Abstract
For every prime power \(q \equiv 7 \; mod \; 16\), there are (q; a, b, c, d)-partitions of GF(q), with odd integers a, b, c, and d, where \(a \equiv \pm 1 \; mod \; 8\) such that \(q=a^2+2(b^2+c^2+d^2)\) and \(d^2=b^2+2ac+2bd\). Many results for the existence of \(4-\{q^2; \; \frac{q(q-1)}{2}; \; q(q-2) \}\) SDS which are simple homogeneous polynomials of parameters a, b, c and d of degree at most 2 have been found. Hence, for each value of q, the construction of SDS becomes equivalent to building a \((q; \; a, \; b, \; c, \; d)\)-partition. Once this is done, the verification of the construction only involves verifying simple conditions on a, b, c and d which can be done manually.
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The authors are grateful to the referees and Prof. Majid Soleimani-damaneh for their helpful comments.
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Communicated by Behruz Tayfeh-Rezaie.
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Appendices
Appendix A
Table of parameters a, b, c and d in the \((q; a, \; b, \; c, \; d)\)-partitions of GF(q) for \(q<1000\)
q | \(k^{*}\) | a | b | c | d | Applicable theorems |
---|---|---|---|---|---|---|
7 | 2 | \(-\)1 | 1 | 1 | 1 | Coro. 5.3 (i), (ii), Coro. 6.3 (i)-(vii), Coro. 7.3 (i), (ii) |
23 | 2 | 1 | 3 | \(-\)1 | \(-\)1 | |
71 | 8 | \(-\)7 | 1 | \(-\)1 | \(-\)3 | Coro. 6.3 (ii) |
103 | 2 | \(-\)7 | 5 | 1 | \(-\)1 | None |
151 | 9 | \(-\)1 | 5 | 7 | \(-\)1 | Coro. 6.3 (ii), (vii) |
167 | 2 | \(-\)1 | 3 | 7 | 5 | |
199 | 13 | \(-\)1 | 3 | 9 | 3 | |
263 | 2 | 9 | \(-\)9 | \(-\)1 | 3 | Coro. 6.3 (ii) |
311 | 4 | 7 | \(-\)1 | 7 | 9 | None |
343 | \(\delta \) + 1 | 7 | 11 | 1 | \(-\)5 | \(\hbox {None}^{**}\) |
359 | 11 | \(-\)9 | 3 | \(-\)7 | \(-\)9 | Coro. 6.3 (ii) |
439 | 9 | 7 | \(-\)5 | 1 | \(-\)13 | Coro. 7.3 (ii) |
487 | 3 | \(-\)1 | \(-\)5 | \(-\)7 | \(-\)13 | None |
503 | 6 | \(-\)17 | \(-\)9 | \(-\)1 | 5 | None |
599 | 11 | \(-\)23 | 3 | \(-\)1 | \(-\)5 | Coro. 6.3 (ii) |
631 | 5 | 1 | 5 | \(-\)17 | 1 | Coro. 6.3 (ii) |
647 | 2 | \(-\)9 | \(-\)15 | 7 | 3 | None |
727 | 2 | \(-\)25 | \(-\)5 | 1 | \(-\)5 | |
743 | 2 | 17 | \(-\)13 | \(-\)7 | \(-\)3 | None |
823 | 3 | \(-\)7 | \(-\)9 | 9 | \(-\)15 | None |
839 | 4 | 17 | \(-\)7 | \(-\)1 | \(-\)15 | None |
887 | 2 | 7 | 9 | \(-\)7 | 17 | None |
919 | 6 | 17 | 15 | \(-\)9 | 3 | Coro. 6.3 (ii) |
