Mumford representation and Riemann Roch space of a divisor on a hyperelliptic curve

For an (imaginary) hyperelliptic curve $ \mathcal{H} $ of genus $g$, with a Weierstrass point $\Omega$, taken as the point at infinity, we determine a basis of the Riemann-Roch space $\mathcal{L}(\Delta + m \Omega)$, where $\Delta$ is of degree zero, directly from the Mumford representation of $\Delta$. This provides in turn a generating matrix of a Goppa code.

Both the Mumford representation of a divisor ∆ of degree zero on a hyperelliptic curve and the Riemann Roch space L(D), where D = ∆ + mΩ, are the subject of a large number of papers, also due to their applications in Coding theory.
But it has not been indicated in the literature that a basis of the latter can be directly found from the former, and it is the aim of the present note to give an explicit basis of L(D), stressing the meaning of the Mumford representation of ∆ in this context.Note that, for a nodal curve, a data structure inspired by the Mumford representation has been used for the same purpose in a recent excellent paper by Le Gluher and Spaenlehauer [4] (the same article also contains a state of the art that we share, wishing to let this paper as short as possible).
Using this basis, one constructs directly a generating matrix of a Goppa code over a hyperelliptic curve defined over a Galois field of characteristic p ≥ 2. We make this for a toy model of MDS codes in Section 3.Although the reduction of a divisor D to its reduced Mumford form might be an inconvenient task, involving the application of the Cantor algorithm (see Remark 1), this difficulty does not occur in the construction of Goppa codes, because in that case one can directly take D in the reduced form D = ∆ + mΩ.

Notations and reduction to the Mumford representation
Let K be the algebraic closure of the field k and let H be a hyperelliptic curve of genus g over k with a rational Weierstrass point Ω.The non-singular curve H is described by an affine equation of the form where f (x) is a polynomial of degree d = 2g + 1, h(x) is a polynomial of degree at most g, and Ω = [0 : 1 : 0] is the point at infinity of and f (x) into f (x) − h 2 (x)/4, transforms the above equation into whereas, if char k = 2, then it is not possible to reduce h(x) to zero.
Let D be a divisor of H. Since its Riemann-Roch space is null both in the cases where D has negative degree, and where D has degree zero and D ∈ Princ(H), whereas L(D) = F −1 0 in the case where D = div(F 0 ), from now on we will assume D has positive degree m, thus for t points P 1 , . . ., P t in H distinct from Ω, with t ≤ g, and a suitable ψ(x, y) ∈ K(H), that is, any divisor class D + Princ(H) ∈ Div(H)/Princ(H) can be reduced to the form In order to extend the use of Mumford representation to divisors of arbitrary degree, we will apply the following: on the curve H, of degree zero and such that l i > 0 for any index i, determines uniquely the polynomial a ls and the polynomial b(x) which is the interpolating polynomial such that b(x t ) = y t (hence b 2 (x) + h(x)b(x) − f (x) is a multiple of a(x) and the degree s − 1 of b(x) is smaller than the degree of a(x)).Conversely, any pair of polynomials a(x) and b(x) such that b 2 (x)+h(x)b(x)−f (x) is a multiple of a(x) and the degree of b(x) is smaller than the degree of a(x) defines such a divisor of degree zero, which is written as ∆ = div(a(x), b(x)).Note that an intersection point of the curve with the x-axis is contained in the support of ∆ if and only if GCD(a(x), a ′ (x), b(x)) = 1.If GCD(a(x), a ′ (x), b(x)) = 1 and the degree of a(x) is not greater than the genus g of the curve (or equivalently, if the support of ∆ contains at most g points which are mutually non-opposite), one says that div(a(x), b(x)) is in Mumford form (or reduced form).We remark here that any divisor D = D 1 − D 2 (with D i effective of degree m i ∈ Z) can be written as in Mumford form, and a suitable function ψ(x, y), obtained with the following argument.First, taking the vertical lines x − x i passing through the points in the support of D 2 we can write Secondly, applying the reduction step in Cantor's algorithm (cf.[2], and [3] in the case where char k = 2), we change D 3 with ), which belong to the same divisor class, where This way deg a ′ (x) < deg a(x), hence after finitely many iterations one gets deg a ′ (x) ≤ g, and one can write where ψ(x, y) is the resulting function of the above reduction.Finally, the function Φ : mapping F onto the product ψ(x, y)F , is an isomorphism.
Up to the latter isomorphism, we will directly assume that D = ∆ + mΩ, m > 0.

