New quantum codes from constacyclic codes over the ring $ R_{k,m} $

Habibul Islam; Om Prakash; Ram Krishna Verma

doi:10.3934/amc.2020097

Article Contents

2022, Volume 16, Issue 1: 17-35. Doi: 10.3934/amc.2020097

This issue Previous Article Additive and linear conjucyclic codes over

$ {\mathbb{F}}_4 $

Next Article Efficient fully CCA-secure predicate encryptions from pair encodings

New quantum codes from constacyclic codes over the ring $ R_{k,m} $

Department of Mathematics, Indian Institute of Technology Patna, Patna- 801 106, India

^* Corresponding author: Om Prakash
^* Corresponding author: Om Prakash

Received: November 30, 2019

Revised: April 30, 2020

Early access: July 2020

Published: February 2022

The research is supported by the University Grants Commission (UGC) and the Council of Scientific & Industrial Research (CSIR), Govt. of India

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Abstract

For any odd prime $ p $, we study constacyclic codes of length $ n $ over the finite commutative non-chain ring $ R_{k,m} = \mathbb{F}_{p^m}[u_1,u_2,\dots,u_k]/\langle u^2_i-1,u_iu_j-u_ju_i\rangle_{i\neq j = 1,2,\dots,k} $, where $ m,k\geq 1 $ are integers. We determine the necessary and sufficient condition for these codes to contain their Euclidean duals. As an application, from the dual containing constacyclic codes, several MDS, new and better quantum codes compare to the best known codes in the literature are obtained.

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Mathematics Subject Classification: 94B15, 94B05, 94B60.

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Table 1. Quantum MDS codes $ [[n,k,d]]_{p^m} $ from constacyclic codes over $ R_{1,m} = \mathbb{F}_{p^m}[u_1]/\langle u_1^2-1\rangle $

$ p^m $	$ n $	$ \gamma $	$ (\delta_0,\delta_1) $	$ f_0(x) $	$ f_1(x) $	$ M $	$ \psi(\mathcal{C}) $	$ [[n,k,d]]_{p^m} $
$ 5 $	$ 2 $	$ -1 $	$ (-1,-1) $	$ 13 $	$ 1 $	$ M_4 $	$ [4,3,2] $	$ [[4,2,2]]_5 $
$ 13 $	$ 6 $	$ u_1 $	$ (1,-1) $	$ 13 $	$ 15 $	$ M_7 $	$ [12,10,3] $	$ [[12,8,3]]_{13} $
$ 11 $	$ 5 $	$ u_1 $	$ (1,-1) $	$ 12 $	$ 14 $	$ M_8 $	$ [10,8,3] $	$ [[10,6,3]]_{11} $
$ 11 $	$ 5 $	$ u_1 $	$ (1,-1) $	$ 12 $	$ 184 $	$ M_8 $	$ [10,7,4] $	$ [[10,4,4]]_{11} $
$ 17 $	$ 8 $	$ u_1 $	$ (1,-1) $	$ 12 $	$ 17 $	$ M_9 $	$ [16,14,3] $	$ [[16,12,3]]_{17} $
$ 23 $	$ 11 $	$ u_1 $	$ (1,-1) $	$ 17 $	$ 13 $	$ M_{10} $	$ [22,20,3] $	$ [[22,18,3]]_{23} $
$ 19 $	$ 9 $	$ u_1 $	$ (1,-1) $	$ 13 $	$ 14 $	$ M_{11} $	$ [18,16,3] $	$ [[18,14,3]]_{19} $
$ 29 $	$ 14 $	$ u_1 $	$ (1,-1) $	$ 1(13) $	$ 13 $	$ M_{12} $	$ [28,26,3] $	$ [[28,24,3]]_{29} $

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Table 3. New Quantum codes $ [[n,k,d]]_{p^m} $ from constacyclic codes over $ R_{1,m} = \mathbb{F}_{p^m}[u_1]/\langle u_1^2-1\rangle $

