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Difference matrices with five rows over finite abelian groups

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Abstract

Let G be a finite group and \(k\geqslant 2\) be an integer. A (Gk, 1)-difference matrix (DM) is a \(k\times |G|\) matrix \(D=(d_{ij})\) with entries from G, such that for all distinct rows x and y, the multiset of differences \(\{d_{xi} d_{yi}^{-1}:1\leqslant i\leqslant |G|\}\) contains each element of G exactly once. This paper examines the existence of difference matrices with five rows over a finite abelian group. It is proved that if G is a finite abelian group and the Sylow 2-subgroup of G is trivial or noncyclic, then a (G, 5, 1)-DM exists, except for \(G \in \{{\mathbb {Z}}_3,\) \({\mathbb {Z}}_2 \oplus {\mathbb {Z}}_2,\) \({\mathbb {Z}}_4 \oplus {\mathbb {Z}}_2,\) \({\mathbb {Z}}_9\}\) and possibly for some groups whose Sylow 2-subgroup lies in \(\{{\mathbb {Z}}_2\oplus {\mathbb {Z}}_2\), \({\mathbb {Z}}_4\oplus {\mathbb {Z}}_2\), \({\mathbb {Z}}_{32} \oplus {\mathbb {Z}}_{2},\) \({\mathbb {Z}}_{16} \oplus {\mathbb {Z}}_{4}\}\), and some cyclic groups of order 9p with p prime.

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Acknowledgements

This research includes computations using the computational cluster Katana supported by Research Technology Services at UNSW Sydney. Also, the third author acknowledges the support from an Australian Government Research Training Program Scholarship and from the School of Mathematics and Statistics, UNSW Sydney. The first, fourth and sixth authors acknowledge support from NSFC under Grant Numbers 11601472, 11871095 and 11771227 respectively. The fifth author was supported by a 2020-2021 AMSI Vacation Research Scholarship.

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Correspondence to R. Julian R. Abel.

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Appendices

Appendix: HPMDs in the proof of Lemma 9

Each element (xy) is simply written as \(x_y\).

A strictly \(({\mathbb {Z}}_2 \oplus {\mathbb {Z}}_{36},{\mathbb {Z}}_2 \oplus 6{\mathbb {Z}}_{36})\)-invariant 5-HPMD of type \(12^{6}\):

$$\begin{aligned} \begin{array}{cccc} (0_{0}, 1_{3}, 1_{2}, 0_{7}, 1_{11}), &{} (0_{0}, 0_{7}, 1_{3},0_{5},0_{10}), &{} (0_{0}, 0_{25}, 0_{26}, 0_{21}, 0_{34}), &{} (0_{0}, 0_{3}, 0_{19},1_{26},1_{10}), \\ (0_{0}, 0_{9}, 1_{28}, 0_{29}, 0_{8}), &{} (0_{0}, 0_{33}, 1_{13},0_{11},0_{28}), &{} (0_{0}, 0_{32}, 0_{23}, 1_{10}, 1_{21}), &{} (0_{0}, 0_{21}, 1_{20},0_{1},1_{22}), \\ (0_{0}, 1_{10}, 1_{ 8}, 1_{27}, 0_{13}), &{} (0_{0}, 1_{29}, 0_{21},0_{25},1_{16}), &{} (0_{0}, 1_{9}, 0_{4}, 0_{26}, 1_{23}), &{} (0_{0}, 0_{14}, 1_{25},1_{35},1_{28}). \\ \end{array} \end{aligned}$$

