . . . , w k ) ∈ F kq . Output: c ( w ) ∈ C = C ( X , D, G ) .1. Set f := w m + · · · + w k m k .2. Compute f = a g (1) + . . . + a r g ( r ) + R G .3. Return c ( w ) := V C ( f − R G ) . This method is more compact compared with the usual encoding via generatormatrix. The total amount of computation is roughly the same and the amount ofnecessary stored data is lower in this method, of order r ( n − k ) against k ( n − k ) whenencoding via generator matrix. More details about this encoding algorithm can befound in [10].2.3. The root diagram.
Let X be as in the previous subsection. Suppose that theone-point AG code C = C X ( D, λP ) has an automorphism σ that fixing the divisors D and G = λP . Suppose also that the order of σ is equal to s , with s = d ( q −
1) forsome d ∈ N . Let C be the submodule of F q [ t ] r associated to C by the automorphism σ . Using the POT ordering we can get that a Gr¨obner basis G = { g , . . . , g r } for C such that g i = (0 , . . . , , g ( i ) i ( t ) , g ( i +1) i ( t ) , . . . , g ( r ) i ( t )), for all i = 1 , . . . , r , see [[10],Proposition II.B.4].Note that, if deg ( g ( i ) i ( t )) = d i , then g ( i ) i ( t ) has d i distinct roots in F ∗ q = F q \ { } . Infact, let q i = ( t | O i | − e i . Note that q i ∈ π − (0 , . . . ,
0) and we have that q i ∈ C . Since | O i | divides s and s divides q −
1, follows that t | O i | − t q − − Q a ∈ F ∗ q ( t − a ).Now, LT ( g ( i ) ) = g ( i ) i ( t ) divides LT ( q i ) = t | O i | −
1, and the claim follows from the fact t q − − q − F q . N GR ¨OBNER BASIS FOR CERTAIN ONE-POINT AG CODES 5
For i = 1 , . . . , r , let R i ⊆ F ∗ q be the set of roots of t | O i | −
1. By a root diagram D C for the code C , we mean a table with r rows. For each i , the boxes on the i -th rowcorrespond to the elements of R i . We mark the roots of g ( i ) i ( t ) on the i -th row with a X in the corresponding box.By Proposition II.C.1 in [10], there is a F q -basis for C in one-to-one correspondencewith the nonstandard monomials in C . That is, terms of the form t ℓ e j appearing asleading terms of some element of C , with ℓ ≤ | O j | −
1. Now, if there are m j emptyboxes on row j of the root diagram, then g ( i ) j ( t ) has | O j | − m j roots and LT ( g ( j ) ) = t | O j |− m j . So, we obtain m j nonstandard monomials t ℓ e j . This fact gives us the followingimportant result. Proposition 2.4. ([11], Proposition 2.3)
The dimension of the code C is equal to thenumber of empty boxes in the root diagram D C . Gr¨obner basis for certain AG codes
Finding a Gr¨obner basis is hard in general. Next, we will see that for certain codesAG this task is simplified by using the concept of root diagram.Let X be as in the previous section and let F q ( X ) be the field of rational functionson X . For a rational point P on X let H ( P ) := { n ∈ N ; ∃ f ∈ F q ( X ) with div ∞ ( f ) = nP } , where N is the set of nonnegative integers and div ∞ ( f ) denotes the divisor of polesof f . The set H ( P ) is a numerical semigroup, called Weierstrass semigroup of X at P and its complement G ( P ) = N \ H ( P ) is called Weierstrass gap set of P . As animportant result, the cardinality of G ( P ) is equal to genus g of X , see Theorem 1.6.8in [14].Let X a,b be the curve defined over F q by affine equation f ( y ) = g ( x ), where f ( T ) , g ( T ) ∈ F q [ T ], deg ( f ) = a and deg ( g ) = b , with a < b and gcd ( a, b ) = 1.Furthermore, suppose that div ∞ ( x ) = aP and div ∞ ( y ) = bP , for some point on X a,b ,and that there exists σ ∈ Aut
