New bound and constructions for geometric orthogonal codes and geometric 180-rotating orthogonal codes

Xiaowei Su; Lidong Wang; Zihong Tian

doi:10.3934/amc.2022078

Article Contents

2022, Volume 16, Issue 4: 961-983. Doi: 10.3934/amc.2022078

This issue Previous Article Some new quantum MDS codes derived from generalized Reed-Solomon codes Next Article on the 2-adic complexity of cyclotomic binary sequences of order three

New bound and constructions for geometric orthogonal codes and geometric 180-rotating orthogonal codes

1.
School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, China
2.
School of Intelligence Policing, China People's Police University, Langfang 065000, China

^*Corresponding author: Zihong Tian
^*Corresponding author: Zihong Tian

Received: March 2022

Revised: July 2022

Early access: September 2022

Published: November 2022

The first author is supported by IPFHB under Grant CXZZBS2021067. The second author is supported by the Key R & D Program of Guangdong province under Grant 2020B030304002 and NSFC under Grant 11901210. The last author is supported by NSFC under Grant 11871019

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Abstract

Geometric orthogonal codes (GOCs) and geometric 180-rotating orthogonal codes (RGOCs) were introduced by Doty and Winslow for their application in DNA origami. So far, only a few classes of GOCs and one class of RGOCs are known. In this paper, we provide the combinatorial descriptions, present the upper bounds for the number of codewords and establish some recursive constructions for GOCs and RGOCs, respectively. As a consequence, the number of codewords in an optimal $ (n\times m, k, k-1) $-GOC and $ (n\times m, k, $ $ k-1) $-RGOC are determined for any positive integers $ n, m $ and $ k $. Some infinite families of $ (n\times m, k, \lambda) $-GOCs and $ (n\times m, k, \lambda) $-RGOCs for $ \lambda<k-1 $ are also obtained.

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Mathematics Subject Classification: Primary: 05B10, 05B40; Secondary: 94C30.

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Table 1. Some existential results of $ (n\times m, k, \lambda) $-GOCs

Parameters	Conditions	Size	Reference
$ (n \times m, 3,1)$	$n\equiv0\pmod {3}, m\equiv3, 6\pmod {12}$ or $m\equiv0\pmod {3}, n\equiv3, 6\pmod {12}$ $(n\times m, 3, 1)$ or $m\equiv0\pmod {3}, n\equiv3, 6\pmod {12}$ or $m\equiv0\pmod {3}, n\equiv3, 6\pmod {12}$	$\frac{2nm-n-m}{3}-1$ (optimal)	[5,30]
$ (n \times m, 3,1)$	otherwise	$\lfloor\frac{2nm-n-m}{3}\rfloor$ (optimal)	[5,30]
$(n\times m, 3, 2)$		$\Phi(n\times m, 3, 2)$ (optimal)	Theorem 3.5
$(n\times m, 4, 1)$	$n, m \equiv1\pmod {6}, n, m\leq6001$ and $n, m\neq13, 19$	$\frac{2nm-n-m}{6}$ (optimal)	Lemma 4.7
$(n\times m, 4, 1)$	$n \equiv1\pmod {6}, n\leq6001, n\neq13, 19$ and $m\geq4, m\not\equiv2\pmod {4}$	$\frac{m(n-1)}{6}$	Corollary 4.19
$(2^x\times 2^y, 4, 2)$	$n\geq3$ and $n=x+y$	$2*J(2^{n}, 4, 2)$	Corollary 4.14
$(2^n\times m, 4, 2)$	$n\geq3$	$2m^2*J(2^n, 4, 2)$	Corollary 4.20
$(n\times m, 4, 3)$		$\Phi (n\times m, 4, 3)$ (optimal)	Theorem 3.5

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Table 2. Some existential results of $ (n\times m, k, \lambda) $-RGOCs

Parameters	Conditions	Size	Reference
$(n\times m, 3, 2)$		$\Psi(n\times m, 3, 2)$ (optimal)	Theorem 3.13
$(n\times m, 4, 2)$	$n \equiv1\pmod {6}, n\leq6001, n\neq13, 19$	$\frac{m^2(n-1)}{6}$	Corollary 4.23
$(n\times m, 4, 3)$		$\Psi(n\times m, 4, 3)$ (optimal)	Theorem 3.13

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