# On Some Cosets of the First-Order Reed-Muller Code with High Minimum Weight

@article{Fontaine1999OnSC, title={On Some Cosets of the First-Order Reed-Muller Code with High Minimum Weight}, author={Caroline Fontaine}, journal={IEEE Trans. Inf. Theory}, year={1999}, volume={45}, pages={1237-1243} }

We study a family of particular cosets of the first-order Reed-Muller code R(1,m): those generated by special codewords, the idempotents. Thus we obtain new maximal weight distributions of cosets of R(1,7) and 84 distinct almost maximal weight distributions of cosets of R(1,9), that is, with minimum weight 240. This leads to crypotographic applications in the context of stream ciphers.

## 51 Citations

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It is shown that there are exactly five weight distributions for optimal cosets of R(1,7) in R(5,7), and that these distributions are related to the degree of the function generating the coset.

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- Computer Science, MathematicsIEEE Trans. Inf. Theory
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A new approach for the study of weight distributions of cosets of the Reed-Muller code of order 1 is introduced, based on the method introduced by Kasami (1968), using Pless (1963) identities, and it is able to distinguish such cosets which have three weights only.

On Cryptographic Properties of the Cosets of

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A new approach for the study of weight distributions of cosets of the Reed-Muller code of order is introduced, based on the method introduced by Kasami in (1), using Pless identities to obtain a condition for a coset to have a "high" minimum weight.

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A systematic algorithm is proposed for the super codes construction base on the m-sequence pairs, which only involves some item permutation within them-sequence, which outperform the conventional codes in terms of decoding error rate.

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Since bent functions can never be balanced, research on super-classes of the class of bent functions, whose elements can have high nonlinearities, but can also be balanced has been investigated.

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