A strictly \(({\mathbb {Z}}_6 \oplus {\mathbb {Z}}_{12},{\mathbb {Z}}_6 \oplus 6{\mathbb {Z}}_{12})\)-invariant 5-HPMD of type \(12^{6}\):

$$\begin{aligned} \begin{array}{cccc} (0_{0}, 1_{3}, 3_{2}, 4_{1}, 3_{11}), &{} (0_{0}, 0_{1}, 3_{3}, 2_{5}, 4_{10}), &{} (0_{0}, 4_{7}, 2_{8}, 4_{3}, 0_{4}), &{} (0_{0}, 4_{9}, 0_{7}, 1_{2}, 5_{10}), \\ (0_{0}, 2_{3}, 1_{10}, 4_{5}, 0_{2}), &{} (0_{0}, 0_{9}, 3_{1}, 0_{11}, 0_{10}), &{} (0_{0}, 4_{2}, 2_{5}, 5_{4}, 3_{9}), &{} (0_{0}, 0_{3}, 5_{2}, 0_{7}, 3_{4}), \\ (0_{0}, 1_{10}, 5_8, 5_3, 0_7), &{} (0_{0}, 5_{5}, 4_{9}, 2_{1}, 1_{4}), &{} (0_{0}, 5_{9}, 4_{10}, 4_{2}, 5_{11}), &{} (0_{0}, 2_{8}, 5_{1}, 1_{5}, 5_{4}). \\ \end{array} \end{aligned}$$

A strictly \((({\mathbb {Z}}_2 \oplus {\mathbb {Z}}_2) \oplus {\mathbb {Z}}_{18}, {\mathbb {Z}}_2 \oplus {\mathbb {Z}}_2 \oplus 6{\mathbb {Z}}_{18})\)-invariant 5-HPMD of type \(12^{6}\):

$$\begin{aligned} \begin{array}{ccc} (00_{0}, 10_{15}, 10_{2}, 00_{1}, 11_{5}), &{} (00_{0}, 00_{1}, 11_{9}, 00_{11}, 00_{10}), &{} (00_{0}, 00_{7}, 01_{8}, 00_{15}, 01_{10}), \\ (00_{0}, 01_{9}, 00_{13}, 10_{8}, 10_{4}), &{} (00_{0}, 00_{3}, 11_{4}, 01_{5}, 00_{2}), &{} (00_{0}, 00_{15}, 10_{1}, 01_{11}, 01_{4}), \\ (00_{0}, 00_{2}, 00_{11}, 11_{16}, 10_{9}), &{} (00_{0}, 01_{3}, 11_{8}, 01_{1}, 11_{4}), &{} (00_{0}, 10_{16}, 10_{8}, 10_{3}, 01_{1}), \\ (00_{0}, 11_{17}, 01_{9}, 01_{13}, 10_{16}), &{} (00_{0}, 11_{15}, 00_{4}, 01_{2}, 10_{11}), &{} (00_{0}, 01_{2}, 10_{13}, 11_{5}, 10_{10}). \\ \end{array} \end{aligned}$$

The construction of the previous three HPMDs was facilitated by first finding suitable \(\bmod \ 2\) values for the first (\({\mathbb {Z}}_2\) or \({\mathbb {Z}}_6\)) coordinate and mod 6 values for the last \(({\mathbb {Z}}_{36},\) \({\mathbb {Z}}_{12}\) or \({\mathbb {Z}}_{18})\) coordinate of each point in the base blocks before doing the main part of the computer search. We used the same mod 2 and mod 6 prespecifications for all three of these HPMDs. The next HPMD was also obtained by first prespecifying these mod 2 and mod 6 values. Prespecifying the mod y values of certain entries (for some small y) is a tool that can sometimes help when searching for other types of designs such as designs obtained from difference families.

A strictly \(({\mathbb {Z}}_2 \oplus {\mathbb {Z}}_{72},{\mathbb {Z}}_2 \oplus 6{\mathbb {Z}}_{72})\)-invariant 5-HPMD of type \(24^{6}\):

