Self-orthogonal codes from equitable partitions of distance-regular graphs

Dean Crnković; Sanja Rukavina; Andrea Švob

doi:10.3934/amc.2022014

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2024, Volume 18, Issue 3: 651-660. Doi: 10.3934/amc.2022014

This issue Previous Article Parameterization of Boolean functions by vectorial functions and associated constructions Next Article On the polycyclic codes over $ \mathbb{F}_q+u\mathbb{F}_q $

Self-orthogonal codes from equitable partitions of distance-regular graphs

Department of Mathematics, University of Rijeka, Croatia

^* Corresponding author: Dean Crnković
^* Corresponding author: Dean Crnković

Received: November 2021

Early access: March 2022

Published: June 2024

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

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Abstract

We give two methods for a construction of self-orthogonal linear codes from equitable partitions of distance-regular graphs. By applying these methods, we construct self-orthogonal codes from equitable partitions of the graph of unitals in $ PG(2,4) $ and the only known strongly regular graph with parameters $ (216,40,4,8) $. Some of the codes obtained are optimal.

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Mathematics Subject Classification: Primary: 05E30, 94B05; Secondary: 05E18, 94B60.

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Table 1. The coefficients $ p_{i,i}^k $ of adjacency matrices for the graph $ \Gamma_{280} $

	$ A_0 $	$ A_1 $	$ A_2 $	$ A_3 $	$ A_4 $
$ A_0 \cdot A_0 $	1	0	0	0	0
$ A_1 \cdot A_1 $	9	0	1	0	0
$ A_2 \cdot A_2 $	72	16	21	15	24
$ A_3 \cdot A_3 $	144	48	84	68	80
$ A_4 \cdot A_4 $	54	6	12	9	12

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Table 2. Self-orthogonal codes from the matrices $ M_i^H $ for $ \Gamma_{280} $

i	$ H\leq P \Gamma L(3,4).Z_2 $	orbit lengths	code	construction
$ 3 $	$ I $	$ 280\times 1 $	$ [280,18,112]_2 $	Cor. 1
$ 3 $	$ Z_3 $	$ 10\times 1,90\times 3 $	$ [100,10,40]_2 $	Cor. 1
$ 3 $	$ Z_3 $	$ 7\times 1,91\times 3 $	$ [98,6,48]_2* $	Cor. 1
$ 3 $	$ Z_3 $	$ 1\times 1,93\times 3 $	$ [94,6,38]_2 $	Cor. 1
$ 3 $	$ Z_5 $	$ 56\times 5 $	$ [56,2,24]_2 $	Cor. 1
$ 3 $	$ E_9 $	$ 1\times 1,9\times 3,28\times 9 $	$ [38,6,16]_2 $	Cor. 1
$ 3 $	$ E_9 $	$ 1\times 1,6\times 3,29\times 9 $	$ [36,2,18]_2 $	Cor. 1
$ 3 $	$ E_9 $	$ 1\times 1,31\times9 $	$ [32,2,16]_2 $	Cor. 1
$ 3 $	$ Z_{15} $	$ 2\times 5,18\times 15 $	$ [20,2,8]_2 $	Cor. 1
$ 3 $	$ E_9:Z_3 $	$ 1\times 1,7\times 9,8\times 27 $	$ [16,2,8]_2 $	Cor. 1
$ 3 $	$ Z_2 $	$ 28\times 1,126\times 2 $	$ [28,6,12]_2* $, $ [126,6,60]_2 $	Thm. 2.5
$ 3 $	$ Z_2 $	$ 10\times 1,135\times 2 $	$ [135,9,56]_2 $	Thm. 2.5
$ 3 $	$ Z_2 $	$ 10\times 1,135\times 2 $	$ [10,6,2]_2 $, $ [135,12,5]_2 $	Thm. 2.5
$ 3 $	$ Z_2 $	$ 8\times 1,136\times 2 $	$ [136,8,56]_2 $	Thm. 2.5
$ 3 $	$ Z_4 $	$ 4\times 1,2\times 2,68\times 4 $, $ 4\times 2,68\times 4 $	$ [68,4,28]_2 $	Thm. 2.5
$ 3 $	$ Z_4 $	$ 6\times 1,1\times 2,68\times 4 $	$ [68,4,32]_2 $	Thm. 2.5
$ 3 $	$ E_4 $	$ 4\times 1,18\times 2,60\times 4 $	$ [18,3,6]_2 $ $ [60,3,30]_2 $	Thm. 2.5
$ 3 $	$ E_4 $	$ 4\times 1,26\times 2,56\times 4 $	$ [26,4,8]_2 $, $ [56,2,32]_2 $	Thm. 2.5
$ 3 $	$ E_4 $	$ 12\times 2,64\times 4 $	$ [64,2,28]_2 $	Thm. 2.5
$ 3 $	$ E_4 $	$ 12\times 2,64\times 4 $	$ [64,2,40]_2 $	Thm. 2.5
$ 3 $	$ E_4 $	$ 2\times 1,11\times 2,64\times 4 $	$ [64,4,30]_2 $	Thm. 2.5
$ 4 $	$ Z_7 $	$ 40\times 7 $	$ [40,12,12]_3 $	Thm. 2.2
$ 4 $	$ Z_3 $	$ 1\times 1,93\times 3 $	$ [93,27,15]_3 $	Thm. 2.5
$ 4 $	$ Z_3 $	$ 7\times 1,91\times 3 $	$ [91,28,18]_3 $	Thm. 2.5
$ 4 $	$ Z_3 $	$ 10\times 1,90\times 3 $	$ [90,25,18]_3 $	Thm. 2.5
$ 4 $	$ E_9 $	$ 1\times 1,31\times 9 $	$ [31,8,6]_3 $	Thm. 2.5
$ 4 $	$ E_9 $	$ 1\times 1,6\times 3,29\times 9 $	$ [29,9,6]_3 $	Thm. 2.5
$ 4 $	$ E_9 $	$ 1\times 1,9\times 3,28\times 9 $	$ [28,7,6]_3 $	Thm. 2.5

