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Cyclic codes over local Frobenius rings of order 16

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    * Corresponding author 

The third author would like to thank TUBITAK (The Scientific and Technological Research Council of Turkey) for their support while writing this paper

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  • We study cyclic codes over commutative local Frobenius rings of order 16 and give their binary images under a Gray map which is a generalization of the Gray maps on the rings of order 4. We prove that the binary images of cyclic codes are quasi-cyclic codes of index 4 and give examples of cyclic codes of various lengths constructed from these techniques including new optimal quasi-cyclic codes.

    Mathematics Subject Classification: Primary: 94B15, 11T71.

    Citation:

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  • Table 1.  One-generator cyclic codes over $\mathbb{F}_{2}\left[u, v\right] /\left\langle u^{2}+v^{2}, uv\right\rangle $

    $n$ $f\left(x\right) $ $\phi \left({C}\right) $
    3 $\left(0, z_{4}, z_{4}\right) $ $\left[12, 2, 8\right] _{2}^{\ast }$
    3 $\left(z_{7}, z_{2}, z_{8}\right) $ $\left[12, 4, 6\right] _{2}^{\ast }$
    3 $\left(z_{1}, z_{8}, z_{6}\right) $ $\left[12, 6, 4\right] _{2}^{\ast }$
    3 $\left(y_{7}, y_{5}, y_{2}\right) $ $\left[12, 8, 3\right] _{2}^{\ast }$
    5 $\left(z_{7}, z_{2}, z_{2}, z_{7}, z_{1}\right) $ $\left[20, 8, 8\right] _{2}^{\ast }$
    5 $\left(y_{4}, y_{1}, y_{3}, y_{2}, y_{5}\right) $ $\left[20, 12, 4\right] _{2}^{\ast }$
    7 $\left(z_{4}, z_{1}, z_{4}, z_{1}, z_{1}, z_{4}, z_{4}\right) $ $\left[28, 3, 16\right] _{2}^{\ast }$
    7 $\left(z_{7}, z_{5}, z_{7}, z_{6}, z_{6}, z_{5}, z_{1}\right) $ $\left[28, 6, 12\right] _{2}^{\ast }$
    7 $\left(z_{6}, z_{5}, z_{6}, z_{3}, z_{3}, z_{5}, z_{4}\right) $ $\left[28, 7, 12\right] _{2}^{\ast }$
    7 $\left(z_{7}, z_{2}, z_{2}, z_{4}, z_{5}, z_{7}, z_{4}\right) $ $\left[28, 8, 10\right] _{2}^{\ast -1}$
    7 $\left(z_{5}, z_{2}, z_{2}, z_{4}, z_{5}, z_{7}, z_{4}\right) $ $\left[28, 12, 8\right] _{2}^{\ast }$
    7 $\left(y_{1}, y_{1}, y_{4}, z_{2}, y_{6}, z_{8}, z_{5}\right) $ $\left[28, 19, 4\right] _{2}^{\ast }$
    7 $\left(y_{2}, y_{4}, y_{4}, z_{8}, y_{1}, z_{5}, z_{1}\right) $ $\left[28, 20, 4\right] _{2}^{\ast }$
    9 $\left(z_{8}, z_{8}, z_{3}, z_{4}, z_{3}, z_{6}, z_{2}, z_{5}, z_{5}\right) $ $\left[36, 16, 8\right] _{2}^{\ast -2}$
    9 $\left(y_{7}, z_{6}, y_{2}, y_{5}, z_{1}, y_{8}, y_{2}, z_{1}, y_{8}\right) $ $\left[36, 20, 6\right] _{2}^{\ast -2}$
    15 $\left(z_{5}, z_{3}, z_{4}, z_{1}, z_{4}, z_{6}, z_{7}, z_{5}, z_{7}, z_{1}, z_{3}, z_{6}, z_{3}, z_{4}, z_{7}\right) $ $\left[60, 20, 16\right] _{2}^{b-1}$
    15 $\left(z_{4}, y_{7}, z_{7}, z_{1}, y_{5}, z_{5}, z_{3}, y_{8}, z_{8}, y_{4}, z_{4}, z_{2}, z_{3}, z_{6}, z_{7}\right) $ $\left[60, 50, 4\right] _{2}^{\ast }$
    17 $\left(z_{7}, z_{1}, z_{2}, z_{8}, y_{2}, y_{7}, y_{7}, z_{1}, z_{3}, y_{2}, y_{3}, z_{6}, z_{3}, y_{4}, y_{4}, y_{3}, z_{6}\right) $ $\left[68, 48, 6\right] _{2}^{\ast -2}$
    21 $\left(z_{4}y_{3}y_{1}y_{8}z_{3}y_{2}z_{3}z_{2}z_{2}y_{2}y_{7}y_{2}y_{5}y_{8}z_{1}z_{2}y_{5}y_{6}y_{1}z_{6}z_{1}\right) $ $\left[84, 58, 8\right] _{2}^{b}$
    21 $\left(y_{2}y_{6}y_{4}y_{7}y_{5}z_{8}z_{3}y_{1}z_{5}y_{8}y_{4}z_{4}y_{3}z_{2}z_{3}z_{7}z_{4}z_{6}z_{8}y_{3}y_{8}\right) $ $\left[84, 62, 6\right] _{2}^{b-2}$
    21 $\left(y_{2}z_{2}z_{6}y_{8}z_{3}y_{3}y_{3}z_{3}z_{1}z_{6}z_{8}z_{2}y_{1}z_{8}z_{7}z_{2}y_{1}y_{4}z_{8}y_{5}z_{7}\right) $ $\left[84, 72, 4\right] _{2}^{b}$
    23 $% (y_{4}y_{4}y_{4}y_{1}y_{6}z_{8}y_{1}z_{6}y_{2}y_{2}z_{1}z_{4}y_{4}y_{1}z_{4}z_{6}y_{1}z_{7}y_{6}z_{1}z_{7}z_{3}z_{6}) $ $\left[92, 66, 8\right] _{2}^{\ast }$
     | Show Table
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    Table 2.  Two-generator cyclic codes over $\mathbb{F}_{2}\left[u, v\right] /\left\langle u^{2}+v^{2}, uv\right\rangle $

