Automorphism groups of some AG codes

We show that in many cases, the automorphism group of a curve and the permutation automorphism group of a corresponding AG code are the same. This generalizes a result of Wesemeyer beyond the case of planar curves.Comment: added a reference, fixed error in remark

On the Automorphism Groups of some AG-Codes Based on Ca;b Curves

*Partially supported by NATO.We study Ca,b curves and their applications to coding theory. Recently, Joyner and Ksir have suggested a decoding algorithm based on the automorphisms of the code. We show how Ca;b curves can be used to construct MDS codes and focus on some Ca;b curves with extra automorphisms, namely y^3 = x^4 + 1, y^3 = x^4 - x, y^3 - y = x^4. The automorphism groups of such codes are determined in most characteristics

Codes from Riemann-Roch Spaces for Y2 = Xp - X over GF(P)

Let Χ denote the hyperelliptic curve y2 = xp - x over a field F of characteristic p. The automorphism group of Χ is G = PSL(2, p). Let D be a G-invariant divisor on Χ(F). We compute explicit F-bases for the Riemann-Roch space of D in many cases as well as G-module decompositions. AG codes with good parameters and large automorphism group are constructed as a result. Numerical examples using GAP and SAGE are also given

Bases and applications of Riemann-Roch Spaces of Function Fields with Many Rational Places

Algebraic geometry codes are generalizations of Reed-Solomon codes, which are implemented in nearly all digital communication devices. In ground-breaking work, Tsfasman, Vladut, and Zink showed the existence of a sequence of algebraic geometry codes that exceed the Gilbert-Varshamov bound, which was previously thought unbeatable. More recently, it has been shown that multipoint algebraic geometry codes can outperform comparable one-point algebraic geometry codes. In both cases, it is desirable that these function fields have many rational places. The prototypical example of such a function field is the Hermitian function field which is maximal. In 2003, Geil produced a new family of function fields which contain the Hermitian function field as a special case. This family, known as the norm-trace function field, has the advantage that codes from it may be defined over alphabets of larger size. The main topic of this dissertation is function fields arising from linearized polynomials; these are a generalization of Geil\u27s norm-trace function field. We also consider applications of this function field to error-correcting codes and small-bias sets. Additionally, we study certain Riemann-Roch spaces and codes arising from the Suzuki function field. The Weierstrass semigroup of a place on a function field is an object of classical interest and is tied to the dimension of associated Riemann-Roch spaces. In this dissertation, we derive Weierstrass semigroups of m-tuples of places on the norm-trace function fields. In addition, we also discuss Weierstrass semigroups from finite graphs

Conjectural permutation decoding of some AG codes

We study the action of a finite group on the Riemann-Roch space of certain divisors on a specific hyperelliptic curve X defined over a finite field with “large ” automorphism group G. If D and E = P1 +...+Pn are G-equivariant divisors on X (Pi ∈ X(F)) then G acts on associated AG code C = C(D,E) by permuting coordinates. This note discusses the permutation decoding of these AG codes. The main “results ” are conjectures regarding the complexity of the permutation decoding of these hyperelliptic codes. The open source GAP error-correcting codes package GUAVA is used to compute examples.