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Table 3. The coefficients $ p_{i,i}^k $ of adjacency matrices for the graph $ \Gamma_{216} $

	$ A_0 $	$ A_1 $	$ A_2 $
$ A_0 \cdot A_0 $	1	0	0
$ A_1 \cdot A_1 $	40	4	8
$ A_2 \cdot A_2 $	175	140	142

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Table 4. Binary self-orthogonal codes from the matrices $ M_1^H $ for the graph $ \Gamma_{216} $, Cor. 1

$ H\leq Aut(\Gamma_{216}) $	orbit lengths	code
$ I $	$ 216\times 1 $	$ [ 216,60,32 ] $
$ Z_3 $	$ 72\times 3 $	$ [ 72, 18, 16 ] $
$ Z_3 $	$ 72\times 3 $	$ [ 72, 16, 16 ] $
$ Z_3 $	$ 9\times 1, 69\times 3 $	$ [ 78, 20, 16 ] $
$ Z_5 $	$ 1\times 1, 43\times 5 $	$ [ 44, 12, 8 ] $
$ E_9 $	$ 6\times 3, 22\times 9 $	$ [ 28, 4, 8 ] $
$ E_9 $	$ 9\times 3, 21\times 9 $	$ [ 30, 6, 8 ] $
$ E_9 $	$ 3\times 3, 23\times 9 $	$ [ 26, 4, 8 ] $
$ Z_9 $	$ 24\times 9 $	$ [ 24, 6, 8 ] $

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Table 5. Binary self-orthogonal codes from the matrices $ M_1^H $ for the graph $ \Gamma_{216} $, Thm. 2.5

