Parameters of hulls of primitive binary narrow-sense BCH codes and their subcodes

Binkai Gong; Chengju Li

doi:10.3934/amc.2022057

Article Contents

2022, Volume 16, Issue 4: 741-752. Doi: 10.3934/amc.2022057

This issue Previous Article Estimate of 4-adic complexity of unified quaternary sequences of length $ 2p $ Next Article Infinite families of $ t $-designs and strongly regular graphs from punctured codes

Parameters of hulls of primitive binary narrow-sense BCH codes and their subcodes

Binkai Gong^1,2, and
Chengju Li^1,2, ,

1.
Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China
2.
National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China

^*Corresponding author: Chengju Li
^*Corresponding author: Chengju Li

Dedicated to the 60th Birthday of Prof. Cunsheng Ding

Received: March 2022

Revised: May 2022

Early access: July 2022

Published: November 2022

The work was supported by the National Natural Science Foundation of China (12071138), the Shanghai Natural Science Foundation (22ZR1419600), the open research fund of National Mobile Communications Research Laboratory (2022D05), and the Shanghai Trusted Industry Internet Software Collaborative Innovation Center

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Abstract

The (Euclidean) hull of a linear code is defined to be the intersection of the code and its Euclidean dual. It is clear that the hulls are self-orthogonal codes, which are an important type of linear codes due to their wide applications in communication and cryptography. Let $ \mathcal C_{(2,n,\delta)} $ be the binary primitive narrow-sense BCH code, where $ n = 2^m-1 $ and $ m $ is a positive integer. In this paper, we will investigate the parameters of the hulls of $ \mathcal C_{(2,n,\delta)} $. The dimension of $ \text{Hull}(\mathcal C_{(2,n,\delta)}) $ will be presented when $ 2 \le \delta \le 2^{\lfloor \frac{m}{2} \rfloor+1}+2^{\lceil \frac{m}{2} \rceil-1}-1 $ with $ m \ge 5 $. Furthermore, we give an improvement on lower bounds of the minimum distances of $ \text{Hull}(\mathcal C_{(2,n,\delta)}) $. We also construct a self-orthogonal subcode of $ \text{Hull}(\mathcal C_{(2,n,\delta)}) $ and investigate the parameters of the self-orthogonal code.

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Mathematics Subject Classification: Primary: 94B05, 11T71; Secondary: 94B15.

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Table 1. Hulls with good parameters

Hull	$ [n,k^H,d^H] $	Optimality
Hull$ (\mathcal C_{(2,31,2)}) $	$ [31,5,16] $	Yes
Hull$ (\mathcal C_{(2,31,4)}) $	$ [31,10,12] $	Yes
Hull$ (\mathcal C_{(2,31,6)}) $	$ [31,15,8] $	Yes
Hull$ (\mathcal C_{(2,31,8)}) $	$ [31,10,12] $	Yes
Hull$ (\mathcal C_{(2,31,12)}) $	$ [31,5,16] $	Yes
Hull$ (\mathcal C_{(2,63,2)}) $	$ [63,6,32] $	Yes
Hull$ (\mathcal C_{(2,63,4)}) $	$ [63,12,24] $	Best known
Hull$ (\mathcal C_{(2,63,12)}) $	$ [63,27,16] $	Best known
Hull$ (\mathcal C_{(2,63,16)}) $	$ [63,15,24] $	Yes
Hull$ (\mathcal C_{(2,63,24)}) $	$ [63,9,28] $	Yes
Hull$ (\mathcal C_{(2,63,28)}) $	$ [63,6,32] $	Yes

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Table 2. Lower bounds on the minimum distances of Hull$ (\mathcal{C}_{(2,n,\delta)}) $

Hull	$ [n,k,d] $	Optimality	$ d^H \ge $	$ (d\ge, d^\bot \ge) $
Hull$ (\mathcal C_{(2,31,2)}) $	$ [31,5,16] $	Yes	$ 16 $	$ (2, 16) $
Hull$ (\mathcal C_{(2,31,4)}) $	$ [31,10,12] $	Yes	$ 8 $	$ (4, 8) $
Hull$ (\mathcal C_{(2,31,6)}) $	$ [31,15,8] $	Yes	$ 8 $	$ (6, 8) $
Hull$ (\mathcal C_{(2,63,2)}) $	$ [63,6,32] $	Yes	$ 32 $	$ (2, 32) $
Hull$ (\mathcal C_{(2,63,4)}) $	$ [63,12,24] $	Best known	$ 16 $	$ (4, 16) $
Hull$ (\mathcal C_{(2,63,8)}) $	$ [63,18,16] $	No	$ 15 $	$ (8, 8) $
Hull$ (\mathcal C_{(2,63,10)}) $	$ [63,21,16] $	No	$ 15 $	$ (10, 8) $
Hull$ (\mathcal C_{(2,63,12)}) $	$ [63,27,16] $	Best known	$ 14 $	$ (12, 8) $

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Table 3. The parameters of $ \mathcal{C'} $ with $ m = 5,6 $

$ m $	$ \delta $	$ k $	$ d $	Optimality of $ \mathcal{C'} $
$ 5 $	$ 2\sim3 $	$ 0 $	$ 31 $	/
	$ 4\sim5 $	$ 5 $	$ 16 $	Yes
	$ 6\sim11 $	$ 10 $	$ 12 $	Yes
	$ 12\sim15 $	$ 5 $	$ 16 $	Yes
	$ 16\sim31 $	$ 0 $	$ 31 $	/
$ 6 $	$ 2\sim3 $	$ 0 $	$ 63 $	/
	$ 4\sim5 $	$ 6 $	$ 32 $	Yes
	$ 6\sim7 $	$ 12 $	$ 24 $	Best known
	$ 8\sim9 $	$ 18 $	$ 16 $	No
	$ 10\sim11 $	$ 21 $	$ 16 $	No
	$ 12\sim13 $	$ 27 $	$ 16 $	Best known
	$ 14\sim15 $	$ 21 $	$ 16 $	No
	$ 16\sim23 $	$ 15 $	$ 24 $	Yes
	$ 24\sim27 $	$ 9 $	$ 28 $	Yes
	$ 28\sim31 $	$ 6 $	$ 32 $	Yes
	$ 32\sim63 $	$ 0 $	$ 63 $	/

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