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Rank r Solutions to the Matrix Equation XAXT = C, A Nonalternate, C Alternate, Over GF(2y).

  • Philip G. Buckhiester (a1)

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Let GF(q) denote a finite field of order q = py , p a prime. Let A and C be symmetric matrices of order n, rank m and order s, rank k, respectively, over GF(q). Carlitz [6] has determined the number N(A, C, n, s) of solutions X over GF(q), for p an odd prime, to the matrix equation

1.1

where n = m. Furthermore, Hodges [9] has determined the number N(A, C, n, s, r) of s × n matrices X of rank r over GF(q), p an odd prime, which satisfy (1.1). Perkin [10] has enumerated the s × n matrices of given rank r over GF(q), q = 2y, such that XXT = 0. Finally, the author [3] has determined the number of solutions to (1.1) in case C = 0, where q = 2y.

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References

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1. Albert, A. A., Symmetric and alternate matrices in an arbitrary field. I, Trans. Amer. Math. Soc. 43 (1938), 386436.
2. Brawley, J. and Carlitz, L., Enumeration of matrices with prescribed row and column sums, Linear Algebra and Appl. (to appear).
3. Buckhiester, P., Gauss sums and the number of solutions to the matrix equation XAXT = 0 over GF(2y), Acta Arith. 23 (1973), 271278.
4. Buckhiester, P., Rank r solutions to the matrix equation XAXT = C, A alternate, over GF(2y), Trans. Amer. Math. Soc. (to appear).
5. Buckhiester, P., Rank r solutions to the matrix equation XAXT = C, A and C nonalternate, over GF(2y), Math. Nachr. (to appear).
6. Carlitz, L., Representations by quadratic forms in a finite field, Duke Math. J. 21 (1954), 123137.
7. Carlitz, L., The number of solutions of certain matric equations over a finite field, Math. Nachr. (to appear).
8. Dai, Zong-duo (Tai Tsung-Tuo), On transitivity of subspaces in orthogonal geometry over fields of characteristic 2, Chinese Math. Acta. 16 (1966), 569584.
9. Hodges, J. H., A symmetric matrix equation over a finite field, Math. Nachr. 30 (1965), 221228.
10. Perkins, J. C., Rank r solutions to the matrix equation XXT = 0 over afield of characteristic two, Math. Nachr. 48 (1971), 6976.
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Rank r Solutions to the Matrix Equation XAXT = C, A Nonalternate, C Alternate, Over GF(2y).

  • Philip G. Buckhiester (a1)

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