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# New quantum codes from skew constacyclic codes

• * Corresponding author: Om Prakash

The research is supported by the Council of Scientific & Industrial Research (CSIR), Govt. of India.

• For an odd prime $p$ and positive integers $m$ and $\ell$, let $\mathbb{F}_{p^m}$ be the finite field with $p^{m}$ elements and $R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell}$. Thus $R_{\ell,m}$ is a finite commutative non-chain ring of order $p^{2^{\ell} m}$ with characteristic $p$. In this paper, we aim to construct quantum codes from skew constacyclic codes over $R_{\ell,m}$. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.

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

 Citation:

• Table 1.  Quantum MDS codes $[[n,k,d]]_{p^m}$ from skew $(\sigma,\gamma)$-constacyclic codes over $R_{1,m} = \mathbb{F}_{p^m}[v_1]/\langle v_1^2-1\rangle$

 $p^m$ $n$ $\gamma$ $(\delta_0,\delta_1)$ $g_0(x)$ $g_1(x)$ $\Phi(C)$ $[[n,k,d]]_{p^m}$ $5^2$ $4$ $v_1$ $(1,-1)$ $21$ $t1$ $[8,6,3]$ $[[8,4,3]]_{25}$ $5^2$ $4$ $-1$ $(-1,-1)$ $(3t+4)(2t+2)1$ $t1$ $[8,5,4]$ $[[8,2,4]]_{25}$ $5^2$ $6$ $v_1$ $(1,-1)$ $(3t+3)(2t+3)1$ $31$ $[12,9,4]$ $[[12,6,4]]_{25}$ $7^2$ $6$ $v_1$ $(1,-1)$ $(3t+5)1$ $(5t+2)1$ $[12,10,3]$ $[[12,8,3]]_{49}$ $11^2$ $8$ $v_1$ $(1,-1)$ $(4t+3)1$ $(10t+1)(6t+9)1$ $[16,13,4]$ $[[16,10,4]]_{121}$ $11^2$ $12$ $v_1$ $(1,-1)$ $(4t+3)1$ $(10t+7)(3t+4)1$ $[24,21,4]$ $[[24,18,4]]_{121}$ $11^2$ $6$ $v_1$ $(1,-1)$ $(4t+3)1$ $(2t+4)(5t+2)1$ $[12,9,4]$ $[[12,6,4]]_{121}$ $13^2$ $4$ $v_1$ $(1,-1)$ $81$ $(12t+11)1$ $[8,6,3]$ $[[8,4,3]]_{169}$ $13^2$ $12$ $v_1$ $(1,-1)$ $(8t+8)1$ $(11t+3)1$ $[24,22,3]$ $[[24,20,3]]_{169}$ $13^2$ $6$ $v_1$ $(1,-1)$ $(5t+3)1$ $(3t+7)1$ $[12,10,3]$ $[[12,8,3]]_{169}$ $17^2$ $8$ $v_1$ $(1,-1)$ $(11t+16)1$ $(15t+13)1$ $[16,14,3]$ $[[16,12,3]]_{289}$ $17^2$ $8$ $v_1$ $(1,-1)$ $(11t+16)(7t+8)1$ $(15t+13)1$ $[16,13,4]$ $[[16,10,4]]_{289}$

Table 2.  New quantum codes $[[n,k,d]]_{p^m}$ from skew $(\sigma,\gamma)$-constacyclic codes over $R_{1,m} = \mathbb{F}_{p^m}[v_1]/\langle v_1^2-1\rangle$

 ${p^m}$ $n$ $\gamma$ $(\delta_0,\delta_1)$ $g_0(x)$ $g_1(x)$ $\Phi(C)$ $[[n,k,d]]_{p^m}$ $[[n',k',d']]_{p^m}$ $3^2$ $12$ $v_1$ $(1,-1)$ $(t+1)(t+2)(t)1$ $2(t+1)1$ $[24,19,4]$ $[[24,14,4]]_9$ $[[24,10,4]]_9$ [24] $3^2$ $8$ $-v_1$ $(-1,1)$ $(t+1)(2t+1)(t+2)(t+2)1$ $(2t+2)(t+1)11$ $[16,9,6]$ $[[16,2,6]]_9$ $[[16,2,5]]_9$ [24] $3^2$ $24$ $1$ $(1,1)$ $(t+1)(2t)1$ $(2t+1)(2t)(t+1)1$ $[48,43,3]$ $[[48,38,3]]_9$ $[[50,30,3]]_9$ [13] $5^2$ $20$ $v_1$ $(1,-1)$ $(2t+1)(3t+1)1$ $(4t+3)1$ $[40,37,3]$ $[[40,34,3]]_{25}$ $[[40,24,3]]_{25}$ [6] $5^2$ $40$ $1$ $(1,1)$ $(2t+3)1$ $(4t+1)2(3t+2)(t+4)1$ $[80,75,3]$ $[[80,70,3]]_{25}$ $[[80,56,3]]_{25}$ [6] $5^2$ $40$ $1$ $(1,1)$ $(t+3)(2t+2)(3t+3)(2t+3)1$ $(4t+1)2(3t+2)(t+4)1$ $[80,72,4]$ $[[80,64,4]]_{25}$ $[[80,48,4]]_{25}$ [6] $7^2$ $28$ $v_1$ $(1,-1)$ $(6t+4)(t+4)(3t+6)(2t+3)(4t+5)1$ $(3t+4)1$ $[56,50,4]$ $[[56,44,4]]_{49}$ $[[56,32,4]]_{49}$ [6] $7^2$ $28$ $-v_1$ $(-1,1)$ $(5t+3)(4t+2)1$ $(6t+4)(t+1)1$ $[56,52,3]$ $[[56,48,3]]_{49}$ $[[56,40,3]]_{49}$ [6]

