Lee Weights of Codes from Elliptic Curves

Degrees of the Elliptic Teichmüller Lift

“…The case n = 1 was proved by Voloch and Walker in [5]. We will get the theorem 1.1 as a special case of the theorem 3.1 below.…”

“…By theorem 4.1 of [5], with G = O (the origin of E), the functions x n and y n are regular except at O (the origin of E), so are of the form R(x 0 ) + y 0 S(x 0 ) for some polynomials R, S ∈ k[x 0 ]. For p = 2, −(y 0 , y 1 , .…”

“…Using the techniques introduced later, we can compute the correct map: We now try to find properties that will allow us to compute explicitly coordinates of the coefficients of the canonical lift and the elliptic Teichmüller. We first observe that a method to compute the second coordinates can be derived from results in [5]. So, we try to obtain the analogues of those results to deduce a way to compute the third coordinates.…”

“…(We have here a small notation problem, since we cannot divide by p. But notice that the polyno- Looking at the orders in the third coordinate, we see that So now we restrict ourselves to p = 2, 3, and then we may assume that E is given by an equation of the form 5) and that the canonical lift is given by…”

“…Voloch and Walker in [5] applied the theory of canonical lifts of elliptic curves to construct error-correcting codes. In that paper, the degrees of some polynomials, that we shall make precise later, have some importance in estimating exponential sums.…”