Factorization of Symmetric Matrices and Trace-Orthogonal Bases in Finite Fields
Abstract
It is shown that every symmetric matrix A, with entries from a finite field F, can be factored over F into $A = BB'$, where the number of columns of B is bounded from below by either the rank $\rho (A)$ of A, or by $1 + \rho (A)$, depending on A and on the characteristic of F This result is applied to show that every finite extension $\Phi $ of a finite field F has a trace-orthogonal basis over F. Necessary and sufficient conditions for the existence of a trace-orthonormal basis are also given. All proofs are constructive, and can be utilized to formulate procedures for minimal factorization and basis construction.
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References
1.
Abraham Lempel, Matrix factorization over ${\rm GF}(2)$ and trace-orthogonal bases of ${\rm GF}(2\sp{n})$, SIAM J. Comput., 4 (1975), 175–186
2.
F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977
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Copyright © 1980 Society for Industrial and Applied Mathematics.
History
Submitted: 13 May 1979
Accepted: 25 February 1980
Published online: 13 July 2006
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