    $n$ $\left. f\left(x\right), g\left(x\right) \right. $ $\phi \left({C}\right) $
    3 $\left. \left(z_{6}, z_{5, }z_{7}\right), \left(z_{8}, z_{8}, z_{8}\right) \right. $ $\left[12, 6, 4\right] _{2}^{\ast }$
    3 $\left. \left(z_{8}, z_{6}, z_{7}\right), \left(z_{5}, z_{3}, z_{6}\right) \right. $ $\left[12, 7, 4\right] _{2}^{\ast }$
    5 $\left. \left(z_{7}, z_{2}, z_{2}, z_{7}, z_{1}\right), \left(z_{3}, z_{3}, z_{3}, z_{7}, z_{1}\right) \right. $ $\left[20, 13, 4\right] _{2}^{\ast }$
    7 $\left. \left(z_{7}, z_{4}, z_{4}, z_{6}, z_{2}, z_{4}, z_{7}\right), \left(z_{8}, z_{5}, z_{1}, z_{1}, z_{4}, z_{2}, z_{2}\right) \right. $ $\left[28, 15, 6% \right] _{2}^{\ast }$
    7 $\left. \left(z_{5}, z_{5}, z_{7}, z_{2}, z_{4}, z_{7}, z_{6}\right), \left(z_{5}, y_{2}z_{8}, y_{5}, y_{1}, y_{1}, z_{7}\right) \right. $ $\left[28, 21, 4% \right] _{2}^{\ast }$
    7 $\left. \left(z_{5}, z_{5}, z_{7}, z_{2}, z_{4}, z_{7}, z_{6}\right), \left(z_{7}, y_{5}, z_{1}, y_{7}, y_{2}, y_{4}, z_{3}\right) \right. $ $\left[28, 22, 4% \right] _{2}^{\ast }$
    15 $\left. \begin{array}{c} \left(z_{5}, z_{3}, z_{4}, z_{1}, z_{4}, z_{6}, z_{7}, z_{5}, z_{7}, z_{1}, z_{3}, z_{6}, z_{3}, z_{4}, z_{7}\right), \\ \left(z_{2}, z_{7}, z_{3}, z_{7}, z_{3}, z_{1}, z_{8}, z_{6}, z_{3}, z_{2}, z_{1}, z_{4}, z_{4}, z_{3}, z_{6}\right)% \end{array}% \right. $ $\left[60, 28, 10\right] _{2}^{b-2}$
    15 $\left. \begin{array}{c} \left(z_{5}, z_{3}, z_{4}, z_{1}, z_{4}, z_{6}, z_{7}, z_{5}, z_{7}, z_{1}, z_{3}, z_{6}, z_{3}, z_{4}, z_{7}\right), \\ \left(z_{4}, y_{1}, z_{3}, z_{7}, z_{8}, y_{2}, z_{4}, z_{5}, y_{2}, y_{5}, y_{5}, y_{7}, y_{8}, y_{6}, z_{4}\right)% \end{array}% \right.$ $\left[60, 44, 6\right] _{2}^{b}$
    17 $\left. \begin{array}{c} \left(z_{7}, z_{1}, z_{2}, z_{8}, y_{2}, y_{7}, y_{7}, z_{1}, z_{3}, y_{2}, y_{3}, z_{6}, z_{3}, y_{4}, y_{4}, y_{3}, z_{6}\right), \\ \left(y_{5}, y_{8}, z_{2}, y_{7}, y_{5}, z_{2}, z_{7}, y_{5}, z_{1}, y_{7}, y_{2}, y_{2}, y_{8}, z_{4}, y_{6}, z_{6}, z_{2}\right)% \end{array}% \right.$ $\left[68, 56, 4\right] _{2}^{b}$
