On additive conjucyclic codes over fields and rings of order $ p^{2} $ using Gaussian integers

Irfan Siap; Taher Abualrub; Arezoo Soufi Karbaski

doi:10.3934/amc.2024041

Article Contents

2025, Volume 19, Issue 4: 1057-1069. Doi: 10.3934/amc.2024041

This issue Previous Article LCD codes constructed from weakly $ p $-self-orthogonal 1-designs Next Article Non-binary expurgated and extended Goppa codes with dihedral or 4-th alternating automorphism groups

On additive conjucyclic codes over fields and rings of order $ p^{2} $ using Gaussian integers

1.
Department of Mathematics and Natural Science, Prince Mohammad Bin Fahd University, Al Khobar 31952, Saudi Arabia
2.
Department of Mathematics & Statistics, American University of Sharjah, Sharjah, UAE
3.
Department of Mathematics Education, Farhangian University, P. O. Box 14665-889, Tehran, Iran

^*Corresponding author: Irfan Siap
^*Corresponding author: Irfan Siap

Received: July 2024

Revised: September 2024

Early access: October 14, 2024

Published online: October 14, 2024

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Abstract

In this paper, structures of conjucyclic codes and their duals over fields and rings of order $ p^{2} $ were studied. The approach was based on realizing additive conjucyclic codes over quotient rings of Gaussian integers, i.e., $ R_{p} = \mathbb{Z}\left[ i\right] /\langle p\rangle $, which has been proven to be an effective choice compared to previous studies on conjucyclic codes. We show that the algebraic structures of these codes are related to those algebraic structures of linear cyclic and linear negacyclic codes over finite fields $ \mathbb{F}_{p} $ of order $ p. $ We also introduce a definition of inner product on $ R_{p} $ and show that duals of additive conjucyclic codes over $ R_{p} $ based on this inner product are also additive conjucyclic codes. As a result, we show that there are no additive self-dual conjucyclic codes over $ R_{p}. $ We then study the properties and structures of additive complementary dual (ACD) conjucyclic codes and maximum distance separable (MDS) conjucyclic codes over $ R_{p}. $ As an application of our study, we construct classes of conjucyclic codes that consist of both ACD and MDS conjucyclic codes with parameters $ \left( n, p^{4k}, n-2k+1\right) . $ Furthermore, we provide many examples of optimal additive conjucyclic codes with the largest Hamming distance among additive conjucyclic codes over $ R_{p}. $

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Mathematics Subject Classification: Primary: 94B05, 94B15, 94B60; Secondary: 11T71.

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Table 1. Conjucyclic codes that are both ACD and MDS codes (L = Length)

p	L	Parameters	p	L	Parameters	p	L	Parameters
11	3	$ (3, 11^4, 2) $	29	15	$ (15, 29^{16} , 8) $	41	21	$ (21, 41^{8}, 18) $
13	7	$ (7, 13^4, 6) $	29	15	$ (15, 29^{20} , 6) $	41	21	$ (21, 41^{12}, 16) $
13	7	$ (7, 13^8, 4) $	29	15	$ (15, 29^{24} , 4) $	41	21	$ (21, 41^{16}, 14) $
13	7	$ (7, 13^{12}, 2) $	29	15	$ (15, 29^{28} , 2) $	41	21	$ (21, 41^{20}, 12) $
17	3	$ (3, 17^4, 2) $	37	19	$ (19, 37^{4}, 18) $	41	21	$ (21, 41^{24}, 10) $
17	9	$ (9, 17^4, 8) $	37	19	$ (19, 37^{8}, 16) $	41	21	$ (21, 41^{28}, 8) $
17	9	$ (9, 17^8, 6) $	37	19	$ (19, 37^{12}, 14) $	41	21	$ (21, 41^{32}, 6) $
17	9	$ (9, 17^{12}, 4) $	37	19	$ (19, 37^{16}, 12) $	41	21	$ (21, 41^{36}, 4) $
17	9	$ (9, 17^{16}, 2) $	37	19	$ (19, 37^{20}, 10) $	41	21	$ (21, 41^{40}, 2) $
19	5	$ (5, 19^4, 4) $	37	19	$ (19, 37^{24}, 8) $	43	11	$ (11, 43^{4}, 10) $
19	5	$ (5, 19^8, 2) $	37	19	$ (19, 37^{28}, 6) $	43	11	$ (11, 43^{8}, 8) $
23	3	$ (3, 23^4, 2) $	37	19	$ (19, 37^{32}, 4) $	43	11	$ (11, 43^{12}, 6) $
29	3	$ (3, 29^4, 2) $	37	19	$ (19, 37^{36}, 2) $	43	11	$ (11, 43^{16}, 4) $
29	5	$ (5, 29^4, 4) $	41	3	$ (3, 41^{4}, 2) $	43	11	$ (11, 43^{20}, 2) $
29	5	$ (5, 29^8, 2) $	41	7	$ (7, 41^{4}, 6) $	47	3	$ (3, 47^{4}, 2) $
29	15	$ (15, 29^4, 14) $	41	7	$ (7, 41^{8}, 4) $	53	3	$ (3, 53^{4}, 2) $
29	15	$ (15, 29^8 , 12) $	41	7	$ (7, 41^{12}, 2) $	53	9	$ (9, 53^{4}, 8) $
29	15	$ (15, 29^{12} , 10) $	41	21	$ (21, 41^{4}, 20) $	53	9	$ (9, 53^{8}, 6) $

