Quasi-symmetric designs on $ 56 $ points

Vedran Krčadinac; Renata Vlahović Kruc

doi:10.3934/amc.2020086

Article Contents

2021, Volume 15, Issue 4: 633-646. Doi: 10.3934/amc.2020086

This issue Previous Article An overview on skew constacyclic codes and their subclass of LCD codes Next Article Correlation distribution of a sequence family generalizing some sequences of Trachtenberg

Quasi-symmetric designs on $ 56 $ points

Vedran Krčadinac^, and
Renata Vlahović Kruc

Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, HR-10000 Zagreb, Croatia

^* Corresponding author: V. Krčadinac
^* Corresponding author: V. Krčadinac

Received: November 2019

Revised: February 2020

Early access: June 2020

Published: November 2021

This work has been fully supported by the Croatian Science Foundation under the project 6732

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Abstract

Computational techniques for the construction of quasi-symmetric block designs are explored and applied to the case with $ 56 $ points. One new $ (56,16,18) $ and many new $ (56,16,6) $ designs are discovered, and non-existence of $ (56,12,9) $ and $ (56,20,19) $ designs with certain automorphism groups is proved. The number of known symmetric $ (78,22,6) $ designs is also significantly increased.

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Mathematics Subject Classification: 05B05, 94B25, 05C69.

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Table 1. The known QSDs with $ v = 56 $

No.	$ v $	$ k $	$ \lambda $	$ r $	$ b $	$ x $	$ y $	NQSD	Ref.
47	56	16	18	66	231	4	8	$ \ge 3 $	[13]
48	56	15	42	165	616	3	6	0	[5]
49	56	12	9	45	210	0	3	?
50	56	21	24	66	176	6	9	0	[5]
51	56	20	19	55	154	5	8	?
52	56	16	6	22	77	4	6	$ \ge 2 $	[24,16]

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Table 3. Dimensions and weight distributions of self-orthogonal binary codes spanned by $ (56,16,6) $ QSDs

	$ \dim $	$ a_0 $	$ a_8 $	$ a_{12} $	$ a_{16} $	$ a_{20} $	$ a_{24} $	$ a_{28} $
$ C_1 $	26	1	91	2 016	152 425	2 939 776	16 194 619	28 531 008
$ C_2 $	26	1	7	$ 2\,016 $	$ 155\,365 $	$ 2\,926\,336 $	$ 16\,224\,019 $	$ 28\,493\,376 $
$ C_3 $	24	1	75	0	40 089	730 368	4 055 835	7 124 480
$ C_{4\hbox{-}6,9,10} $	22	1	15	0	9 933	183 168	1 012 515	1 783 040
$ C_{7,11\hbox{-}13} $	25	1	75	672	77 721	1 465 984	8 103 963	14 257 600
$ C_8 $	25	1	75	960	75 417	1 474 048	8 087 835	14 277 760
$ C_{14} $	22	1	15	0	10 701	178 560	1 024 035	1 767 680
$ C_{15} $	23	1	15	288	19 917	361 216	2 040 867	3 544 000
$ C_{16} $	23	1	15	96	19 917	365 056	2 028 579	3 561 280
$ C_{17} $	24	1	75	160	39 833	728 704	4 062 235	7 115 200
$ C_{18} $	22	1	15	64	9 677	183 424	1 012 771	1 782 400
$ C_{19,21,24} $	22	1	15	16	10 061	182 080	1 015 459	1 779 040
$ C_{20,22} $	22	1	15	64	10 445	178 816	1 024 291	1 767 040
$ C_{23} $	25	1	75	1 280	74 905	1 470 720	8 100 635	14 259 200
$ C_{25} $	25	1	75	992	77 209	1 462 656	8 116 763	14 239 040
$ C_{26} $	27	1	139	4 992	307 161	5 848 832	32 477 083	56 941 312
$ C_{27} $	27	1	99	4 304	305 873	5 872 320	32 406 731	57 039 072
$ C_{28,29} $	27	1	99	4 112	307 409	5 866 944	32 417 483	57 025 632
$ C_{30} $	26	1	147	1 008	158 529	2 920 512	16 231 467	28 485 536
$ C_{31,32,34,35} $	27	1	147	3 696	309 057	5 862 976	32 423 979	57 018 016
$ C_{33,39} $	27	1	147	4 976	307 009	5 849 664	32 475 179	56 943 776
$ C_{36} $	26	1	75	2 240	153 241	2 931 200	16 218 395	28 498 560
$ C_{37,38} $	27	1	75	4 416	305 817	5 871 616	32 408 859	57 036 160

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Table 2. Dimensions and weight distributions of self-orthogonal binary codes spanned by $ (56,16,18) $ QSDs

	$ \dim $	$ a_0 $	$ a_8 $	$ a_{12} $	$ a_{16} $	$ a_{20} $	$ a_{24} $	$ a_{28} $
$ C_{1,2} $	23	1	75	0	21 657	353 536	2 059 035	3 520 000
$ C_3 $	19	1	0	0	1 722	19 936	134 085	212 800
$ C_4 $	23	1	15	216	20 493	359 200	2 044 899	3 538 960

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Table 4. Distribution of the known $ (56,16,6) $ QSDs and symmetric $ (78,22,6) $ designs by full automorphism group order

$ \| \mathrm{Aut}\| $	#$ (56,16,6) $	#$ (78,22,6) $
$ 168 $	$ 1 $	$ 2 $
$ 78 $	$ 0 $	$ 1 $
$ 48 $	$ 876 $	$ 1664 $
$ 24 $	$ 1 $	$ 378 $
$ 21 $	$ 1 $	$ 2 $
$ 16 $	$ 228 $	$ 456 $
$ 12 $	$ 303 $	$ 606 $
$ 6 $	$ 0 $	$ 32 $

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Table 5. An updated table of QSDs with $ v = 56 $

No.	$ v $	$ k $	$ \lambda $	$ r $	$ b $	$ x $	$ y $	NQSD
47	56	16	18	66	231	4	8	$ \ge 4 $
48	56	15	42	165	616	3	6	0
49	56	12	9	45	210	0	3	?
50	56	21	24	66	176	6	9	0
51	56	20	19	55	154	5	8	?
52	56	16	6	22	77	4	6	$ \ge 1410 $

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