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Article Contents

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

• * Corresponding author: Om Prakash

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

• 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.

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

 Citation:

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

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

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]

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]

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]

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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Tables(5)