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}
}
  • C. Fontaine
  • Published 1 May 1999
  • Computer Science, Mathematics
  • IEEE Trans. Inf. Theory
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. 

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References

SHOWING 1-10 OF 31 REFERENCES
Orphans of the first order Reed-Muller codes
If C is a code, an orphan is a coset that is not a descendant. Orphans arise naturally in the investigation of the covering radius. Case C has only even-weight vectors and minimum distance of at
Orphan structure of the first-order Reed-Muller codes
The covering radius of the (215, 16) Reed-Muller code is at least 16276
Two interesting cosets of the first-order Reed-Muller code of block length 2^{15} are described. They provide counterexamples to a conjecture on the covering radius.
Weight distributions of the cosets of the (32, 6) Reed-Muller code
In this paper we present the weight distribution of all 2^26 cosets of the (32,6) first-order Reed-Muller code. The code is invariant under the complete affine group, of order 32 \times 31 \times 30
On A Fast Correlation Attack on Certain Stream Ciphers
TLDR
A new algorithm for the recovery of the initial state of a linear feedback shift register when a noisy output sequence is given and the results show the importance of low-weight checks and show that the complexity of the recovery problem grows less than exponentially with the length of the shift register.
Highly Nonlinear Balanced Boolean Functions with a Good Correlation-Immunity
We study a corpus of particular Boolean functions: the idempotents. They enable us to construct functions which achieve the best possible tradeoffs between the cryptographic fundamental properties:
Covering radius of RM (1, 9) in RM (3, 9)
TLDR
It is proved that the distance of the first order Reed-Muller code of length 512 to any cubic is at most 240 and new properties about Fourier coefficients are given.
On the cosets of the simplex code
Shift-register synthesis and BCH decoding
  • J. Massey
  • Computer Science
    IEEE Trans. Inf. Theory
  • 1969
It is shown in this paper that the iterative algorithm introduced by Berlekamp for decoding BCH codes actually provides a general solution to the problem of synthesizing the shortest linear feedback
Two New Classes of Bent Functions
  • C. Carlet
  • Mathematics, Computer Science
    EUROCRYPT
  • 1993
TLDR
A new class of bent functions on (GF(2)n ( n even) is introduced and it is proved that this class is not included in one of the known classes of bent function, and that, when n equals 6, it covers the whole set ofbent functions of degree 3.
...
...