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Table 2. The case: $ p = 4k+1 $ with best possible parameters

Parameters	$ \mathbf{\alpha} $	Generator polynomials
$(12, 5^{16}, 4)$	$2+i$	$(0,1)(3,1)(4,0)(1,3)(4,1)(0,0)^7$
$(19, 5^{18}, 8)$	$2+i$	$(4,1)(0,0)(3,2)(4,2)(3,2)(0,0)^2(2,1)\\ (2,3)(2,1)(4,1)(0,0)^8$
$(22, 5^{22}, 8)$	$2+i$	$(0,1)(4,4)(0,4)(1,3)(4,2)(1,1)(3,4)\\ (2,2)(4,4)(2,3)(2,1)(4,1)(0,0)^{10}$
$(26, 5^{16}, 14)$	$2+i$	$(2,1)(1,3)(4,4)(3,0)(2,4)(4,1)(4,4)(1,4)(4,0)(0,0)\\ (1,0)(4,1)(0,4)(1,4)(2,4)(2,0)(0,4)(4,2)(4,1)(0,0)^7$
$(29, 5^{28}, 12)$	$2+i$	$(2,1)(0,2)(3,0)(0,2)(1,0)(4,3)(4,1)(3,2)(4,2)\\ (2,1)(4,3)(4,0)(2,2)(2,0)(2,2)(4,1)(0,0)^{13}$
$(31, 5^{24}, 14)$	$2+i$	$(2,1)(1,1)(1,2)(2,1)(2,3)(1,2)(2,0)(3,4)(1,1)(1,3)(0,3)\\ (2,4)(0,2)(0,3)(4,0)(2,2)(0,0)^2(3,4)(4,1)(0,0)^{11}$
$(31, 5^{36}, 9)$	$2+i$	$(2,1)(2,3)^2(0,3)(2,0)(4,3)(0,1)(2,0)\\ (0,2)(4,2)^2(1,0)(0,3)(4,1)(0,0)^{17}$
$(31, 5^{42}, 7)$	$2+i$	$(4,1)(3,1)(0,1)(3,0)(0,4)(2,4)(2,3)(4,0)\\ (1,3)(0,4)(4,1)(0,0)^{19}$
$(31, 5^{48}, 5)$	$2+i$	$(2,1)(1,2)(0,2)(2,4)(4,2)(1,2)(4,1)(0,0)^{23}$
$(38, 5^{36}, 13)$	$2+i$	$(4,1)(0,2)(0,0)(3,3)(4,1)(0,1)(3,4)(4,4)(2,3)(0,3)(4,2)\\ (3,3)(2,3)(0,4)(3,4)(1,1)(4,1)(0,3)(0,0)(2,2)(4,1)(0,0)^{17}$
$(38, 5^{40}, 11)$	$2+i$	$(2,1)(2,2)(1,4)(0,1)(4,2)(0,0)(3,2)(0,4)(2,1)(4,4)\\ (4,1)(0,4)(4,2)(0,0)(3,2)(0,1)(3,4)(2,2)(4,1)(0,0)^{19}$

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Table 3. The case: $ p = 4k+3 $ with best possible parameters

Parameters	Generator polynomials
$(7, 3^{12}, 2)$	$(2, 1) (1, 1) (0, 0)^5$
$(10,3^{16}, 2)$	$(2, 1) (0, 0) (1, 1) (0, 0)^7$
$(13, 3^{6}, 9)$	$(1, 1) (2, 2) (1, 2) (0, 2) (2, 1) (2, 2) (1, 0) (1, 1) (1, 0) (0, 1) (1, 1) (0, 0)^2$
$(13, 3^{8}, 7)$	$(2, 1) (0, 1) (2, 1) (2, 1) (0, 0) (1, 2) (0, 1) (2, 0) (1, 0) (1, 1) (0, 0)^3$
$(13, 3^{18}, 3)$	$(1, 1) (0, 0) (2, 2) (2, 1) (1, 1) (0, 0)^8$
$(13, 3^{12}, 6)$	$(2, 1) (0, 2) (2, 0) (2, 0) (1, 2) (1, 2) (0, 2) (1, 1) (0, 0)^5$
$(23, 3^{22}, 9)$	$(1, 1) (0, 1) (0, 0) (1, 1) (0, 1) (1, 1)^2 (2, 2) (1, 0)(2,2) (0, 0)\\(2, 0) (1, 1) (0, 0)^{10}$

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