$ {p^m} $	$ n $	$ \gamma $	$ (\delta_0,\delta_1) $	$ f_0(x) $	$ f_1(x) $	$ M $	$ \psi(\mathcal{C}) $	$ [[n,k,d]]_{p^m} $	$ [[n',k',d']]_{p^m} $
$ 3 $	$ 12 $	$ -1 $	$ (-1,-1) $	$ 112 $	$ 122 $	$ M_1 $	$ [24,20,4] $	$ [[24,16,4]]_3 $	$ [[26,16,4]]_3 $ [9]
$ 3 $	$ 15 $	$ u_1 $	$ (1,-1) $	$ 11111 $	$ 12121 $	$ M_1 $	$ [30,22,6] $	$ [[30,14,6]]_3 $	$ [[31,13,6]]_3 $ [9]
$ 3 $	$ 18 $	$ u_1 $	$ (1,-1) $	$ 12021 $	$ 10201 $	$ M_2 $	$ [36,28,3] $	$ [[36,20,3]]_3 $	$ - $
$ 3 $	$ 30 $	$ 1 $	$ (1,1) $	$ 11 $	$ 12 $	$ M_2 $	$ [60,58,2] $	$ [[60,56,2]]_3 $	$ [[60,54,2]]_3 $ [12]
$ 3 $	$ 36 $	$ 1 $	$ (1,1) $	$ 12 $	$ 12 $	$ M_2 $	$ [72,70,2] $	$ [[72,68,2]]_3 $	$ [[72,66,2]]_3 $[12, 13]
$ 5 $	$ 10 $	$ u_1 $	$ (1,-1) $	$ 131 $	$ 12 $	$ M_4 $	$ [20,17,3] $	$ [[20,14,3]]_5 $	$ [[22,14,3]]_5 $[10]
$ 5 $	$ 10 $	$ u_1 $	$ (1,-1) $	$ 1441 $	$ 143122 $	$ M_4 $	$ [20,12,5] $	$ [[20,4,5]]_5 $	$ [[19,1,5]]_5 $[10]
$ 5 $	$ 11 $	$ u_1 $	$ (1,-1) $	$ 124114 $	$ 114431 $	$ M_4 $	$ [22,12,7] $	$ [[22,2,7]]_5 $	$ [[22,2,5]]_5 $[10, 17]
$ 5 $	$ 12 $	$ u_1 $	$ (1,-1) $	$ 10224 $	$ 12041 $	$ M_4 $	$ [24,16,5] $	$ [[24,8,5]]_5 $	$ [[23,6,5]]_5 $[10]
$ 5 $	$ 15 $	$ u_1 $	$ (1,-1) $	$ 1003001 $	$ 1003421 $	$ M_4 $	$ [30,18,6] $	$ [[30,6,6]]_5 $	$ [[60,8,6]]_5 $[6]
$ 5 $	$ 15 $	$ u_1 $	$ (1,-1) $	$ 1003001 $	$ 11021 $	$ M_4 $	$ [30,20,4] $	$ [[30,10,4]]_5 $	$ [[30,10,2]]_5 $[6]
$ 5 $	$ 20 $	$ 1 $	$ (1,1) $	$ 1034 $	$ 12 $	$ M_4 $	$ [40,36,3] $	$ [[40,32,3]]_5 $	$ [[40,24,3]]_5 $ [30]
$ 5 $	$ 22 $	$ u_1 $	$ (1,-1) $	$ 13024212034 $	$ 111212 $	$ M_4 $	$ [44,29,8] $	$ [[44,14,8]]_5 $	$ [[44,4,8]]_5 $ [30]
$ 5 $	$ 30 $	$ u_1 $	$ (1,-1) $	$ 13431 $	$ 13 $	$ M_4 $	$ [60,55,3] $	$ [[60,50,3]]_5 $	$ [[60,48,3]]_5 $ [29]

