Hulls of $\mathbb Z_p\mathbb Z_p[v] $-cyclic codes and construction of EAQECCs

Awadhesh Kumar Shukla; Om Prakash Pandey; Vipul Mishra; Sachin Pathak; Ashish Kumar Upadhyay

doi:10.3934/amc.2025037

Article Contents

2026, Volume 20: 209-238. Doi: 10.3934/amc.2025037

This volume Previous Article Construction of wide-gap low-hit-zone frequency-hopping sequence sets with strictly optimal partial hamming correlation based on chinese remainder theorem Next Article A fast multiplication algorithm and RLWE–PLWE equivalence for the maximal real subfield of the $ 2^r p^s $-th cyclotomic field

Hulls of $\mathbb Z_p\mathbb Z_p[v] $-cyclic codes and construction of EAQECCs

1.
Department of Mathematics, Banaras Hindu University, Varanasi, Uttar Pradesh 221005, India
2.
Department of Mathematics, University of Allahabad, Prayagraj, Uttar Pradesh, 211002 India

^*Corresponding author: Ashish Kumar Upadhyay
^*Corresponding author: Ashish Kumar Upadhyay

Received: February 2025

Revised: July 2025

Early access: August 11, 2025

Published online: August 11, 2025

The authors are not supported by any funding.

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Abstract

In this paper, we study the hulls of $ \mathbb{Z}_p\mathbb{Z}_p[v] $- cyclic codes for any prime $ p $, where $ v^2 = v $. Firstly, we derive the generator polynomials of the hulls of separable cyclic codes (SCCs) over $ \mathbb{Z}_p\mathbb{Z}_p[v] $ and determine their dimensions using the concept of a generating function in combinatorics. We also enumerate the SCCs over $ \mathbb{Z}_p\mathbb{Z}_p[v] $ with hulls of a fixed dimension. Next, we present the generator polynomials of the hulls of non-separable cyclic codes (NSCCs) over $ \mathbb{Z}_2\mathbb{Z}_2[v] $ with coprime odd block lengths and determine their dimensions. Additionally, we enumerate the NSCCs over $ \mathbb{Z}_2\mathbb{Z}_2[v] $ with hulls of a fixed dimension. As an application, we use the hulls of NSCCs over $ \mathbb{Z}_2\mathbb{Z}_2[v] $ to construct some good entanglement-assisted quantum error-correcting codes (EAQECCs).

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Mathematics Subject Classification: Primary: 94B05; Secondary: 94B15.

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Table 1. NSCCs of length $ (7,15) $ in $ \mathfrak R_{\lambda, \gamma} $, whose hulls have the fixed dimension $ 4 $

Generator of $ \mathfrak C $	Generator of $ \text{Hull} (\mathfrak C)$
$ \langle(x^4+x^3+x^2+1\mid 0),(x^3+x+1\mid \xi_1(x-1)+\xi_2(x^2+x+1))\rangle $	$ \langle(x^4+x^3+x^2+1\mid 0),(x^3+x+1\mid \xi_1(x^{15}-1)+\xi_2(x^{14}+x^{13}+\cdots+x+1))\rangle $
$ \langle(x^4+x^3+x^2+1\mid 0),(x^3+x+1\mid \xi_1(x^4+x^3+x^2+x+1)+\xi_2(x^2+x+1))\rangle $	$ \langle(x^4+x^3+x^2+1\mid 0),(x^3+x+1\mid \xi_1(x^{15}-1)+\xi_2(x^{14}+x^{13}+\cdots+x+1))\rangle $
$ \langle(x^4+x^3+x^2+1\mid 0),(x^3+x+1\mid \xi_1(x-1)+\xi_2(x^4+x^3+x^2+x+1))\rangle $	$ \langle(x^4+x^3+x^2+1\mid 0),(x^3+x+1\mid \xi_1(x^{15}-1)+\xi_2(x^{14}+x^{13}+\cdots+x+1))\rangle $
$ \langle(x^4+x^3+x^2+1\mid 0),(x^3+x+1\mid \xi_1(x^4+x^3+x^2+x+1)+\xi_2(x^4+x^3+x^2+x+1))\rangle $	$ \langle(x^4+x^3+x^2+1\mid 0),(x^3+x+1\mid \xi_1(x^{15}-1)+\xi_2(x^{14}+x^{13}+\cdots+x+1))\rangle $
$ \langle(x^4+x^2+x+1\mid 0),(x^3+x^2+1\mid \xi_1(x-1)+\xi_2(x^2+x+1))\rangle $	$ \langle(x^4+x^2+x+1\mid 0),(x^3+x^2+1\mid \xi_1(x^{15}-1)+\xi_2(x^{14}+x^{13}+\cdots+x+1))\rangle $
$ \langle(x^4+x^2+x+1\mid 0),(x^3+x^2+1\mid \xi_1(x^4+x^3+x^2+x+1)+\xi_2(x^2+x+1))\rangle $	$ \langle(x^4+x^2+x+1\mid 0),(x^3+x^2+1\mid \xi_1(x^{15}-1)+\xi_2(x^{14}+x^{13}+\cdots+x+1))\rangle $
$ \langle(x^4+x^2+x+1\mid 0),(x^3+x^2+1\mid \xi_1(x-1)+\xi_2(x^4+x^3+x^2+x+1))\rangle $	$ \langle(x^4+x^2+x+1\mid 0),(x^3+x^2+1\mid \xi_1(x^{15}-1)+\xi_2(x^{14}+x^{13}+\cdots+x+1))\rangle $
$ \langle(x^4+x^2+x+1\mid 0),(x^3+x^2+1\mid \xi_1(x^4+x^3+x^2+x+1)+\xi_2(x^4+x^3+x^2+x+1))\rangle $	$ \langle(x^4+x^2+x+1\mid 0),(x^3+x^2+1\mid \xi_1(x^{15}-1)+\xi_2(x^{14}+x^{13}+\cdots+x+1))\rangle $

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