$$\begin{aligned} \begin{array}{cccc} (0_{0}, 1_{65}, 1_{2}, 0_{1}, 1_{10}), &{} (0_{0}, 0_{23}, 1_{1}, 1_{3}, 1_{4}),&{} (0_{0}, 0_{3}, 1_{16}, 1_{31},0_{62}), &{} (0_{0}, 0_{25}, 0_{33}, 1_{38}, 1_{70}),\\ (0_{0}, 0_{67}, 0_{50}, 0_{21},1_{46}), &{} (0_{0}, 1_{33}, 1_{5}, 0_{43}, 1_{14}),&{} (0_{0}, 0_{38}, 1_{55}, 0_{40},0_{33}), &{} (0_{0}, 0_{26}, 1_{23}, 1_{58}, 1_{37}),\\ (0_{0}, 0_{56}, 0_{34}, 0_{15},0_{11}), &{} (0_{0}, 0_{41}, 1_{9}, 0_{25}, 0_{52}),&{} (0_{0}, 1_{49}, 0_{68}, 0_{10},1_{21}), &{} (0_{0}, 0_{13}, 1_{57}, 0_{65}, 1_{20}).\\ \end{array} \end{aligned}$$

The 24 base blocks are now obtained by multiplying each of the above 12 base blocks by 1 and \(-1\).

A strictly \(({\mathbb {Z}}_4 \oplus {\mathbb {Z}}_{24},2{\mathbb {Z}}_4 \oplus 3{\mathbb {Z}}_{24})\)-invariant 5-HPMD of type \(16^{6}\):

$$\begin{aligned} \begin{array}{cccc} (0_{0}, 0_{2}, 0_{10}, 1_{3}, 3_{16}), &{} (0_{0}, 1_{10}, 1_{0}, 2_{16}, 2_{23}),&{} (0_{0}, 1_{0}, 1_{4}, 2_{22}, 3_{17}), &{} (0_{0}, 3_{22}, 2_{23}, 1_{17}, 1_{12}),\\ (0_{0}, 1_{11}, 3_{16}, 0_{19}, 3_{15}), &{} (0_{0}, 3_{2}, 3_{15}, 0_{16}, 3_{19}),&{} (0_{0}, 1_{13}, 1_{12}, 0_{22}, 2_{8}), &{} (0_{0}, 2_{22}, 1_{7}, 3_{3}, 1_{20}). \end{array} \end{aligned}$$

The 16 base blocks are obtained by multiplying each of the above 8 base blocks by 1 and \(-1\).

Appendix: TWhFrames in the proof of Lemma 10

A strictly \(({\mathbb {Z}}_{8} \oplus {\mathbb {Z}}_{4} \oplus {\mathbb {Z}}_2, 4{\mathbb {Z}}_{8} \oplus 2{\mathbb {Z}}_4 \oplus {\mathbb {Z}}_2)\)-invariant TWhFrame:

$$\begin{aligned} \begin{array}{ccc} ((6,1,0), (5,0,0), (4,1,1), (7,2,1)),&{} ((2,3,0), (3,0,1), (2,0,1), (0,3,0)),\\ ((2,1,0), (3,3,1), (5,1,1), (7,0,1)),&{} ((6,2,1), (1,2,0), (3,1,1), (3,2,0)),\\ ((5,3,1), (1,0,1), (6,0,1), (3,3,0)),&{} ((2,0,0), (4,1,0), (1,2,1), (5,1,0)),\\ ((5,2,0), (2,3,1), (6,0,0), (7,3,1)),&{} ((1,3,1), (2,1,1), (2,2,1), (5,0,1)),\\ ((1,0,0), (3,2,1), (0,3,1), (7,3,0)),&{} ((7,0,0), (7,1,0), (6,1,1), (1,3,0)),\\ ((0,1,1), (6,2,0), (3,0,0), (6,3,0)),&{} ((4,3,0), (6,3,1), (7,1,1), (1,1,1)),\\ ((2,2,0), (7,2,0), (0,1,0), (1,1,0)),&{} ((3,1,0), (5,3,0), (5,2,1), (4,3,1)). \end{array} \end{aligned}$$