967 | 2 | \(-\)17 | \(-\)13 | \(-\)7 | 11 | None |
983 | 2 | \(-\)7 | \(-\)3 | \(-\)17 | 13 | None |
Appendix B. 34 Equivalence Subsets of the Probable Sets
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\(\mid I \mid = 1\), 1 equivalence subset \(\{ 0 \}\);
-
\(\mid I \mid = 3\), 6 equivalence subsets:
$$\begin{aligned} \{ 0, 1, 2 \}&\equiv 11 \{ 0, 3, 6 \}, \\ \{ 0, 1, 3 \}&\equiv 13 \{ 0, 1, 6 \} + 3 \equiv 15 \{ 0, 1, 14 \} + 1 \equiv 3 \{ 0, 1, 11 \}, \\ \{ 0, 1, 4 \}&\equiv 3 \{ 0, 1, 5 \} + 1 \equiv 13 \{ 0, 1, 12 \} + 4 \equiv 15 \{ 0, 1, 13 \} + 1, \\ \{ 0, 1, 7 \}&\equiv 9 \{ 0, 1, 10 \} + 7 \equiv 3 \{ 0, 2, 5 \} + 1 \equiv 11 \{ 0, 2, 13 \} +1, \\ \{ 0, 2, 4\}&\equiv 13 \{ 0, 4, 10 \}, \\ \{ 0, 2, 6 \}&\end{aligned}$$ -
\(\mid I \mid = 5\), 17 equivalence subsets:
$$\begin{aligned} \{ 0, 1, 2, 3, 4 \}&\equiv 3 \{ 0, 1, 5, 6, 11 \} + 1, \\ \{ 0, 1, 2, 3, 5 \}&\equiv 15 \{ 0, 1, 2, 3, 14 \}\\&\quad +3 \equiv 11 \{ 0, 1, 4, 7, 10 \} + 5 \equiv 3 \{ 0, 1, 5, 10, 11 \}+2, \\ \{ 0, 1, 2, 3, 6 \}&\equiv 15 \{ 0, 1, 2, 3, 13 \} + 3 \equiv 3 \{ 0, 1, 2, 6, 11 \} \equiv 13 \{ 0, 1, 2, 7, 12 \} + 6, \\ \{ 0, 1, 2, 3, 7 \}&\equiv 15 \{ 0, 1,2,3,12 \}+3 \equiv 5 \{0, 1, 3, 6, 13 \}+2 \equiv 13 \{ 0, 1, 4, 6, 11 \} + 3, \\ \{ 0, 1, 2, 4, 5 \}&\equiv 15 \{ 0, 1, 2, 13, 14 \} + 2 \equiv 11 \{ 0, 1, 4, 7, 13 \} + 5 \equiv 5 \{ 0, 1, 4, 10, 13 \}, \\ \{ 0, 1, 2, 4, 6 \}&\equiv 15 \{ 0, 1, 2, 12, 14 \} + 2 \equiv 5 \{ 0, 1, 3, 7, 13 \} + 1 \equiv 11 \{ 0, 1, 4, 10, 14 \}+6, \\ \{ 0, 1, 2, 4, 7 \}&\equiv 3 \{ 0, 1, 2, 5, 11 \}+1 \equiv 13 \{ 0, 1, 2, 7, 13 \} + 7 \equiv 15 \{ 0, 1, 2, 11, 14\} +2, \\ \{ 0, 1, 2, 4, 11 \}&\equiv 9 \{ 0, 1, 2, 7, 14 \} + 2 \equiv 11 \{ 0, 1, 3, 6, 12 \} \equiv 3 \{ 0, 1, 5, 11, 14 \}+1, \\ \{ 0, 1, 2, 4, 13 \}&\equiv 15 \{ 0, 1, 2, 5, 14 \} + 2 \equiv 13 \{ 0, 1, 4, 5, 10 \} \equiv 3 \{ 0, 1, 4, 5, 11 \} + 1, \\ \{ 0, 1, 2, 4, 14 \}&\equiv 13 \{ 0, 1, 2, 6, 12\} + 4, \\ \{ 0, 1, 2, 5, 6 \}&\equiv 15 \{ 0, 1, 2, 12, 13 \} + 2 \equiv 11 \{ 0, 1, 3, 4, 7 \}+5 \equiv 5\{ 0, 1, 3, 4, 13 \} + 1, \\ \{ 0, 1, 2, 5, 7 \}&\equiv 15 \{ 0, 1, 2, 11, 13 \} + 2\equiv 5 \{ 0, 1, 3, 6, 7 \}+2 \equiv 11 \{ 0, 1, 4, 6, 7 \} +5, \\ \{ 0, 1, 2, 5, 12 \}&\equiv 15 \{ 0, 1, 2, 6, 13 \} + 2 \equiv 13 \{ 0, 1, 3, 7, 12 \} +5 \equiv 5\{ 0, 1, 4, 5, 14 \} -4, \\ \{ 0, 1, 2, 6, 7 \}&\equiv 15 \{ 0, 1, 2, 11, 12 \} + 2 \equiv 5 \{ 0, 1, 3, 4, 6 \} + 2 \equiv 11 \{ 0, 1, 3, 4, 14 \} +6, \\ \{ 0, 1, 2, 7, 11 \}&\equiv 11 \{ 0, 1, 3, 5, 6 \}), \\ \{ 0, 1, 3, 5, 7 \}&\equiv 11 \{ 0, 1, 4, 6, 10 \} + 5 \equiv 3 \{ 0, 1, 7, 11, 13 \} \equiv 9 \{ 0, 1, 10, 12, 14 \} +7, \\ \{ 0, 1, 3, 5, 14 \}&\equiv 9 \{ 0, 1, 3, 12, 14 \} + 5\equiv 11 \{ 0, 1, 4, 10, 11 \} + 5 \equiv 3\{ 0, 1, 6, 7, 13 \} -2; \end{aligned}$$ -