    17 $\left. \begin{array}{c} \left(z_{7}, z_{1}, z_{2}, z_{8}, y_{2}, y_{7}, y_{7}, z_{1}, z_{3}, y_{2}, y_{3}, z_{6}, z_{3}, y_{4}, y_{4}, y_{3}, z_{6}\right), \\ \left(z_{1}, z_{5}, y_{3}, y_{8}, z_{6}, z_{4}, z_{2}, z_{6}, y_{7}, y_{5}, z_{3}, z_{2}, y_{2}, z_{3}, z_{7}, z_{6}, y_{2}\right)% \end{array}% \right.$ $\left[68, 57, 4\right] _{2}^{b}$
    21 $\left. \begin{array}{c} \left(z_{4}y_{3}y_{1}y_{8}z_{3}y_{2}z_{3}z_{2}z_{2}y_{2}y_{7}y_{2}y_{5}y_{8}z_{1}z_{2}y_{5}y_{6}y_{1}z_{6}z_{1}\right), \\ \left(z_{3}z_{6}z_{3}z_{1}z_{1}z_{8}z_{3}z_{1}z_{6}z_{6}z_{4}z_{2}z_{3}z_{5}z_{8}z_{4}z_{5}z_{4}z_{7}z_{1}z_{7}\right)% \end{array}% \right.$ $\left[84, 63, 6\right] _{2}^{\ast -2}$
    21 $\left. \begin{array}{c} \left(z_{4}y_{3}y_{1}y_{8}z_{3}y_{2}z_{3}z_{2}z_{2}y_{2}y_{7}y_{2}y_{5}y_{8}z_{1}z_{2}y_{5}y_{6}y_{1}z_{6}z_{1}\right), \\ \left(z_{6}y_{4}y_{8}z_{3}z_{7}z_{3}y_{5}y_{7}y_{5}z_{7}y_{7}z_{8}y_{4}z_{8}y_{3}y_{2}y_{4}z_{5}y_{3}z_{1}y_{2}\right)% \end{array}% \right.$ $\left[84, 75, 4\right] _{2}^{\ast }$
    21 $\left. \begin{array}{c} \left(z_{4}y_{3}y_{1}y_{8}z_{3}y_{2}z_{3}z_{2}z_{2}y_{2}y_{7}y_{2}y_{5}y_{8}z_{1}z_{2}y_{5}y_{6}y_{1}z_{6}z_{1}\right), \\ \left(y_{6}y_{5}z_{7}y_{5}y_{5}z_{1}y_{2}y_{4}y_{6}z_{6}z_{4}y_{3}z_{8}z_{1}z_{5}y_{4}y_{4}z_{7}z_{8}z_{8}z_{6}\right)% \end{array}% \right.$ $\left[84, 76, 4\right] _{2}^{\ast }$
    23 $\left. \begin{array}{c} (y_{4}y_{4}y_{4}y_{1}y_{6}z_{8}y_{1}z_{6}y_{2}y_{2}z_{1}z_{4}y_{4}y_{1}z_{4}z_{6}y_{1}z_{7}y_{6}z_{1}z_{7}z_{3}z_{6}), \\ \left(z_{5}z_{3}z_{1}z_{7}z_{6}z_{3}z_{5}z_{1}z_{3}z_{2}z_{5}z_{2}z_{2}z_{4}z_{7}z_{1}z_{1}z_{3}z_{2}z_{8}z_{3}z_{8}z_{4}\right)% \end{array}% \right. $ $\left[92, 78, 4\right] _{2}^{b-1}$
     | Show Table
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    Table 3.  Some cyclic codes over $\mathbb{Z}_{4}\left[u\right] /\left\langle u^{2}-2\right\rangle $