D,G ( X a,b ), where G = λP for some positive integer λ ,given by σ ( x ) = αx and σ ( y ) = α t y , for some positive integer t and some α ∈ F ∗ q withorder equal to ord ( α ) := ν . Finally, assume that H ( P ) = h a, b i .Consider the one-point AG code C X a,b ( D, λP ). Let D = P + . . . + P n and Supp ( D ) = O ∪ . . . ∪ O r ∪ O r +1 ∪ O r + s be the decomposition of the support of D intodisjoint orbits under the action of σ . Note that, by definition of σ , if Q = (0 , η ) ∈ O i ,for some η ∈ F q , then O i = { (0 , η ) , (0 , α t η ) , . . . , (0 , α t.t i ) } , where t i is the smallestnonnegative integer such that α t. ( t i +1) = 1. Analogously, if Q = ( ω, ∈ O i , forsome ω ∈ F q , then O i = { ( ω, , ( αω, , . . . , ( α ν − ω, } . Let O r +1 , . . . , O r + s be theorbits that contains rational points of the form (0 , η ) or ( ω, r rows of the root diagram D C for the code C X a,b ( D, λP ), the results for thelast s rows are similar can be obtained in particular cases. For each i = 1 , . . . , r ,suppose that O i = { P i, , P i, , . . . , P i, | O i |− } , where P i, = ( x i , y i ), with x i = 0 and F. FORNASIERO AND G. TIZZIOTTI y i = 0, and P i,j = σ j ( P i, ) = ( α j x i , α jt y i ). So, by the definition of σ follows that | O | = . . . = | O r | = ord ( α ) = ν . Assume that, for each i = 1 , . . . , r , there exists poly-nomials M i ( y ) such that the orbit O i is the intersection of X with the curve M i ( y ) = 0and, for all i , M i ( y ) is a non-zero constant when restricted to each of the orbits O k , k = i . For 1 ≤ i ≤ r and 0 ≤ j ≤ | O i | − ν −
1, assume also that there arepolynomials B i,j ( x, y ) such that B i,j ( x, y ) vanishes at each point of O i except P i,j . Lemma 3.1.
For i = 1 , . . . , r and j = 0 , . . . , | O i | − , let M i ( y ) and B i,j ( x, y ) be asabove. Then, div ∞ ( M i ) = ( ρ b ) P and div ∞ ( B i,j ) = ( ρ a + ρ b ) P , where ρ , ρ and ρ are non-negative integers.Proof. We have that div ∞ ( x ) = aP and div ∞ ( y ) = bP . Then, the result follows fromthe fact that M i ( y ) and B i,j ( x, y ) are polynomials. (cid:3) (cid:3) Let ρ , ρ and ρ be as the previous lemma. So, for λ ≤ ( ρ a + ρ b ) + r ( ρ b ), wecan get the following information about the root diagram D C . Proposition 3.2.
Let C X a,b ( D, λP ) and σ be as above. Let D C be the root diagram for C X a,b ( D, λP ) . Fix i , ≤ i ≤ r , and let ρ , ρ and ρ be as above. Therefore,1) if λ ≥ ( i − ρ b ) , then the i -th row of D C is not full, in the sense that notevery boxes composing the i -th row are marked with X ;2) if λ ≥ ( ρ a + ρ b ) + ( i − ρ b ) , then the row is empty, in the sense that noneof the boxes composing the i -th row is marked with X .Proof. Let C ≤ F q [ t ] r be the submodule associated to C X a,b ( D, λP ).1) Suppose that λ ≥ ( i − ρ b ). By Lemma 3.1, the function F i ( x, y ) = M ( x, y ) · · · M i − ( x, y )belongs to L ( λP ) and hence ev ( F i ) ∈ C X a,b ( D, λP ). By computing ev ( F i ), we observethat C contains an element of the form (0 , . . . , , h i ( t ) , . . . , h r ( t )) with i − h i ( t ) = P | O i |− j =0 F i ( P i,j ) t j . Since F i ( P i,j ) = M ( P i,j ) · · · M i − ( P i,j ) = constant c = 0 , we have h i ( t ) = c. P | O i |− j =0 t j and thus h (1) = 0 as | O i | divides q −
1. Therefore the i -throw of D C is not full, since g ( i ) i ( t ) divides h i ( t ).2) Now, suppose λ ≥ ( ρ + ρ b ) + ( i − ρ b ). So, by Lemma 3.1, G i ( x, y ) = B i, ( x, y ) F i ( x, y ) ∈ L ( λP ) and G i ( Q ) = 0 for Q ∈ O ∪ O ∪ . . . ∪ O i − . Moreover, G i ( Q ) = 0 for all Q ∈ O i \ { P i, } . Then the element of C corresponding to ev ( F ′ i )verifies h ( t ) = h ( t ) = . . . = h i − ( t ) = 0 and h i ( t ) = G i ( P i, ) = c = 0. Thus, C contains the element (0 , . . . , , c, h i +1 ( t ) , . . . , h r ( t )) and follows that the i -th row of D C is empty. (cid:3) (cid:3) N GR ¨OBNER BASIS FOR CERTAIN ONE-POINT AG CODES 7
Let N be the number of rational points on X a,b , by Riemman-Roch Theorem,follows that if λ < N , then the dimension of the one-point AG code C X a,b ( D, λP ) isequal to the dimension of the Riemann-Roch space L ( λP ). In this case, we completethe informations about the root diagram D C . Theorem 3.3.