A strictly \(({\mathbb {Z}}_{16} \oplus {\mathbb {Z}}_{2}^2,8{\mathbb {Z}}_{16}\oplus {\mathbb {Z}}_2^2)\)-invariant TWhFrame:

$$\begin{aligned} \begin{array}{cc} ((9,1,1), (6,0,1), (2,0,1), (13,1,0)),&{} ((10,1,1), (1,1,1), (11,0,1), (12,0,0)),\\ ((4,1,1), (9,1,0), (15,1,0), (6,1,1)),&{} ((3,0,1), (5,0,0), (7,0,1), (10,1,0)),\\ ((9,0,1), (14,0,1), (3,1,0), (12,0,1)),&{} ((13,1,1), (1,0,1), (14,1,0), (10,0,1)),\\ ((7,1,0), (3,1,1), (4,0,0), (2,1,1)),&{} ((4,0,1), (15,1,1), (11,0,0), (13,0,0)),\\ ((7,0,0), (6,1,0), (1,0,0), (2,0,0)),&{} ((9,0,0), (11,1,0), (5,0,1), (15,0,1)),\\ ((14,1,1), (11,1,1), (12,1,0), (13,0,1)),&{} ((14,0,0), (5,1,0), (3,0,0), (15,0,0)),\\ ((6,0,0), (12,1,1), (5,1,1), (2,1,0)),&{} ((10,0,0), (4,1,0), (7,1,1), (1,1,0)). \end{array} \end{aligned}$$

A strictly \(({\mathbb {Z}}_{54} \oplus {\mathbb {Z}}_{2},9{\mathbb {Z}}_{54} \oplus {\mathbb {Z}}_2)\)-invariant TWhFrame:

$$\begin{aligned} \begin{array}{ccccc} (17_0,11_0,7_0,39_0),&{} (28_1,12_0,2_0,22_1),&{} (21_0,49_0,20_1,35_0),&{} (33_0,37_0,44_0,25_0),&{} (23_1,28_0,52_1,49_1),\\ (43_0,40_1,19_0,5_1),&{} (2_1,42_1,31_0,41_1),&{} (42_0,35_1,20_0,19_1),&{} (31_1,14_0,51_1,10_1),&{} (53_1,15_1,10_0,12_1),\\ (39_1,51_0,43_1,4_1),&{} (23_0,1_1,26_1,3_0),&{} (1_0,34_1,8_0,50_0),&{} (7_1,37_1,47_0,6_1),&{} (29_1,21_1,24_1,16_0),\\ (13_1,33_1,30_0,53_0),&{} (52_0,48_1,17_1,46_1),&{} (46_0,16_1,38_0,48_0),&{} (29_0,34_0,14_1,40_0),&{} (8_1,25_1,32_0,38_1),\\ (50_1,3_1,4_0,15_0),&{} (11_1,44_1,32_1,13_0),&{} (22_0,47_1,5_0,6_0),&{} (24_0,26_0,30_1,41_0). \end{array} \end{aligned}$$

A strictly \(({\mathbb {Z}}_{16} \oplus {\mathbb {Z}}_{8}, 4{\mathbb {Z}}_{16} \oplus 2{\mathbb {Z}}_8)\)-invariant TWhFrame:

$$\begin{aligned} \begin{array}{ccccc} (0_1,7_1,1_3,5_2),&{} (2_1,9_5,11_3,15_6),&{} (4_1,3_1,13_7,9_6),&{} (6_1,5_5,7_7,3_2),&{} (6_0,9_0,8_1,11_0),\\ (2_0,5_4,12_1,15_4),&{} (3_3,14_3,13_0,8_7),&{} (4_7,15_3,6_4,1_7),&{} (6_2,5_1,1_2,15_7),&{} (11_6,10_1,14_6,12_7),\\ (1_6,10_7,14_2,8_5),&{} (14_1,7_6,3_5,13_4),&{} (0_3,7_5,7_4,2_6),&{} (11_7,2_5,10_0,5_6),&{} (15_2,14_4,14_7,1_1),\\ (13_3,14_5,6_6,3_0),&{} (5_3,7_3,10_5,10_6),&{} (15_1,13_5,2_7,2_2),&{} (12_5,6_5,9_3,1_4),&{} (6_3,0_7,11_1,3_6),\\ (7_0,13_2,14_0,9_1),&{} (9_2,3_4,10_2,15_5),&{} (9_7,11_5,10_3,7_2),&{} (4_5,6_7,13_1,10_4),&{} (1_0,0_5,1_5,15_0),\\ (11_4,12_3,3_7,5_0),&{} (13_6,4_3,5_7,11_2),&{} (2_3,9_4,2_4,8_3). \end{array} \end{aligned}$$