    $n$ $f\left(x\right) $ $\phi _{\mathbb{Z}_{4}}\left({C}\right) $ $\phi \left({C}\right) $
    3 $\left(00|02|02\right) $ $\left(3, 4^{0}2^{2}, 8\right) $ $\left[12, 2, 8\right] _{2}^{\ast }$
    3 $\left(03|03|03\right) $ $\left(3, 4^{1}2^{1}, 6\right) $ $\left[12, 3, 6\right] _{2}^{\ast }$
    3 $\left(01|00|01\right) $ $\left(3, 4^{2}2^{3}, 6\right) $ $\left[12, 7, 4\right] _{2}^{\ast }$
    5 $\left(21|21|21|21|21\right) $ $\left(10, 4^{1}2^{1}, 10\right) $ $% \left[20, 3, 10\right] _{2}^{\ast -1}$
    5 $\left(01|22|01|01|23\right) $ $\left(10, 4^{4}2^{4}, 4\right) $ $% \left(20, 2^{12}, 4\right) _{2}^{\ast }$
    5 $\left(03|03|21|23|00\right) $ $\left(10, 4^{4}2^{5}, 4\right) $ $% \left[20, 13, 4\right] _{2}^{\ast }$
    7 $\left(00|02|00|02|02|02|00\right) $ $\left(14, 4^{0}2^{3}, 8\right) $ $\left[28, 3, 16\right] _{2}^{\ast }$
    7 $\left(23|21|21|23|23|23|21\right) $ $\left(14, 4^{1}2^{4}, 8\right) $ $\left[28, 6, 12\right] _{2}^{\ast }$
    7 $\left(22|23|00|01|23|01|22\right) $ $\left(14, 4^{3}2^{6}, 8\right) $ $\left[28, 12, 8\right] _{2}^{\ast }$
    7 $\left(11|13|23|20|02|33|00\right) $ $\left(14, 4^{6}2^{3}, 6\right) $ $\left(28, 2^{15}, 6\right) _{2}^{\ast }$
    7 $\left(23|00|30|01|11|33|02\right) $ $\left(14, 4^{8}2^{3}, 4\right) $ $\left(28, 2^{19}, 4\right) _{2}^{\ast }$
    7 $\left(33|30|11|22|12|00|20\right) $ $\left(14, 4^{9}2^{3}, 4\right) $ $\left(28, 2^{21}, 4\right) _{2}^{\ast }$
    7 $\left(32|01|33|33|11|21|03\right) $ $\left(14, 4^{9}2^{4}, 4\right) $ $\left(28, 2^{22}, 4\right) _{2}^{\ast }$
    9 $\left(33|23|33|32|03|30|13|21|32\right) $ $\left(18, 4^{10}2^{6}, 4\right) $ $\left(36, 2^{26}, 4\right) _{2}^{\ast }$
    9 $\left(12|02|11|32|21|10|30|20|32\right) $ $\left(18, 4^{10}2^{7}, 4\right) $ $\left(36, 2^{27}, 4\right) _{2}^{\ast }$
    17 $\left(02|20|01|30|32|32|02|03|31|31|32|21|33|20|21\right) $ $\left(34, 4^{24}2^{9}, 4\right) $ $\left(68, 2^{57}, 4\right) _{2}^{\ast }$
     | Show Table
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    Table 4.  Some cyclic codes over $\mathbb{Z}_{16}$