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Table 4. New Quantum codes $ [[n,k,d]]_{p^m} $ from constacyclic codes over $ R_{1,m}=\mathbb{F}_{p^m}[u_1]/\langle u_1^2-1\rangle $

$ {p^m} $	$ n $	$ \gamma $	$ (\delta_0,\delta_1) $	$ f_0(x) $	$ f_1(x) $	$ M $	$ \psi(\mathcal{C}) $	$ [[n,k,d]]_{p^m} $	$ [[n',k',d']]_{p^m} $
$ 5 $	$ 60 $	$ 1 $	$ (1,1) $	$ 13 $	$ 12 $	$ M_4 $	$ [120,118,2] $	$ [[120,116,2]]_5 $	$ [[120,114,2]]_5 $ [13, 29]
$ 5 $	$ 70 $	$ u_1 $	$ (1,-1) $	$ 134444431 $	$ 13 $	$ M_4 $	$ [140,131,3] $	$ [[140,122,3]]_5 $	$ [[140,116,3]]_5 $ [30]
$ 7 $	$ 7 $	$ u_1 $	$ (1,-1 ) $	$ 151 $	$ 121 $	$ M_5 $	$ [14,10,3] $	$ [[14,6,3]]_7 $	$ [[14,2,3]]_7 $[33]
$ 7 $	$ 7 $	$ u_1 $	$ (1,-1) $	$ 1436 $	$ 1331 $	$ M_5 $	$ [14,8,4] $	$ [[14,2,4]]_7 $	$ [[14,2,3]]_7 $[33]
$ 7 $	$ 14 $	$ 1 $	$ (1,1) $	$ 1661 $	$ 16 $	$ M_5 $	$ [28,24,3] $	$ [[28,20,3]]_7 $	$ [[27,17,3]]_7 $[29]
$ 7 $	$ 14 $	$ 1 $	$ (1,1) $	$ 15026 $	$ 11 $	$ M_5 $	$ [28,23,4] $	$ [[28,18,4]]_7 $	$ [[27,15,4]]_7 $[29]
$ 7 $	$ 21 $	$ u_1 $	$ (1,-1) $	$ 1054214515 $	$ 1515511 $	$ M_6 $	$ [42,27,7] $	$ [[42,12,7]]_7 $	$ [[37,1,7]]_7 $[10]
$ 7 $	$ 84 $	$ 1 $	$ (1,1) $	$ 12 $	$ 13 $	$ M_6 $	$ [168,166,2] $	$ [[168,164,2]]_7 $	$ [[168,162,2]]_7 $[13]
$ 9 $	$ 8 $	$ 1 $	$ (1,1) $	$ 1w^3w^3 $	$ 1w^2 $	$ M_3 $	$ [16,13,3] $	$ [[16,10,3]]_9 $	$ [[16,8,3]]_9 $[30]
$ 9 $	$ 8 $	$ u_1 $	$ (1,-1) $	$ 1w^7ww^6 $	$ 10w^201 $	$ M_3 $	$ [16,9,5] $	$ [[16,2,5]]_9 $	$ [[17,1,4]]_9 $[10]
$ 9 $	$ 12 $	$ u_1 $	$ (1,-1) $	$ 102w^60w^2 $	$ 1w^3 $	$ M_3 $	$ [24,18,4] $	$ [[24,10,4]]_9 $	$ [[24,8,4]]_9 $[30]
$ 11 $	$ 15 $	$ u_1 $	$ (1,-1) $	$ 1(10)382(10)9 $	$ 19(12)39 $	$ M_8 $	$ [30,20,6] $	$ [[30,10,6]]_{11} $	$ [[30,10,5]]_{11} $[25]
$ 11 $	$ 26 $	$ -1 $	$ (-1,-1) $	$ 1342443749481 $	$ 1849473442431 $	$ M_8 $	$ [52,28,10] $	$ [[52,4,10]]_{11} $	$ [[52,4,8]]_{11} $[25]
$ 11 $	$ 33 $	$ u_1 $	$ (1,1) $	$ 191(10)2(10) $	$ 11 $	$ M_8 $	$ [66,60,4] $	$ [[66,54,4]]_{11} $	$ [[66,52,4]]_{11} $ [29]
$ 11 $	$ 33 $	$ u_1 $	$ (1,1) $	$ 191949191 $	$ 11 $	$ M_8 $	$ [66,57,5] $	$ [[66,48,5]]_{11} $	$ [[57,39,5]]_{11} $ [29]