A strictly \(({\mathbb {Z}}_{32} \oplus {\mathbb {Z}}_{4}, 8{\mathbb {Z}}_{32} \oplus {\mathbb {Z}}_4)\)-invariant TWhFrame:

$$\begin{aligned} \begin{array}{ccccc} (17_1,30_1,11_3,5_1),&{} (27_2,14_0,17_0,23_0),&{} (31_2,2_2,21_2,11_0),&{} (19_2,22_0,25_2,15_2),&{} (9_2,5_0,6_2,23_3),\\ (15_1,19_1,2_1,17_2),&{} (3_3,23_1,22_1,21_0),&{} (23_2,11_2,26_0,25_1),&{} (19_0,28_2,22_3,7_1),&{} (14_3,5_3,27_0,10_0),\\ (18_1,25_3,31_0,30_2),&{} (9_3,2_3,12_0,13_0),&{} (19_3,17_3,10_2,21_1),&{} (27_1,29_3,20_2,9_1),&{} (13_3,27_3,20_0,15_3),\\ (15_0,29_2,6_1,1_2),&{} (31_1,10_3,1_1,4_2),&{} (18_0,29_0,4_0,7_3),&{} (30_3,3_1,12_1,25_0),&{} (28_3,1_3,26_1,7_2),\\ (9_0,29_1,30_0,28_1),&{} (1_0,21_3,6_0,4_3),&{} (7_0,11_1,12_2,26_3),&{} (31_3,3_2,20_1,2_0),&{} (12_3,18_2,13_2,6_3),\\ (20_3,14_2,3_0,10_1),&{} (28_0,18_3,13_1,22_2),&{} (4_1,26_2,5_2,14_1). \end{array} \end{aligned}$$

A strictly \(({\mathbb {Z}}_{64} \oplus {\mathbb {Z}}_{2}, 8{\mathbb {Z}}_{64} \oplus {\mathbb {Z}}_2)\)-invariant TWhFrame:

$$\begin{aligned} \begin{array}{ccccc} (50_1,21_0,41_0,47_0),&{} (34_1,37_0,25_1,63_1),&{} (33_1,62_1,10_1,4_0),&{} (23_1,26_1,46_0,20_1),&{} (7_0,18_1,19_0,60_1),\\ (6_0,27_1,58_1,49_0),&{} (42_1,53_1,22_0,63_0),&{} (1_0,44_0,45_0,54_0),&{} (23_0,51_0,9_0,36_1),&{} (9_1,5_1,59_0,54_1),\\ (58_0,22_1,12_1,39_1),&{} (1_1,61_0,19_1,14_1),&{} (39_0,51_1,12_0,30_0),&{} (11_1,31_0,38_0,52_0),&{} (2_0,14_0,7_1,25_0),\\ (55_0,35_0,60_0,46_1),&{} (13_1,20_0,47_1,17_1),&{} (13_0,38_1,43_1,41_1),&{} (43_0,50_0,45_1,15_0),&{} (30_1,55_1,28_1,26_0),\\ (17_0,34_0,36_0,3_0),&{} (29_1,44_1,10_0,11_0),&{} (15_1,62_0,28_0,61_1),&{} (18_0,3_1,5_0,4_1),&{} (29_0,42_0,57_1,35_1),\\ (21_1,2_1,49_1,59_1),&{} (57_0,6_1,53_0,31_1),&{} (33_0,52_1,37_1,27_0). \end{array} \end{aligned}$$

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Pan, R., Abel, R.J.R., Bunjamin, Y.A. et al. Difference matrices with five rows over finite abelian groups. Des. Codes Cryptogr. 90, 367–386 (2022). https://doi.org/10.1007/s10623-021-00981-6

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