New quantum codes from metacirculant graphs via self-dual additive $\mathbb{F}_4$-codes

Padmapani Seneviratne; Martianus Frederic Ezerman

doi:10.3934/amc.2021073

Article Contents

2023, Volume 17, Issue 1: 288-297. Doi: 10.3934/amc.2021073

This issue Previous Article Constructions of optimal rank-metric codes from automorphisms of rational function fields Next Article

New quantum codes from metacirculant graphs via self-dual additive $\mathbb{F}_4$-codes

Padmapani Seneviratne^1, and
Martianus Frederic Ezerman^2, ,

1.
Department of Mathematics, Texas A&M University-Commerce, 2600 South Neal Street, Commerce TX 75428, USA
2.
School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371

^* Corresponding author: Martianus Frederic Ezerman
^* Corresponding author: Martianus Frederic Ezerman

Received: May 2021

Early access: January 2022

Published: December 2022

Nanyang Technological University Grant Number 04INS000047C230GRT01 supports the research carried out by M. F. Ezerman

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Abstract

We use symplectic self-dual additive codes over $ \mathbb{F}_4 $ obtained from metacirculant graphs to construct, for the first time, $ \left[\kern-0.15em\left[ {\ell, 0, d} \right]\kern-0.15em\right] $ qubit codes with parameters $ (\ell,d) \in \{(78, 20), (90, 21), (91, 22), (93,21),(96,22)\} $. Secondary constructions applied to the qubit codes result in many new qubit codes that perform better than the previous best-known.

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Mathematics Subject Classification: Primary: 94B25, 81P73; Secondary: 05C75.

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Figure 1. The Petersen Graph as $ \Gamma(2,5,2,\{1,4\}, \{0\}) $

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Figure 2. $ G_{12}: = \Gamma(2, 6, 5, \{ 3 \},\{ 0, 3, 4, 5 \}) $ of the dodecacode $ \mathcal{D} $.

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Table 1. New codes from modifying $ Q_{78} $

Code	Parameters	Propagation rule
$ Q_{78, 1} $	$ \left[\kern-0.15em\left[ {77,0,19} \right]\kern-0.15em\right]_2 $	Puncture $ Q_{78} $ at $ \{78\} $
$ Q_{78,2} $	$ \left[\kern-0.15em\left[ {77,1,19} \right]\kern-0.15em\right]_2 $	Shorten $ Q_{78} $ at $ \{78\} $
$ Q_{78,3} $	$ \left[\kern-0.15em\left[ {78,1,19} \right]\kern-0.15em\right]_2 $	Lengthen $ Q_{78,2} $ by $ 1 $
$ Q_{78,4} $	$ \left[\kern-0.15em\left[ {76,2,18} \right]\kern-0.15em\right]_2 $	Shorten $ Q_{78} $ at $ \{77, 78\} $
$ Q_{78,5} $	$ \left[\kern-0.15em\left[ {76,1,18} \right]\kern-0.15em\right]_2 $	Subcode of $ Q_{78,4} $
$ Q_{78,6} $	$ \left[\kern-0.15em\left[ {77,2,18} \right]\kern-0.15em\right]_2 $	Lengthen $ Q_{78,4} $ by $ 1 $
$ Q_{78,7} $	$ \left[\kern-0.15em\left[ {75,3,17} \right]\kern-0.15em\right]_2 $	Shorten $ Q_{78} $ at $ \{76,77,78\} $
$ Q_{78,8} $	$ \left[\kern-0.15em\left[ {76,3,17} \right]\kern-0.15em\right]_2 $	Lengthen $ Q_{78,7} $ by $ 1 $
$ Q_{78,9} $	$ \left[\kern-0.15em\left[ {75, 2, 17} \right]\kern-0.15em\right]_2 $	Subcode of $ Q_{78,7} $

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Table 2. Properties of the Graphs

$ G $	$ d_{\rm min}(G) $	$ \nu(G) $	$ \gamma(G) $	$ \|{\rm Aut}(G)\| $	$ G $	$ d_{\rm min}(G) $	$ \nu(G) $	$ \gamma(G) $	$ \|{\rm Aut}(G)\| $
$ G_{12} $	$ 6 $	$ 5 $	$ 4 $	$ 24 $	$ G_{78} $	$ 20 $	$ 41 $	$ 7 $	$ 78 $
$ G_{27,1} $	$ 9 $	$ 16 $	$ 6 $	$ 27 $	$ G_{90} $	$ 21 $	$ 42 $	$ 7 $	$ 90 $
$ G_{27,2} $	$ 9 $	$ 10 $	$ 4 $	$ 27 $	$ G_{91} $	$ 22 $	$ 44 $	$ 7 $	$ 546 $
$ G_{36,1} $	$ 12 $	$ 13 $	$ 6 $	$ 72 $	$ G_{93} $	$ 22 $	$ 28 $	$ 4 $	$ 186 $
$ G_{36,2} $	$ 12 $	$ 13 $	$ 4 $	$ 72 $	$ G_{96} $	$ 22 $	$ 35 $	$ 6 $	$ 96 $

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