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Table 5. New Quantum codes $ [[n,k,d]]_{p^m} $ from constacyclic codes over $ R_{1,m} = \mathbb{F}_{p^m}[u_1]/\langle u_1^2-1\rangle $

$ {p^m} $	$ n $	$ \gamma $	$ (\delta_0,\delta_1) $	$ f_0(x) $	$ f_1(x) $	$ M $	$ \psi(\mathcal{C}) $	$ [[n,k,d]]_{p^m} $	$ [[n',k',d']]_{p^m} $
$ 13 $	$ 6 $	$ u_1 $	$ (1,-1 ) $	$ 17(12) $	$ 17(10) $	$ M_7 $	$ [12,8,4] $	$ [[12,4,4]]_{13} $	$ [[12,4,3]]_{13} $[13]
$ 13 $	$ 8 $	$ u_1 $	$ (1,-1 ) $	$ 15 $	$ 155(12) $	$ M_7 $	$ [16,12,3] $	$ [[16,8,3]]_{13} $	$ [[16,8,2]]_{13} $[11]
$ 13 $	$ 9 $	$ u_1 $	$ (1,-1 ) $	$ 1(10) $	$ 1003 $	$ M_7 $	$ [18,14,3] $	$ [[18,10,3]]_{13} $	$ [[12,4,3]]_{13} $[11]
$ 13 $	$ 12 $	$ u_1 $	$ (1,-1) $	$ 12 $	$ 102 $	$ M_7 $	$ [24,21,3] $	$ [[24,18,3]]_{13} $	$ [[24,16,3]]_{13} $[30]
$ 13 $	$ 13 $	$ u_1 $	$ (1,-1) $	$ 1(11)1 $	$ 121 $	$ M_7 $	$ [26,22,3] $	$ [[26,18,3]]_{13} $	$ [[36,20,3]]_{13} $[25]
$ 13 $	$ 13 $	$ u_1 $	$ (1,-1) $	$ 1(11)1 $	$ 14641 $	$ M_7 $	$ [26,20,5] $	$ [[26,14,5]]_{13} $	$ [[24,8,5]]_{13} $[13]
$ 13 $	$ 13 $	$ u_1 $	$ (1,-1) $	$ 1(11)1 $	$ 15(10)(10)51 $	$ M_7 $	$ [26,19,6] $	$ [[26,12,6]]_{13} $	$ [[24,4,6]]_{13} $[13]
$ 13 $	$ 18 $	$ u_1 $	$ (1,-1) $	$ 13 $	$ 12 $	$ M_7 $	$ [36,34,2] $	$ [[36,32,2]]_{13} $	$ [[36,30,2]]_{13} $[13]
$ 13 $	$ 18 $	$ u_1 $	$ (1,-1) $	$ 130780(12)(10) $	$ 120830(12)(11) $	$ M_7 $	$ [36,22,6] $	$ [[36,8,6]]_{13} $	$ [[36,8,4]]_{13} $[25]
$ 17 $	$ 8 $	$ u_1 $	$ (1,-1) $	$ 168 $	$ 15 $	$ M_9 $	$ [16,13,3] $	$ [[16,10,3]]_{17} $	$ [[16,8,3]]_{17} $ [30]
$ 17 $	$ 12 $	$ u_1 $	$ (1,-1) $	$ 14 $	$ 124 $	$ M_9 $	$ [24,21,3] $	$ [[24,18,3]]_{17} $	$ [[24,18,2]]_{17} $[13]
$ 17 $	$ 16 $	$ u_1 $	$ (1,-1) $	$ 1(14)311 $	$ 1010(10)0(14) $	$ M_9 $	$ [32,22,7] $	$ [[32,12,7]]_{17} $	$ [[32,12,6]]_{17} $[13]
$ 17 $	$ 16 $	$ u_1 $	$ (1,-1) $	$ 1(15)(12)24(13)(12) $	$ 10(12)040501 $	$ M_9 $	$ [32,18,10] $	$ [[32,4,10]]_{17} $	$ [[32,4,8]]_{17} $[13]
$ 17 $	$ 24 $	$ u_1 $	$ (1,-1) $	$ 16(12)(16) $	$ 17 $	$ M_9 $	$ [48,44,3] $	$ [[48,40,3]]_{17} $	$ [[48,36,3]]_{17} $[13]
$ 17 $	$ 24 $	$ u_1 $	$ (1,-1) $	$ 1(14)9(10)9 $	$ 1(11) $	$ M_9 $	$ [48,43,3] $	$ [[48,38,4]]_{17} $	$ [[48,30,4]]_{17} $[13]

