The Construction of Regular Hadamard Matrices by Cyclotomic Classes

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.

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	Coro.  6.3 (ii), (iii), Coro. 7.3 (iii)
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	Coro. 5.3 (ii),Coro.  6.3 (v)
199	13	$-$1	3	9	3	Coro. 5.3 (i), Coro.  6.3 (i)
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	Coro. 5.3 (i), Coro.  6.3 (i), Coro. 7.3 (iii)
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

1. *Generator $g=\omega + k$.
2. **$\delta$ is a generator of GF(343) and satisfies $\delta ^3=\delta +5$. Regular Hadamard matrices of order $4.7^{2r}$ have been constructed by [11] for all $r \ge 1$

Appendix B. 34 Equivalence Subsets of the Probable Sets

• $\mid I \mid = 1$, 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.