$ H\leq Aut(\Gamma_{216}) $	orbit lengths	code
$ Z_2 $	$ 6\times 1,105\times 2 $	$ [ 105, 26, 20 ] $
$ Z_2 $	$ 48 \times 1, 84\times 2 $	$ [ 48, 12, 12 ], [ 84, 24, 16 ] $
$ Z_2 $	$ 12\times 1,102\times 2 $	$ [ 102, 28, 16 ] $
$ Z_2 $	$ 10\times 1,103\times 2 $	$ [ 103, 30, 16 ] $
$ Z_4 $	$ 24\times 2, 42\times 4 $	$ [ 24, 4, 8 ], [ 42, 12, 8 ] $
$ Z_4 $	$ 6\times 2, 51\times 4 $	$ [ 51, 14, 8 ] $
$ E_4 $	$ 18\times 2, 45\times 4 $	$ [ 45, 10, 8 ] $
$ E_4 $	$ 12\times 1, 36\times 2, 33\times 4 $	$ [ 36, 8, 8 ], [ 33, 10, 8 ] $
$ E_4 $	$ 12\times 2, 48\times 4 $	$ [ 48, 12, 12 ] $
$ E_4 $	$ 36\times 2, 36\times 4 $	$ [ 36, 10, 8 ], [ 36, 4, 8 ] $
$ E_4 $	$ 4\times 1, 26\times 2, 40 \times 4 $	$ [ 26, 6, 6 ], [ 40, 10, 12 ] $
$ E_4 $	$ 2\times 1, 15\times 2, 46\times 4 $	$ [ 46, 12, 8 ] $
$ Z_4 $	$ 4\times 1, 4\times 2, 51\times 4 $	$ [ 51, 14, 8 ] $
$ E_4 $	$ 4\times 1, 10\times 2, 48\times 4 $	$ [ 48, 12, 12 ] $
$ Z_4 $	$ 2\times 1, 5\times 2, 51\times 4 $	$ [ 51, 14, 8 ] $
$ E_4 $	$ 2\times 1, 11\times 2, 48\times 4 $	$ [ 48, 12, 12 ] $
$ E_4 $	$ 2\times 1, 13\times 2, 47\times 4 $	$ [ 47, 14, 8 ] $
$ E_4 $	$ 4\times 1, 28\times 2, 39\times 4 $	$ [ 39, 12, 8 ],[ 28, 6, 6 ] $
$ S_3 $	$ 6\times 3, 33\times 6 $	$ [ 33, 4, 12 ] $
$ Z_6 $	$ 16\times 3, 28\times 6 $	$ [ 28, 8, 8 ] $
$ Z_6 $	$ 2\times 3, 35\times 6 $	$ [ 35, 6, 8 ] $
$ S_3 $	$ 10\times 3, 31\times 6 $	$ [ 31, 8, 8 ] $
$ S_3 $	$ 12\times 3, 30\times 6 $	$ [ 30, 6, 8 ] $
$ Z_6 $	$ 16\times 3, 28\times 6 $	$ [ 28, 6, 8 ], [ 16, 4, 8 ] * $
$ S_3 $	$ 10\times 3, 31\times 6 $	$ [ 31, 8, 8 ] $
$ Z_6 $	$ 4\times3, 34\times 6 $	$ [ 34, 8, 8 ] $
$ S_3 $	$ 10\times 3, 31\times 6 $	$ [ 31, 9, 8 ] $
$ Q_8 $	$ 12\times 4, 21\times 8 $	$ [ 21, 4, 8 ] $
$ E_8 $	$ 4\times 2, 24\times 4, 14\times 8 $	$ [ 14, 4, 4 ] $
$ D_8 $	$ 6\times 2, 21\times 4, 15\times 8 $	$ [ 15, 4, 8 ]*, [ 21, 4, 8 ] $
$ E_8 $	$ 18\times 2, 21\times 4, 12\times 8 $	$ [ 12, 4, 4 ] $
$ Z_4\times Z_2 $	$ 18\times 4, 18\times 8 $	$ [ 18, 4, 4 ] $
$ E_8 $	$ 4\times 2, 10\times 4, 21\times 8 $	$ [ 21, 6, 8 ]* $
$ E_8 $	$ 6\times 2, 9\times 4, 21\times 8 $	$ [ 21, 4, 8 ] $
$ Z_4\times Z_2 $	$ 2\times 2, 17\times 4, 18\times 8 $	$ [ 18, 4, 8 ] * $
$ D_8 $	$ 2\times 2, 11\times 4, 21\times 8 $	$ [ 21, 4, 8 ] $
$ E_8 $	$ 8\times 2, 14\times 4, 18\times 8 $	$ [ 18, 4, 8 ] * $
$ Z_4\times Z_2 $	$ 4\times 2, 16\times 4, 18\times 8 $	$ [ 18, 4, 8 ]* $
$ D_8 $	$ 4\times 1, 4\times 2, 21\times 4, 15\times 8 $	$ [ 21, 4, 8 ] $
$ D_8 $	$ 24\times 4, 15\times 8 $	$ [ 15, 4, 4 ] $
$ E_8 $	$ 10\times 2, 13\times 4, 18\times 8 $	$ [ 18, 4, 8 ]* $
$ Z_4\times Z_2 $	$ 2\times 2, 17\times 4, 18\times 8 $	$ [ 18, 4, 4 ] $
$ D_8 $	$ 2\times 2, 9\times 4, 22\times 8 $	$ [ 22, 6, 8 ] $
$ D_8 $	$ 4\times 2, 10\times 4, 21\times 8 $	$ [ 21, 4, 8 ] $
$ D_8 $	$ 2\times 2, 11\times 4, 21\times 8 $	$ [ 21, 6, 8 ]* $
$ Z_8 $	$ 12\times 4, 21\times 8 $	$ [ 21, 6, 4 ] $
$ D_8 $	$ 2\times 1, 5\times 2, 23\times 4, 14\times 8 $	$ [ 23, 4, 8 ] $
$ E_8 $	$ 2\times 1, 9\times 2, 19\times 4, 15\times 8 $	$ [ 19, 4, 4 ] $
$ Z_4\times Z_2 $	$ 2\times 2, 5\times 4, 24\times 8 $	$ [ 24, 6, 8 ] $
$ D_8 $	$ 2\times 2, 13\times 4, 20\times 8 $	$ [ 20, 4, 4 ] $
$ D_8 $	$ 4\times 2, 16\times 4, 18\times 8 $	$ [ 18, 6, 4 ] $
$ Z_6\times Z_2 $	$ 4\times 3, 12\times 6, 11\times 12 $	$ [ 12, 4, 4 ] $
$ Z_{12} $	$ 8\times 6, 14\times 12 $	$ [ 14, 4, 4 ] $
$ D_{12} $	$ 2\times 3, 9\times 6, 13\times 12 $	$ [ 13, 4, 4 ] $
$ Z_{12} $	$ 2\times 6, 17\times 12 $	$ [ 17, 4, 4 ] $
$ Z_3:Z_4 $	$ 2\times 3, 1\times 6, 17\times 12 $	$ [ 17, 4, 4 ] $
$ D_{12} $	$ 4\times 3, 12\times 6, 11\times 12 $	$ [ 11, 4, 4 ] $
$ D_{12} $	$ 4\times 3, 12\times 6, 11\times 12 $	$ [ 11, 3, 4 ] $

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