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Additive polycyclic codes over $\mathbb{F}_{4}$ induced by binary vectors and some optimal codes

• *Corresponding author: Nuh Aydin
• In this paper, we study the structure and properties of additive right and left polycyclic codes induced by a binary vector $a$ in $\mathbb{F}_{2}^{n}.$ We find the generator polynomials and the cardinality of these codes. We also study different duals for these codes. In particular, we show that if $C$ is a right polycyclic code induced by a vector $a\in \mathbb{F}_{2}^{n}$, then the Hermitian dual of $C$ is a sequential code induced by $a.$ As an application of these codes, we present examples of additive right polycyclic codes over $\mathbb{F}_{4}$ with more codewords than comparable optimal linear codes as well as optimal binary linear codes and optimal quantum codes obtained from additive right polycyclic codes over $\mathbb{F}_{4}.$

Mathematics Subject Classification: Primary: 94B60, 58F17; Secondary: 94B05.

 Citation:

• Table 1.  Additive codes $[n, k, d]_4$ v.s. BKLC $[n, k, d]_4$ with smaller dimension

 $[n, \frac{2k+1}{2}, d]_4$ $[n, k, d]_4$ $[n, k+1, d-1]_4$ $\langle \alpha g_1 +g_2, b\rangle$ Multinomial $[ 7, 9/2, 3]_4$ $[ 7, 4, 3 ]_4$ $[ 7, 5, 2 ]_4$ $\langle\alpha (x^2 + x + 1)+x, x^3 + x^2 + 1\rangle$ $x^7 + x^6 + x^5 + x^3 + 1$ $[7, 7/2, 4]_4$ $[7, 3, 4]_4$ $[7, 4, 3]_4$ $\langle\alpha (x + 1)+ x^4 + x^3 + x^2 + x, x^6 + \\x^5 + x^4 + x^3 + x^2 + x + 1\rangle$ $x^7 + 1$ $[22, 37/2, 3]_4$ $[22, 18, 3]_4$ $[22, 19, 2]_4$ $\langle\alpha (x + 1)+ x^3 + x, x^6 + x^4 + x^3 + \\x + 1\rangle$ $x^{22} + x^{19} + x^{15} + x^{14} + x^{13} + x^8 + \\x^7 + x^6 + x^4 + x^2 + x + 1$ $[23, 39/2, 3]_4$ $[23, 19, 3]_4$ $[23, 20, 2]_4$ $\langle\alpha (x + 1)+ x^4 + x^2, x^6 + x^5 + 1\rangle$ $x^{23} + x^{22} + x^{21} + x^{15} + x^{13} + x^{11} + \\x^{10} + x^9 + x^8 + x^7 + x^3 + x^2+ x + 1$ $[24, 41/2 , 3]_4$ $[24, 20, 3]_4$ $[24, 21, 2]_4$ $\langle\alpha (x^2 + x + 1)+ x^2 + x, x^5 + x^4 + \\x^3 + x + 1\rangle$ $x^{24} + x^{21} + x^{20} + x^{19} + x^{18} + x^{16} +\\ x^{14} + x^8 + x^5 + x^4 + x^3 + x^2 1,$ $[25, 43/2, 4]_4$ $[25, 21, 3]_4$ $[25, 22, 2]_4$ $\langle\alpha (x + 1)+x^4 + x^2, x^6 + x^5 + 1\rangle$ $x^{25} + x^{24} + x^{22} + x^{21} + x^{19} + x^{18} + \\x^{15} + x^{13} + x^{12} + x^{11} + x^{10} +x^9 + \\x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + 1$ $[26, 45/2, 3]_4$ $[26, 22, 3]_4$ $[26, 23, 2]_4$ $\langle\alpha (x + 1)+x^4 + x^3 + x, x^6 + x^5 + \\x^3 + x^2 + 1\rangle$ $x^{26} + x^{24} + x^{21} + x^{16} + x^{15} + x^{13} + \\x^{12} + x^{11} + x^{10} + x^9 + x^7 +x^3 + \\x^2 + 1$ $[27, 47/2, 3]_4$ $[27, 23, 3]_4$ $[27, 24, 2]_4$ $\langle\alpha (x^2 + x + 1)+ x^2 + x, x^5 + x^4 + \\x^3 + x + 1\rangle$ $x^{27} + x^{26} + x^{23} + x^{21} + x^{19} + x^{18} + \\x^{17} + x^{14} + x^{13} + x^{11} + x^{10} +x^7 + \\x^3 + x + 1$ $[27, 47/2, 3]_4$ $[27, 23, 3]_4$ $[27, 24, 2]_4$ $\langle\alpha (x^2 + x + 1)+ x^2 + x, x^5 + x^4 + \\x^3 + x + 1\rangle$ $x^{27} + x^{26} + x^{23} + x^{21} + x^{19} + x^{18} + \\x^{17} + x^{14} + x^{13} + x^{11} + x^{10} +x^7 + \\x^3 + x + 1$ $[28, 49/2, 3, ]_4$ $[28, 24, 3]_4$ $[28, 25, 2]_4$ $\langle\alpha (x^2 + x + 1)+ x^3 + x^2 + x, x^5 + \\x^3 + x^2 + x + 1\rangle$ $x^{28} + x^{27} + x^{24} + x^{23} + x^{22} + x^{20} + \\x^{17} + x^{16} + x^{15} + x^{14} + x^{13} +x^{11} + \\x^{10} + x^8 + x^7 + x^5 + x^3 + x + 1$ $[29, 51/2, 3]_4$ $[29, 25, 3]_4$ $[29, 26, 2]_4$ $\langle\alpha (x + 1)+ x^2 + x + 1, x^6 + x^5 + \\x^4 + x + 1\rangle$ $x^{29} + x^{25} + x^{22} + x^{21} + x^{19} + x^{15} + \\x^{14} + x^{13} + x^{11} + x^{10} + x^6 +x^3 + \\x + 1$ $[30, 53/2, 3]_4$ $[30, 26, 3]_4$ $[30, 27, 2]_4$ $\langle\alpha (x + 1)+ x^3 + x^2 + 1, x^6 + x^5 + \\x^3 + x^2 + 1\rangle$ $x^{30} + x^{27} + x^{26} + x^{25} + x^{24} + x^{20} + \\x^{19} + x^{18} + x^{16} + x^{11} + x^{10} +x^9 + \\x^8 + x^7 + x^2 + 1$ $[31, 55/2, 3]_4$ $[31, 27, 3]_4$ $[31, 28, 2]_4$ $\langle\alpha ( x + 1)+ x^4 + x + 1, x^6 + x^5 + \\x^3 + x^2 + 1\rangle$ $x^{31} + x^{25} + x^{23} + x^{22} + x^{20} + x^{18} + \\x^{17} + x^{16} + x^{15} + x^{14} + x^{13} +x^{12} + \\x^{10} + x^9 + x^8 + x^7 + x^6 + x^4 + x^3 +1$

Table 2.  Optimal binary linear codes $[n, k, d]_2$ obtained from quaternary additive codes based on $W$, $T$, and $L$