Let D C be the root diagram for C X a,b ( D, λP ) . If there is i ∈ { , . . . , r } such that ( i − ρ b ) ≤ λ < ( ρ a + ρ b ) + ( i − ρ b ) , then the i -th row of D C is neither full, nor empty, and the complement of the setof roots marked on row i of the diagram is the set E i = { α − ( β + γb ) ∈ F ∗ q | ≤ β ≤ b − , ≤ γ ≤ ρ − , ( i − ρ b ) + βa + γb ≤ λ } .Proof. Let C ≤ F q [ t ] r be the submodule associated to C X a,b ( D, λP ). Let D i ⊂ F ∗ q bethe set of non marked boxes in row i , where 1 ≤ i ≤ r . We will show that D i = E i . Let F i ( y ) be as in the previous proposition and consider f i ( x, y ) = F i ( y ) x β y γ . By Lemma3.1 and the conditions over β and γ given in the definition of E i , we have that f i ( x, y ) ∈ L ( λP ). So, associated to f i ( x, y ) we get an element h = ( h ( t ) , . . . , h r ( t )) ∈ C . Since F i ( Q ) = 0 for all Q ∈ O ∪ . . . ∪ O i − , follows that h k ( t ) = 0, for k = 1 , . . . , i − P i,j = σ ( P i, ) = ( α j x i , α tj y i ) ∈ O i . Thus, f i ( P i,j ) = F i ( P i,j ) α jβ x βi α tjγ y γi = F i ( P i,j ) x βi y γi α j ( β + tγ ) . Now, F i ( P i,j ), x βi and y γi are all non-zero constants and inde-pendents of j . Taking b i = F i ( P i,j ) x βi y γi = 0, we have h i ( t ) = | O i |− X j =0 f i ( P i,j ) t j = [ | O i | − b i | O i |− X j =0 ( α β + tγ t ) j whose roots are all distinct from α − ( β + tγ ) . Consequently, α − ( β + tγ ) is not a root of g ( i ) i ( t ) and hence E i ⊆ D i .By Proposition 2.4, dim ( C X a,b ( D, λP )) = P ♯D i . Since H ( P ) = h a, b i and λ < N ,we have that dim ( C X a,b ( D, λP )) = ♯ { ( β, γ ) ∈ N ; 0 ≤ β ≤ b − βa + γb ≤ λ } .Let b E i = { ( β, γ ) ∈ N | ≤ β ≤ b − , ≤ γ ≤ ρ − , ( i − ρ b ) + βa + γb ≤ λ } .Thus, ♯ { ( β, γ ) ∈ N ; 0 ≤ β ≤ b − βa + γb ≤ λ } = P ♯ b E i and, since P ♯ b E i = ♯ P E i , follows that P ♯D i = P ♯E i . Therefore, E i = D i . (cid:3) (cid:3) Let F i ( y ) be as above. Then, we have that F i ( Q ) = c i ∈ F ∗ q , for all Q ∈ O i . Withthe conditions of the above theorem, fix an index i , 1 ≤ i ≤ r , where the row i of D C is neither full, nor empty. Let α k , α k , . . . , α k ℓ be the roots marked on the row i andlet p ( t ) = Q ℓj =1 ( t − α k j ) be the unique monic polynomial of degree ℓ with these roots.Note that, including zeroes for powers of t higher than the number of roots, we canwrite p ( t ) = P | O i |− j =0 a j t j , where a j = 0 for j > ℓ . Consider the function f i ( x, y ) = F i ( y ) c i | O i |− X j =0 a j B i,j ( x, y ) B i,j ( P i,j ) F. FORNASIERO AND G. TIZZIOTTI
Then, by definition of F i ( y ) and B i,j ( x, y ), it is clear that f i ( x, y ) ∈ L ( λP ) and itsassociated module element h ∈ C has i − i -th component h i ( t ) equal to p ( t ).So, using the same procedures used in [11] and [4]: ◦ RootDiagram[ i ] : returns a list of the roots corresponding to the marked boxesin line i of D C ; ◦ Boxes[ i ] : the number of boxes in row i of D C , that is T Boxes[ i ] = | O i | ; ◦ Evaluate[ i, point ] : a procedure