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A characterization of the number of roots of linearized and projective polynomials in the field of coefficients
Results of this type are proved which characterizes the number of roots using the rank of a matrix that is smaller than the Dickson matrix, of a recursive nature. Expand
Proof of a Conjecture on the Sequence of Exceptional Numbers, Classifying Cyclic Codes and APN Functions
The conjecture states that every exceptional number has the form $2^i+1$ or $4^i-2^ i+1$, which is equivalent to 1, 2, 3, or 4. Expand
New families of quadratic almost perfect nonlinear trinomials and multinomials
Two new infinite families of APN functions are introduced, one on fields of order 2^2^k for k not divisible by 2, and the other on field of order2^3^k with polynomials between three and k+2 terms. Expand
A few more functions that are not APN infinitely often
We consider exceptional APN functions on ${\bf F}_{2^m}$, which by definition are functions that are not APN on infinitely many extensions of ${\bf F}_{2^m}$. Our main result is that polynomialExpand
Double-Error-Correcting Cyclic Codes and Absolutely Irreducible Polynomials over GF(2)
Codewords of weight ≤ 4 in certain cyclic codes of length n = 2s − 1, parameterized by an odd integer t, can be related to zeros of certain projective plane curves gt(X, Y, Z). Some families of theseExpand
There Is No 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues Problem via Hitting Set Enumeration
An exhaustive computer search for 16-clue sudoku puzzles and did not find any, thus proving that the answer is indeed 17, which is one of Karp’s 21 classic NP-complete problems. Expand
Construction of bent functions from near-bent functions
The first ever examples of non-weakly-normal bent functions in dimensions 10 and 12 are given, which demonstrates the significance of the construction of bent functions from near-bent functions in dimension 2m-1. Expand
On the Function Field Sieve and the Impact of Higher Splitting Probabilities: Application to Discrete Logarithms in F21971
A binary field variant of the Joux-Lercier medium-sized Function Field Sieve is proposed, which results not only in complexities as low as \(L_{q^n}(1/3,(4/9)^{1/ 3})\) for computing arbitrary logarithms, but also in an heuristic polynomial time algorithm for finding the discrete logariths when the field has a subfield of an appropriate size. Expand
Quasi-Symmetric Designs and Codes Meeting the Grey-Rankin Bound
  • G. McGuire
  • Mathematics, Computer Science
  • J. Comb. Theory, Ser. A
  • 1 May 1997
We give a characterization of codes meeting the Grey?Rankin bound. When the codes have even length, the existence of such codes is equivalent to the existence of certain quasi-symmetric designs. WeExpand
On the Nonlinearity of Exponential Welch Costas Functions
The nonlinearity of the exponential Welch Costas functions is studied using the Fourier transform on Z m, and a plausible connection of the non linearity to the class number of a quadratic field is suggested. Expand