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Table 2. Matrix Encoding

$ M $	$ GL_2(\mathbb{F}_{p^m}) $	$ MM^t=cI_2 $	$ M $	$ GL_2(\mathbb{F}_{p^m}) $	$ MM^t=cI_2 $
${M_1} = \left[ {\begin{array}{*{20}{c}} 2&1\\ 2&2 \end{array}} \right]$	$ GL_2(\mathbb{F}_3) $	$ M_1M_1^t=2I_2 $	${M_7} = \left[ {\begin{array}{*{20}{c}} 3&3\\ 3&{10} \end{array}} \right]$	$ GL_2(\mathbb{F}_{13}) $	$ M_7M_7^t=5I_2 $
${M_2} = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&2 \end{array}} \right]$	$ GL_2(\mathbb{F}_3) $	$ M_2M_2^t=2I_2 $	${M_8} = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&{10} \end{array}} \right]$	$ GL_2(\mathbb{F}_{11}) $	$ M_8M_8^t=2I_2 $
${M_3} = \left[ {\begin{array}{*{20}{c}} w&{ - 1}\\ 1&w \end{array}} \right]$	$ GL_2(\mathbb{F}_9) $	$ M_3M_3^t=(1+w^2)I_2 $	${M_9} = \left[ {\begin{array}{*{20}{c}} 2&2\\ 2&{15} \end{array}} \right]$	$ GL_2(\mathbb{F}_{17}) $	$ M_9M_9^t=8I_2 $
${M_4} = \left[ {\begin{array}{*{20}{c}} 1&4\\ 1&1 \end{array}} \right]$	$ GL_2(\mathbb{F}_{5}) $	$ M_4M_4^t=2I_2 $	$ {M_{10}} = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&{22} \end{array}} \right]$	$ GL_2(\mathbb{F}_{23}) $	$ M_{10}M_{10}^t=2I_2 $
$ {M_5} = \left[ {\begin{array}{*{20}{c}} 3&4\\ 3&3 \end{array}} \right] $	$ GL_2(\mathbb{F}_{7}) $	$ M_5M_5^t=4I_2 $	$ {M_{11}} = \left[ {\begin{array}{*{20}{c}} 2&2\\ 2&{17} \end{array}} \right] $	$ GL_2(\mathbb{F}_{19}) $	$ M_{11}M_{11}^t=8I_2 $
$ {M_6} = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&6 \end{array}} \right] $	$ GL_2(\mathbb{F}_{7}) $	$ M_6M_6^t=2I_2 $	$ {M_{12}} = \left[ {\begin{array}{*{20}{c}} 2&2\\ 2&{27} \end{array}} \right] $	$ GL_2(\mathbb{F}_{29}) $	$ M_{12}M_{12}^t=8I_2 $

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