Yves Edel

Quantum twisted codes

A major contribution of [1] is a reduction of the problem of correcting errors in quantum computations to the construction of codes in binary symplectic spaces. This mechanism is known as the additive or stabilizer construction. We consider an obvious generalization of these quantum codes in the symplectic geometry setting and obtain general constructions using our theory of twisted BCH-codes (also known as Reed-Solomon subfield subcodes). This leads to families of quantum codes with good parameters. Moreover the generator matrices of these codes can be described in a canonical way.

A new APN function which is not equivalent to a power mapping

A new almost-perfect nonlinear function (APN) on F(2/sup 10/) which is not equivalent to any of the previously known APN mappings is constructed. This is the first example of an APN mapping which is not equivalent to a power mapping.

41 is the Largest Size of a Cap in PG(4,4)

We settle the question of the maximal size of caps in PG(4, 4), with the help of a computer program.

Extensions of Generalized Product Caps

We give some variants of a new construction for caps. As an application of these constructions, we obtain a 1216-cap in PG(9,3) a 6464-cap in PG(11,3) and several caps in ternary affine spaces of larger dimension, which lead to better asymptotics than the caps constructed by Calderbank and Fishburn [1]. These asymptotic improvements become visible in dimensions as low as 62, whereas the bound from Calderbank and Fishburn [1] is based on caps in dimension 13,500.

The Classification of the Largest Caps in AG(5, 3)

We prove that 45 is the size of the largest caps in AG(5,3), and such a 45-cap is always obtained from the 56-cap in PG(5,3) by deleting an 11-hyper-plane.

Construction of digital nets from BCH-codes

We establish a link between the theory of error-correcting codes and the theory of (t, m, s)-nets. This leads to the fundamental problem of net embeddings of linear codes. Our main result is the construction of four infinite families of digital (t, m, s)-nets based on BCH- codes.

Sequences in abelian groups G of odd order without zero-sum subsequences of length exp(G)

We present a new construction for sequences in the finite abelian group

The Largest Cap in AG (4, 4) and Its Uniqueness

C_{n}^r