Felix Ulmer

Skew-cyclic codes

We generalize the notion of cyclic codes by using generator polynomials in (non commutative) skew polynomial rings. Since skew polynomial rings are left and right euclidean, the obtained codes share most properties of cyclic codes. Since there are much more skew-cyclic codes, this new class of codes allows to systematically search for codes with good properties. We give many examples of codes which improve the previously best known linear codes.

Galois groups of second and third order linear differential equations

Abstract Using the representation theory of groups, we are able to give simple necessary and sufficient conditions regarding the structure of the Galois groups of second and third order linear differential equations. These allow us to give simple necessary and sufficient conditions for a second order linear differential equation to have liouvillian solutions and for a third order linear differential equation to have liouvillian solutions or be solvable in terms of second order equations. In many cases these conditions also allow us to determine the group.

Liouvillian and algebraic solutions of second and third order linear differential equations

Abstract In this paper we show that the index of a 1-reducible subgroup of the differential Galois group of an ordinary homogeneous linear differential equation L(y) = 0 yields the best possible bound for the degree of the minimal polynomial of an algebraic solution of the Riccati equation associated to L(y) = 0. For an irreducible third order equation we show that this degree belongs to {3,6,9,21,36}. When the Galois group is a finite primitive group, we reformulate and generalize work of L. Fuchs to show how to compute the minimal polynomial of a solution instead of the minimal polynomial of the logarithmic derivative of a solution. These results lead to an effective algorithm to compute Liouvillian solutions of second and third order linear differential equations.

Skew Constacyclic Codes over Galois Rings

We generalize the construction of linear codes via skew polynomial rings by using Galois rings instead of finite fields as coefficients. The resulting non commutative rings are no longer left and right Euclidean. Codes that are principal ideals in quotient rings of skew polynomial rings by a two sided ideals are studied. As an application, skew constacyclic self-dual codes over GR(4, 2) are constructed. Euclidean self-dual codes give self-dual Z(4)-codes. Hermitian self-dual codes yield 3-modular lattices and quasi-cyclic self-dual Z(4)-codes.

Necessary conditions for liouvillian solutions of (third order) linear differential equations

In this paper we show how group theoretic information can be used to derive a set of necessary conditions on the coefficients ofL(y) forL(y=0 to have a liouvillian solution. The method is used to derive (and improve in one case) the necessary conditions of the Kovacic algorithm and to derive an explicit set of necessary conditions for third order differential equations.

Linear codes using skew polynomials with automorphisms and derivations

In this work the definition of codes as modules over skew polynomial rings of automorphism type is generalized to skew polynomial rings, whose multiplication is defined using an automorphism and a derivation. This produces a more general class of codes which, in some cases, produce better distance bounds than module skew codes constructed only with an automorphism. Extending the approach of Gabidulin codes, we introduce new notions of evaluation of skew polynomials with derivations and the corresponding evaluation codes. We propose several approaches to generalize Reed-Solomon and BCH codes to module skew codes and for two classes we show that the dual of such a Reed-Solomon type skew code is an evaluation skew code. We generalize a decoding algorithm due to Gabidulin for the rank metric and derive families of Maximum Distance Separable and Maximum Rank Distance codes.

Codes as Modules over Skew Polynomial Rings

In previous works we considered codes defined as ideals of quotients of skew polynomial rings, so called Ore rings of automorphism type. In this paper we consider codes defined as modules over skew polynomial rings, removing therefore some of the constraints on the length of the skew codes defined as ideals. The notion of BCH codes can be extended to this new approach and the skew codes whose duals are also defined as modules can be characterized. We conjecture that self-dual skew codes defined as modules must be constacyclic and prove this conjecture for the Hermitian scalar product and under some assumptions for the Euclidean scalar product. We found new [56, 28, 15], [60,30,16], [62,31,17], [66,33,17] Euclidean self-dual skew codes and new [50,25,14], [58,29,16] Hermitian self-dual skew codes over F 4 , improving the best known distances for self-dual codes of these lengths over F 4 .

Linear differential equations and products of linear forms

Abstract We show that liouvillian solutions of an n th-order linear differential equation L ( y ) = 0 are related to semi-invariant forms of the differential Galois group of L ( y ) = 0 which factor into linear forms. The logarithmic derivative of such a form F , evaluated in the solutions of L ( y ) = 0, is the first coefficient of a polynomial P ( u ) whose zeros are logarithmic derivatives of solutions of L ( y ) = 0. Together with the Brill equations, this characterization allows one to efficiently test if a semi-invariant corresponds to such a coefficient and to compute the other coefficients of P ( u ) via a factorization of the form F .

Linear Differential Operators for Polynomial Equations

Given a squarefree polynomial P?k0x,y ], k0a number field, we construct a linear differential operator that allows one to calculate the genus of the complex curve defined by P= 0 (when P is absolutely irreducible), the absolute factorization of P over the algebraic closure of k0, and calculate information concerning the Galois group of P over ___ k0(x) as well as overk0 (x).

Skew codes of prescribed distance or rank

In this paper we generalize the notion of cyclic code and construct codes as ideals in finite quotients of non-commutative polynomial rings, so called skew polynomial rings of automorphism type. We propose a method to construct block codes of prescribed rank and a method to construct block codes of prescribed distance. Since there is no unique factorization in skew polynomial rings, there are much more ideals and therefore much more codes than in the commutative case. In particular we obtain a [40, 23, 10]4 code by imposing a distance and a [42,14,21]8 code by imposing a rank, which both improve by one the minimum distance of the previously best known linear codes of equal length and dimension over those fields. There is a strong connection with linear difference operators and with linearized polynomials (or q-polynomials) reviewed in the first section.