Table 3.  New quantum codes from skew $(\sigma,\gamma)$-constacyclic codes over $R_{\ell,m}$

 $\ell$ $n$ $\gamma$ $(\delta_0,\delta_1,\dots, \delta_{2^{\ell}-1})$ $g_0(x),g_1(x),\dots,g_{2^{\ell}-1}(x)$ $\Phi(C)$ $[[n,k,d]]_{p^m}$ $1$ $16$ $1$ $(1,1)$ $(t+1)1,(2t+1)111$ $[32,27,4]$ $[[32,22,4]]_{9}$ $2$ $16$ $1$ $(1,1,1,1)$ $tt1,(t+1)1,(t+1)1,(t+2)(2t+1)(2t+2)1$ $[64,57,4]$ $[[64,50,4]]_{9}$ $2$ $16$ $1$ $(1,1,1,1)$ $(t+1)1,(t+1)1,(t+1)1,(2t)(t+2)11$ $[64,58,3]$ $[[64,52,3]]_{9}$ $2$ $24$ $-1+v_1-v_2-v_1v_2$ $(1,1,-1,1)$ $(2t)1,(t+1)t1,tt1,t1$ $[96,90,3]$ $[[96,84,3]]_{9}$ $2$ $24$ $-1+v_1-v_2-v_1v_2$ $(1,1,-1,1)$ $(2t)1,1(2t+2)1,(t+2)11,(2t+2)(2t+2)11$ $[96,88,4]$ $[[96,80,4]]_{9}$ $3$ $16$ $1$ $(1,1,1,1,1,1,1,1)$ $1,1,1,(2t+2)1,1,tt1,(2t+1)(2t+2)(2t+1)1,(t+2)21$ $[128,120,4]$ $[[128,112,4]]_{9}$ $1$ $16$ $1$ $(1,1)$ $t1,1(3t)1$ $[32,28,3]$ $[[32,24,3]]_{25}$ $1$ $18$ $-1$ $(-1,-1)$ $2(2t+1)(4t+2)1,31$ $[36,32,3]$ $[[36,28,3]]_{25}$ $1$ $20$ $1$ $(1,1)$ $(t+3)1,(2t+2)(t+3)(4t+4)t1$ $[40,35,4]$ $[[40,30,4]]_{25}$ $1$ $24$ $1$ $(1,1)$ $(2t+3)(t+1)1,(4t+3)(t+1)(4t+1)1$ $[48,43,4]$ $[[48,38,4]]_{25}$ $2$ $16$ $1$ $(1,1,1,1)$ $t1,t1,t1,(4t+1)(2t)(3t+4)1$ $[64,58,4]$ $[[64,52,4]]_{25}$ $2$ $30$ $1$ $(1,1,1,1)$ $(2t+2)1,(3t+4)1,(2t+2)(4t+3)1,(2t+2)(3t+3)(t+1)1$ $[120,113,3]$ $[[120,106,3]]_{25}$ $2$ $24$ $1$ $(1,1,1,1)$ $(4t)1,(3t+1)t1,(t+3)21,(2t)1$ $[96,90,4]$ $[[96,84,4]]_{25}$ $2$ $24$ $1$ $(1,1,1,1)$ $(2t+4)1,2(2t+2)1,21,(3t+1)1$ $[96,91,3]$ $[[96,86,3]]_{25}$ $3$ $16$ $1$ $(1,1,1,1,1,1,1,1)$ $1,1,1,(3t+1)1,(2t+3)(t+1)1,(2t+2)(2t+3)(2t+1)1,(t+4)1$ $[128,121,4]$ $[[128,114,4]]_{25}$ $2$ $24$ $4-3v_1+3v_2-3v_1v_2$ $(1,-1,1,1)$ $t1,(2t+6)(3t)1,(4t+1)(3t+6)1,(3t+6)1$ $[96,90,4]$ $[[96,84,4]]_{49}$ $2$ $24$ $1$ $(1,1,1,1)$ $t1,(4t+2)1,(6t+3)(2t+5)1,(2t+5)1$ $[96,91,3]$ $[[96,86,3]]_{49}$ $3$ $16$ $1$ $(1,1,1,1,1,1,1,1)$ $1,1,1,(6t+4)1,1,(2t+4)(5t+4)1,(3t+4)(6t)(4t+4)1,(6t+4)(4t+2)1$ $[128,120,4]$ $[[128,112,4]]_{49}$ $3$ $18$ $1$ $(1,1,1,1,1,1,1,1)$ $1,1,1,(2t+6)1,1,(2t+6)(t+1)1,(t+4)(t+2)(6t+6)1,(3t+5)(5t+5)11$ $[144,135,3]$ $[[144,126,3]]_{49}$ $3$ $16$ $1$ $(1,1,1,1,1,1,1,1)$ $1,1,1,(7t+8)1,1,(t+4)(10t+8)1,(10t+3)(3t+1)11,(t+8)(2t+1)1$ $[128,120,4]$ $[[128,112,4]]_{121}$
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