“Paley Construction”的概念、定义、习题解题方法及翻译-数学辞典

单词

Paley Construction

释义

Paley Construction

Hadamard Matrices can be constructed using Galois Field GF() when and is Odd. Pick a representation Relatively Prime to . Then by coloring white (where is the Floor Function) distinct equally spaced Residues mod (, ,, ...; , , , ...; etc.) in addition to 0, a Hadamard Matrix is obtained if thePowers of (mod ) run through . For example,

is of this form with

and

. Since

, we are dealing with GF(11), so pick

and compute itsResidues (mod 11), which are

Picking the first

Residues and adding 0 gives: 0, 1, 2, 4, 5, 8, which shouldthen be colored in the Matrix obtained by writing out the Residues increasing to theleft and up along the border (0 through

, followed by

), then adding horizontal and vertical coordinates to getthe residue to place in each square.

can be trivially constructed from . cannot be built up from smallerMatrices, so use . Only the first form can be used, with and . We therefore use GF(19), and color 9 Residues plus 0 white. can beconstructed from .

Now consider a more complicated case. For , the only form having is the first, so use theGF() field. Take as the modulus the Irreducible Polynomial , written 1021. A four-digit number canalways be written using only three digits, since and . Now look at the modulistarting with 10, where each digit is considered separately. Then

Taking the alternate terms gives white squares as 000, 001, 020, 021, 022, 100, 102, 110, 111, 120, 121, 202, 211, and221.
References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 107-109 and 274, 1987.

Beth, T.; Jungnickel, D.; and Lenz, H. Design Theory, 2nd ed. rev. Cambridge, England: Cambridge University Press, 1998.

Geramita, A. V. Orthogonal Designs: Quadratic Forms and Hadamard Matrices. New York: Marcel Dekker, 1979.

Kitis, L. ``Paley's Construction of Hadamard Matrices.''http://www.mathsource.com/cgi-bin/MathSource/Applications/Mathematics/0205-760.

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