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| PEOPLE |
| Sidel'nikov Vladimir Michilovich |
| (1940–2008) |
| Professor |
| Doctor of physico-mathematical sciences (1981) |
Coding theory, Cryptography, Decoding, Some issues in algebra, Packing and covering problems (geometry), Theory of sequences, Theory of Orthogonal Polynomials, Spherical Codes, Designs.
I. Upper bounds on the cardinalities of codes in various metric spaces. E.g., in 1973 Sidelnikov substantially improved the Blichfeldt classical upper bound on the density of ball packings in Euclidean space. This was the first improvement since 1929. Also, Sidelnikov substantially improved the Elias–Bassalygo estimate of cardinalities of codes in Hamming spaces.
II. In 1971 new bounds were obtained for crosscorelation and autococorrelation p-ary sequences. Some of these bounds are tight. Some others are best known nowadays.
III. Decoding error correcting codes. Sidelnikov initiated research of decoding techniques for the case when the number of errors exceeds half of the code distance and proposed the first algorithm for decoding the Reed–Solomon code in this setting. He also proved some related results for Reed–Muller codes.
IV. New spherical codes and spherical design. Sidelnikov defined a new family of finite groups formed by unitary $p^m\times p^m$-matrices and proved some properties of this groups. These groups were used to construct new spherical codes and spherical designs. Special attention is paid to the case $ð=2$. In this case matrices are orthogonal and the orbit of any initial point on the unit sphere in the Euclidean space $\mathbb R^{2^m}$ is a spherical design of strength 7. This and earlier results allow to construct infinite sequence of the spherical designs of strengths 7, 9 and 11 on the unit sphere in the Euclidean space.
V. Ternary codes of length $(3^m-1)/2$ correcting up to 2 errors. In 1986 Sidelnikov and Gachkov constructed ternary quasiperfect codes having some amazing properties. Since then, no new examples of infinite sequences of quasiperfect codes were discovered.
VI. Cryptoanalysis of public-key cryptosystems. In 1992 Sidelnikov and Shestakov showed insecurity of one variant of McEliece public-key cryptosystem. This nontrivial result is appreciated as achievement in mathematical cryptography.
VII. New key agreement scheme. In 1993 Sidel'nikov et al. proposed a new construction of cryptographic protocol for generating private keys based on noncommutative groups.