which takes as input the coefficients { a k } ofthe unique monic polynomial over F q having the marked elements on a row number i as roots and a point P i,j on O i , and evaluates the function f i ( x, y ) as above at a pointP i,j ; we get an analogous algorithm that computes a non-reduced POT Gr¨obner basisfor the submodule C associated to C X a,b ( D, λP ) and thus to apply the systematicencoding given in Subsection 2.2 to the AG codes C = C X a,b ( D, λP ). Algorithm 3.4.Input: the root diagram D C , the N rational points P i,j of Supp ( D ) = O ∪ . . . ∪ O r ∪ O r +1 ∪ O r + s . Output: a non-reduced Gr¨obner basis G = { g (1) , g (2) , . . . , g ( r + s ) } of C .1. G := {} for i from 1 to r + s do if T —RootDiagram[ i ]— < T Boxes[ i ] then for k from 1 to r + s do g ( i ) k := 06. if k ≥ i then for j from 0 to T Boxes[ k ] − do g ( i ) k := g ( i ) k + T Evaluate[ i , P k,j ] t j e k end for end if end for else g ( i ) := ( t T Boxes[ i ] − e i end if G := G ∪ { g ( i ) } end for return G We note that this algorithm has the same computational complexity as the orig-inal one developed by Little, Saints and Heegard in [11]. It is much lower than thecomplexity of general Gr¨obner basis algorithms, since we only make use of interpola-tion problems and evaluation of functions. In particular we do not use divisions norreductions that would increase the complexity, as in the case of Buchberger’s algorithm.
N GR ¨OBNER BASIS FOR CERTAIN ONE-POINT AG CODES 9 Examples
The curve X q r . Let X q r be the curve defined over F q r by the affine equation y q + y = x q r +1 , where q is a prime power and r an odd integer. Note that when r = 1 the curve is justthe Hermitian curve. The curve X q r has genus g = q r ( q − /
2, one single singularpoint P ∞ = (0 : 1 : 0) at infinity and others q r +1 rational points. Thus, this curveis a maximal curve over F q r because its number of rational points equals the upperHasse-Weil bound, namely equals q r + 1 + 2 gq r . Furthermore, H ( P ∞ ) = h q, q r + 1 i ,see [13], and(4.1) σ : x αxy α q r +1 y with α ∈ F ∗ q r such that α ( q r +1)( q − = 1, is an automorphism of X q r , see [9]. Notethat σ has order ( q r + 1)( q − σ divides q r − σ above the q r +1 rational pointson X q r are disposed in q ( q r − + · · · + q ) + 2 orbits, where q ( q r − + · · · + q ) of them haslength ( q r + 1)( q −
1) and the remaining two orbits, one has length q − σ , it is clear that: · σ (0 ,
0) = (0 , · all the q − , b ), with b = 0, form an orbit with length q − σ (0 , b ) = (0 , α q r +1 b ) and α ∈ F ∗ q r is such that α ( q r +1)( q − = 1; · the others q r +1 − q = q ( q r + 1)( q r −
1) rational points ( x, y ) ∈ X q r , with x = 0and y = 0, are arranged in q ( q r − + · · · + q ) orbits of length ( q r + 1)( q − r = q ( q r − + · · · + q ) and α be as in (4.1). Let F ∗ q r = h a i and t ∈ { , , . . . , q r − } be such that α = a t . So, given P i, = ( a t i , a l i ) ∈ O i , the others points P i,j on O i are P i,j = σ j ( P i, ) = ( a t i + jk , a l i + jk ( q r +1) ), with j ∈ { , . . . , ( q r + 1)( q − − } . Then, for i = 1 , . . . , r and j = 0 , . . . , ( q r + 1)( q − −