Main theorem
In the following theorem we determine a basis of L(D), with D = ∆ + mΩ., and ∆ = div(u(x), v(x)) is in Mumford representation, with t := deg u(x) ≤ g.Also, the kind of unexpected varying, according to m, of its dimension becomes manifest: in order to determine dim L(D), in , for p = 2.
Proof.In order to compute div(Ψ(x, y)), recall that deg v(x) < deg u(x) ≤ g and that, in the case where p = 2, deg h(x) ≤ g, as well.
2 is smaller than the degree of f (x), hence there are d = 2g + 1 intersection points of the curve y + v(x) + h(x) = 0 and H in the affine plane, the remaining d(l − 1) intersection points coinciding with Ω.More precisely, t intersection points in the affine plane belong to the support of the divisor ∆ = div u(x), w(x) in Mumford representation, where w where W is the effective divisor of degree d − t, whose support consists of the remaining intersection points in the affine plane.Note that, in the case t = 0, the divisor ∆ has the Mumford representation (1, 0) and the degree of W is d = 2g + 1 and the support of W coincides with the intersections of H with the curve y + h(x) = 0. On the other hand, the intersection of u(x) = 0 and H is simply and, if t = 0, then ∆ = (1, 0) and Ψ(x, y) = y + h(x), whence div(Ψ(x, y)) = div (y + h(x)) − div (1) = W − dΩ, thus in both cases the equality (2.1) holds.Hence Let m ≥ d − t, that is, the case where Ψ(x, y) ∈ L(D).First we consider the cases where either t = 0 (hence m ≥ d = 2g + 1), or t = 1 (hence m ≥ d − 1), or t ≥ 2 and m ≥ d − 2, as in these cases we know, by the theorem of Riemann-Roch, that the dimension of L(D) is m − g + 1.Thus, in order to prove that it is sufficient to note that, for each of those values of the parameters i and j, these functions belong to and the claim will follow from dimensional reasons.Now, as well as are effective divisors, hence the functions belong to L(D).
Secondly, we consider the case where d − t ≤ m < d − 2. In this case, the dimension of L(D) is not necessarily m − g + 1, but still Ψ(x, y) ∈ L(D).

2
. Of course, L ǫ+1 ≤ L ǫ , and we will see that dim(L ǫ+1 ) = dim(L ǫ ) − 1.Indeed, by (2.4) and (2.5), the functions x i , Ψ(x, y)x j of L ǫ belong to L ǫ+1 as long as i ≤ mǫ+1−t 2 , and and our assertion is proved.In particular, we found that dim where the missing function is, once for one, x i or Ψ(x, y)x j , because d is odd and changes the parity of Now we consider the cases where m < d − t, that is, the cases where, by ( Note that appending the value ǫ = 0, that is, considering also the case where m = m 0 = d − t, by (2.3) we have L 0 = x i , Ψ(x, y) , with 0 ≤ i ≤ m0−t 2 .
In order to give the equation of a hyperelliptic curve H realizing the above code as an AG-Goppa code, defined by D = div(u(x), v(x)) + 15Ω by evaluating the functions in L(D) on the above five points (x s , y s ), we note that the genus g of H must be equal at least to the degree of u(x).With g equal to the degree of u(x), hence with the degree of H equal to 23, we need eight further points, because H passes through the five points (x s , y s ) and through the eleven points (in the affine plane) of the support of div(u(x), v(x)).Choose arbitrarily eight pairs (x s , y s ) (now with s = 6, . . ., 13) such that u(x s ) = 0, for instance (48, 80), (58, 91), (64, 88), (89, 16), (95, 33), (53, 4), (51, 85), (71,35).With this choice, the curve H defined by the equation