 $[n, k, d]_2$ Map $\langle \alpha g_1 +g_2, b\rangle$ Multinomial $[ 7, 2, 4]_2^{*\circ}$ L $\langle\alpha (x^5 + x^4 + x + 1) + 1, x^2 + x + 1 \rangle$ $x^7 + x^4 + x^3 + 1$ $[ 10, 4, 4]_2$ L $\langle\alpha (x + 1)+x + 1, x^6 + x^5 + x^4 + \\x + 1\rangle$ $x^{10} + x^9 + x^8 + x^6 + x + 1$ $[ 12, 5, 4]_2$ L $\langle\alpha (x^2 + x + 1)+x^2 + x + 1, x^7 + \\x^3 + x^2 + x + 1\rangle$ $x^{12} + x^6 + x^5 + x^4 + 1$ $[ 16, 9, 4]_2^*$ T $\langle\alpha (x^7 + x^3 + x^2 + x + 1)+x^4 + \\x^3 + x^2 + x + 1, x^9 + x + 1 \rangle$ $x^{16} + x^{12} + x^{11} + x^{10} + x^9 + x^8 + \\x^7 + x^4 + 1$ $[ 17, 9, 5]_2$ T $\langle\alpha (x^8 + x^5 + x^4 + x^3 + 1)+x^7 + \\x^6 + x^4 + x^3 + x, x^9 + x^8 + x^6 + \\x^5 + x^4 + x^3 + x^2 + x + 1\rangle$ $x^{17} + x^{16} + x^{13} + x^{12} + x^{11} + x^{10} + \\x^9 + x^8 + x^7 + x^4 + x^2 + x + 1$ $[ 20, 11, 5]_2$ W $\langle\alpha (x+1)+x^6 + x^4 + x^2 + x, x^8 + \\x^6 + x^5 + x^4 + x^2 + x + 1\rangle$ $x^{10} + x^7 + x^5 + x^3 + x + 1$ $[ 26, 17, 4]_2$ W $\langle\alpha (x+1)+x^4 + x^3 + x^2, x^8 + x^5 + \\x^3 + x + 1\rangle$ $x^{13} + x^{10} + x^6 + x^3 + x + 1$ $[ 35, 24, 5]_2^*$ T $\langle\alpha (x^{11} + x^9 + x^8 + x^6 + x^4 + x^3 + \\x^2 + x + 1)+x^{22} + x^{16} + x^{15} + x^{14} + \\x^{13} + x^{10} + x^9 + x^8 + x^7 + x^5 + x + \\1 , x^{24} + x^{21} + x^{19} + x^{16} + x^{13} + \\x^{12} + x^{10} + x^8 + x^7 + x^6 + x^2 + x +1\rangle$ $x^{35} + x^{33} + x^{30} + x^{29} + x^{26} + x^{23} + \\x^{21} + x^{20} + x^{19} + x^{14} + x^{13} +x^{12} + \\x^{11} + x^9 + x^8 + x^7 + x^6 + x^4 + \\x^3 + x^2 + 1$ $[ 49, 39, 4]_2^*$ T $\langle\alpha ( x^{10} + x^8 + x^6 + x^4 + x^2 + x + \\1)+x , x^5 + x^4 + x^3 + x^2 + 1 \rangle$ $x^{49} + x^{46} + x^{45} + x^{44} + x^{40} + x^{39} + \\x^{38} + x^{36} + x^{35} + x^{31} + x^{30} + x^{29} + \\x^{28} + x^{27} + x^{24} + x^{23} + x^{20} + x^{19} + \\x^{17} + x^{15} + x^{12} + x^{11} + x^9 + x^8 + \\x^4 + x + 1$ $[ 62, 51, 4]_2$ W $\langle\alpha ( x + 1)+x^8 + x^4 + x + 1 , x^10 + \\x^9 + x^8 + x^4 + 1 \rangle$ $x^{31} + x^{29} + x^{28} + x^{27} + x^{26} + x^{25} + \\x^{23} + x^{15} + x^{14} + x^{13} + x^{12} +x^9 + \\x^8 + x^7 + x^3 + x^2 + x + 1$ $[ 98, 86, 4]_2$ W $\langle\alpha (x^2+ x + 1)+x^8 + x^7 + x^6 + x^3 + \\1 , x^{10} + x^8 + x^6 + x^4 + x^2 + x + 1\rangle$ $x^{49} + x^{48} + x^{46} + x^{45} + x^{43} + x^{39} + \\x^{36} + x^{31} + x^{27} + x^{25} + x^{24} +x^{22} + \\x^{21} + x^{20} + x^{19} + x^{18} + x^{17} + x^{14} + \\x^7 + x^6 + x^3 + x + 1$