1, we get(4.2) M i ( y ) := q − Y j =0 ( y − a l i + jk ( q r +1) ) = y q − − a l i ( q − , and(4.3) B i,j ( x, y ); = q − Y s =1 ( y − a l i + k ( q r +1)( j + s ) ) ( q r +1) − Y s =1 ( x − a t i + k ( j + s ) ) . Since div ∞ ( x ) = qP ∞ and div ∞ ( y ) = ( q r + 1) P ∞ , we have that • div ∞ ( M i ( y )) = ( q − q r + 1) P ∞ , that is, M i ( y ) ∈ L (( q − q r + 1) P ∞ ), for all i = 1 , . . . , r ; • div ∞ ( B i,j ( x, y )) = (( q − q r + 1) + q (( q r + 1) − P ∞ , that is, B i,j ∈ L ( q.q r +( q − q r + 1)) P ∞ ), for all 1 ≤ i ≤ r ) e 0 ≤ j ≤ ( q r + 1)( q − − • a = q and b = q r + 1; • P = P ∞ ; • div ∞ ( x ) = qP ∞ and div ∞ ( y ) = ( q r + 1) P ∞ ; • H ( P ∞ ) = h q, q r + 1 i ; • ρ = q − ρ = q r and ρ = q − C X q r ( D, λP ∞ ) and then the Gr¨obner basis for the module C associated to C X q r ( D, λP ∞ ) by Algorithm 3.4.4.2. A Quotient of the Hermitian curve.
Let X m de the curve defined over F q by theaffine equation y q + y = x m , where q is a prime power and m > q + 1. This curve has genus g = ( q − m − /
2, a single point at infinity, denoted by P ∞ , and others q (1+ m ( q − X m ia a maximal curve and in [12], G. Matthewsstudied Weierstrass semigroup and algebraic codes over this codes. As a result presentby Matthews we have that H ( P ∞ ) = h m, q i .Let F ∗ q = h α i and k such that mk = q + 1. Then,(4.4) τ : x → α k xy → α q +1 y is an automorphism of X m of order m ( q − q − τ above the q (1 + m ( q − X m are disposed in q + 2 orbits, where q of them haslength m ( q −
1) and the remaining two orbits, one has length q − r = q and the first r orbits given by points on X m of the form P = ( a, b )with a, b = 0. So, for each i = 1 , . . . , r , given P i, = ( α ℓ i , α t i ) ∈ O i , the others points P i,j on O i are P i,j = σ j ( P i, ) = ( α ℓ i + jk , α t i + j ( q +1) ), with j ∈ { , . . . , m ( q − − } . Thatis, O i = { P i,j = ( α ℓ i + jk , α t i + j ( q +1) ) ; j = 0 , . . . , m ( q − − } . Then, for i = 1 , . . . , r and j = 0 , , . . . , m ( q − −
1, we get M i ( y ) = q − Y j =0 ( y − α t i + j ( q +1) ) N GR ¨OBNER BASIS FOR CERTAIN ONE-POINT AG CODES 11 and B i,j ( x, y ) = q − Y s =0 ,s = j ( y − α t i + s ( q +1) ) m ( q − − Y s =0 ,s = j ( x − α ℓ i + sk ) . So, since div ∞ ( x ) = qP ∞ and div ∞ ( y ) = mP ∞ , follows that • div ∞ ( M i ( y )) = ( q − mP ∞ , that is, M i ( y ) ∈ L (( q − mP ∞ ), for all i = 1 , . . . , r ; • div ∞ ( B i,j ( x, y )) = (( q − m + ( m − q ) P ∞ , that is, B i,j ∈ L (( m − q + ( q − m ) P ∞ ), for all 1 ≤ i ≤ r ) e 0 ≤ j ≤ m ( q − − • a = q and b = m ; • P = P ∞ ; • ( x ) ∞ = qP ∞ and ( y ) ∞ = mP ∞ ; • H ( P ∞ ) = h q, m i ; • ρ = q − ρ = q − ρ = m − C X m ( D, λP ∞ ) and thenthe Gr¨obner basis for the module C associated to C X m ( D, λP ∞ ). References [1] W. Adams and P. Loustaunau,
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