By Toyokazu Hiramatsu, Günter Köhler (auth.)

This booklet grew out of our lectures given within the Oberseminar on 'Cod ing concept and quantity concept' on the arithmetic Institute of the Wiirzburg collage in the summertime Semester, 2001. The coding the ory combines mathematical attractiveness and a few engineering difficulties to an strange measure. the main good thing about learning coding concept is the wonderful thing about this actual mix of arithmetic and engineering. during this publication we want to introduce a few sensible difficulties to the mathematics ematician and to handle those as an important a part of the advance of contemporary quantity idea. The ebook involves 5 chapters and an appendix. bankruptcy 1 might typically be dropped from an introductory process linear codes. In Chap ter 2 we speak about a few kin among the variety of suggestions of a diagonal equation over finite fields and the load distribution of cyclic codes. bankruptcy three starts off through reviewing a few simple proof from elliptic curves over finite fields and modular varieties, and exhibits that the load distribution of the Melas codes is represented by way of the hint of the Hecke operators performing on the distance of cusp types. bankruptcy four is a scientific examine of the algebraic-geometric codes. for a very long time, the learn of algebraic curves over finite fields was once the province of natural mathematicians. within the interval 1977 - 1982, V. D. Goppa chanced on an grand connection among the speculation of algebraic curves over fi nite fields and the speculation of q-ary codes.

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**Example text**

Consider an algebraic curve C in A 2 defined by an equation f(x, y) = 0, and let P = (a, b) be a point on the curve C. If at least one of the derivatives fx or fy is not zero in P, then P is called a simple or nonsingular point of the curve C. A curve is called nonsingular or smooth if all the points are nonsingular. In this case, the curve has a tangent at P with equation fx{P)(x - a) for all of its points P = + fy(P)(y - b) =0 (a, b). The definitions for a projective plane curve are similar. From now on, we assume all curves to be smooth and projective.

The line connecting P and Q has slope 1/2, so its equation is 1 Y ="2 x + 4. 1 Substituting this into the equation for E gives X 3 - 1 2 4"x - 4x + 1 = o. This must have x = - 2 and x = 2 as roots, so it factors as Therefore, by taking the line through the two known solutions (-2,3), (2,5), we have found the rational solution (1/4,33/8) for our elliptic curve E. 1. If (x, y) is a point on the elliptic curve E, then the point (x, -y) will also be a point on E. This is clear from the symmetry of E about the x-axis.

Hence a· b = 0, and a E C. ))1.. ~ Res (C). ) 2 (Res (C) )1... 3 Let F 4 = 22 , q = 2, = {O, 1, a, a2 }, n = 2, k = 1, we obtain Res (C) = C n Fl = { ( ~)} Thus k = dim C = 1, but dim Res (C) = O. satisfied with ((Res (C))1.. ) = F~. 2 The weight distribution of the Melas codes Let q = 2m > 4. Let a be a primitive element ofF q . Let m(x) E F2[X] be the minimal polynomial of a and m_(x) E F2[X] that of a-I. The ideal in the ring F2[X]/(X q- 1 - 1) which is generated by m(x)m_(x) is called a Melas code, and it is denoted by M(q).