Table 3.  Optimal quantum codes $[[n, k, d]]_4$ from self-dual/self-orthogonal/dual-containing binary linear codes obtained from quaternary additive codes

 $[[n, k, d]]_4$ Map $\langle \alpha g_1 +g_2, b\rangle$ Multinomial $[[ 7, 1, 3]]_4$ T $\langle\alpha (x^3+x^2 + 1)+ x, x^4 + x^3 + x^2 + x + 1 \rangle$ $x^7 + x^4 + x^3 + x + 1$ $[[ 10, 8, 2]]_4$ T $\langle\alpha (x + 1)+ 1, x^2 + x + 1\rangle$ $x^{10} + x^9 + x^7 + x^5 + x^4 + x^2 + x + 1$ $[[ 15, 7, 3]]_4$ T $\langle\alpha (x^4 + x + 1)+ x^2 + x + 1, x^4 + x^3 + 1 \rangle$ $x^{15} + x^{10} + x^9 + x^7 + x^6 + x^5 + \\x^3 + x^2 + 1$ $[[ 30, 26, 2]]_4$ T $\langle\alpha (x^2 + x + 1)+ x^8 + x^6 + 1, x^{12} + x^{11} + \\x^8 + x^7 + x^5 + x^4 + x^2 + x + 1 \rangle$ $x^{30} + x^{29} + x^{27} + x^{25} + x^{24} + x^{23} + \\x^{22} + x^{21} + x^{20} + x^{17} + x^{13} + x^{11} + \\x^{10} + x^9 + x^6 + x^5 + x^2 + x + 1$ $[[ 35, 29, 2]]_4$ L $\langle\alpha ( x + 1)+ x + 1, x^3 + x + 1 \rangle$ $x^{35} + x^{34} + x^{33} + x^{31} + x^{27} + x^{25} + \\x^{24} + x^{22} + x^{20} + x^{18} + x^{17} + x^{16} + \\x^{15} + x^{11} + x^5 + x^4 + x + 1$ $[[ 36, 30, 2]]_4$ W $\langle\alpha ( x + 1)+ 1, x^2 + x + 1 \rangle$ $x^{18} + x^{16} + x^{13} + x^{11} + x^9 + x^8 + \\x^7 + x^4 + x^3 + 1$ $[[ 48, 42, 2]]_4$ W $\langle\alpha ( x + 1)+ 1, x^2 + x + 1 \rangle$ $x^{24} + x^{22} + x^{16} + x^{15} + x^{14} + x^{13} + \\x^{11} + x^{10} + x^7 + x^6 + x^4 + 1$ $[[ 56, 50, 2]]_4$ L $\langle\alpha ( x + 1)+ 1, x^3 + x + 1 \rangle$ $x^{56} + x^{51} + x^{49} + x^{48} + x^{47} + x^{41} + \\x^{38} + x^{36} + x^{31} + x^{30} + x^{29} + x^{28} + \\x^{26} + x^{18} + x^{16} + x^{15} + x^{14} + x^{12} + \\x^{10} + x^9 + x^5 + x^2 + x + 1$ $[[ 72, 66, 2]]_4$ W $\langle\alpha ( x + 1)+ 1, x^2 + x + 1 \rangle$ $x^{36} + x^{35} + x^{32} + x^{31} + x^{30} + x^{29} + \\x^{26} + x^{22} + x^{21} + x^{20} + x^{18} +x^{16} + \\x^{13} + x^{10} + x^9 + x^2 + x + 1$ $[[ 84, 76, 2]]_4$ W $\langle\alpha ( x + 1)+ 1, x^3 + x + 1 \rangle$ $x^{42} + x^{41} + x^{40} + x^{39} + x^{37} + x^{35} + \\x^{33} + x^{30} + x^{29} + x^{28} + x^{27} +x^{25} + \\x^{21} + x^{15} + x^{14} + x^{13} + x^{11} + x^{10} + \\x^9 + x^6 + x^5 + x^2 + x + 1$
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