\(\mid I \mid = 7\), 10 equivalence subsets:
$$\begin{aligned} \{0, 1, 2, 3, 4, 5, 6\}&\equiv 3 \{0, 1, 2, 6, 7, 11, 12\}), \\ \{0, 1, 2, 3, 4, 5, 7\}&\equiv 15 \{0, 1, 2, 3, 4, 5, 14\}+5\equiv 3 \{0, 1, 2, 5, 6, 11, 12 \}+1 \\&\equiv 13 \{0, 1, 2, 6, 7, 12, 13\}+7, \\ \{0, 1, 2, 3, 4, 6, 7\}&\equiv 15 \{0, 1, 2, 3, 4, 13, 14\}+4\equiv 3 \{0, 1, 2, 5, 6, 7, 11 \}+1 \\&\equiv 13 \{0, 1, 2, 5, 6, 7, 12\}+6, \\ \{0, 1, 2, 3, 4, 6, 13\}&\equiv 15 \{0, 1, 2, 3, 4, 7, 14\}+4\equiv 13 \{0, 1, 2, 3, 6, 7, 12 \}+6 \\&\equiv 3 \{0, 1, 2, 3, 7, 12, 13\}-3, \\ \{0, 1, 2, 3, 4, 7, 13\}&\equiv 3 \{0, 1, 2, 4, 5, 6, 11\}+1, \\ \{0, 1, 2, 3, 5, 6, 7\}&\equiv 15 \{0, 1, 2, 3, 12, 13, 14\}+3\equiv 13 \{0, 1, 2, 5, 7, 11, 12 \}+6 \\&\equiv 3 \{0, 1, 2, 6, 7, 11, 13\}, \\ \{0, 1, 2, 3, 5, 6, 12\}&\equiv 15 \{0, 1, 2, 3, 7, 13, 14\}+3\equiv 5 \{0, 1, 2, 4, 5, 11, 14 \}-4 \\&\equiv 11 \{0, 1, 2, 4, 7, 13, 14\}+6, \\ \{0, 1, 2, 3, 5, 7, 12\}&\equiv 5 \{0, 1, 2, 3, 6, 7, 13\}+2\equiv 11 \{0, 1, 2, 3, 6, 12, 13 \}+1 \\&\equiv 15 \{0, 1, 2, 3, 7, 12, 14\}+3, \\ \{0, 1, 2, 3, 5, 7, 14\}&\equiv 9 \{0, 1, 2, 3, 5, 12, 14\}+5\equiv 3 \{0, 1, 3, 6, 7, 12, 13 \}-2 \\&\equiv 11 \{0, 1, 4, 5, 10, 12, 14\}+7, \\ \{0, 1, 2, 4, 5, 7, 11\}&\equiv 11 \{0, 1, 2, 4, 6, 7, 13\}+5\equiv 3 \{0, 1, 2, 5, 11, 12, 14 \}+1 \\&\equiv 9 \{0, 1, 2, 7, 11, 13, 14\}+2. \end{aligned}$$
To simplify the notation we omit “\(mod \; 16\)”, and for each equivalence class we only enumerate the probable sets, which occupy positions at the head in lexicographic order from 0 to 15, and the remainder unlisted in the class, might be represented as \(I+k\) or \(qI+k\) with an integer k and an enumerated probable set I in these classes.
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Xia, T., Xia, M. & Seberry, J. The Construction of Regular Hadamard Matrices by Cyclotomic Classes. Bull. Iran. Math. Soc. 47, 601–625 (2021). https://doi.org/10.1007/s41980-020-00402-9
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DOI: https://doi.org/10.1007/s41980-020-00402-9