Patrick Solé

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Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson , Kerdock, Preparata, Goethals, and Delsarte-Goethals . It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z4, the integers mod 4(More)
Introduces a new image coding scheme using lattice vector quantization. The proposed method involves two steps: biorthogonal wavelet transform of the image, and lattice vector quantization of wavelet coefficients. In order to obtain a compromise between minimum distortion and bit rate, we must truncate and scale the lattice suitably. To meet this goal, we(More)
A new algebraic approach to quasi-cyclic codes is introduced. The key idea is to regard a quasi-cyclic code over a field as a linear code over an auxiliary ring. By the use of the Chinese Remainder Theorem (CRT), or of the Discrete Fourier Transform (DFT), that ring can be decomposed into a direct product of fields. That ring decomposition in turn yields a(More)
Boolean bent functions were introduced by Rothaus (1976) as combinatorial objects related to difference sets, and have since enjoyed a great popularity in symmetric cryptography and low correlation sequence design. In this paper direct links between Boolean bent functions, generalized Boolean bent functions (Schmidt, 2006) and quaternary bent functions(More)
Type II Z 4-codes are introduced as self-dual codes over the integers modulo 4 containing the all-one vector and with euclidean weights multiple of 8: Their weight enumerators are characterized by means of invariant theory. A notion of extremality for the euclidean weight is introduced. Their binary images under the Gray map are formally self-dual with even(More)
The ring decomposition technique of part I is extended to the case when the factors in the direct product decomposition are no longer fields but arbitrary chain rings. This includes not only the case of quasi-cyclic codes over rings but also the case of quasi-cyclic codes over fields whose co-index is no longer prime to the characteristic of the field. A(More)
In [1] the authors introduced a lattice invariant called “Secrecy Gain” which measures the confusion experienced by a passive eavesdropper on the Gaussian Wiretap Channel. We study, here, the behavior of this invariant for unimodular lattices by using tools from Modular Forms and show that, for some families of unimodular lattices, indexed by(More)