    $n$ $f\left(x\right) $ ${C}$ $\phi \left({C}\right) $
    3 $\left(8, 8, 0\right) $ $\left[3, 2^{2}, 8\right] $ $\left(12, 2^{2}, 8\right) _{2}^{\ast }$
    3 $\left(6, 6, 6\right) $ $\left[3, 8^{1}, 6\right] $ $\left(12, 2^{3}, 6\right) _{2}^{\ast }$
    3 $\left(12, 4, 8\right) $ $\left[3, 4^{2}2^{1}, 4\right] $ $\left(12, 2^{5}, 4\right) _{2}^{\ast }$
    3 $\left(4, 6, 6\right) $ $\left[3, 8^{2}, 4\right] $ $\left(12, 2^{6}, 4\right) _{2}^{\ast }$
    3 $\left(2, 2, 4\right) $ $\left[3, 8^{2}2^{1}, 4\right] $ $\left(12, 2^{7}, 4\right) _{2}^{\ast }$
    3 $\left(11, 3, 12\right) $ $\left[5, 8^{4}, 4\right] $ $\left(12, 2^{11}, 2\right) _{2}^{\ast }$
    5 $\left(14, 10, 6, 6, 12\right) $ $\left[5, 8^{4}, 4\right] $ $\left(20, 2^{12}, 4\right) _{2}^{\ast }$
    5 $\left(12, 14, 8, 4, 2\right) $ $\left[5, 8^{4}2^{1}, 4\right] $ $\left(20, 2^{13}, 4\right) _{2}^{\ast }$
    5 $\left(13, 6, 7, 10, 12\right) $ $\left[5, 16^{4}, 2\right] $ $\left(20, 2^{16}, 2\right) _{2}^{\ast }$
    5 $\left(7, 5, 3, 11, 14\right) $ $\left[5, 16^{4}2^{1}, 2\right] $ $% \left(20, 2^{17}, 2\right) _{2}^{\ast }$
    7 $\left(10, 12, 6, 10, 2, 8, 0\right) $ $\left[7, 8^{3}, 10\right] $ $% \left(28, 2^{9}, 10\right) _{2}^{\ast }$
    7 $\left(2, 12, 12, 6, 12, 14, 6\right) $ $\left[7, 8^{3}2^{3}, 8\right] $ $% \left(28, 2^{12}, 8\right) _{2}^{\ast }$
    7 $\left(13, 11, 7, 13, 1, 5, 7\right) $ $\left[7, 16^{1}8^{3}, 7\right] $ $% \left(28, 2^{13}, 7\right) _{2}^{\ast -1}$
    7 $\left(1, 4, 14, 7, 11, 7, 4\right) $ $\left[7, 16^{3}2^{3}, 6\right] $ $% \left(28, 2^{15}, 6\right) _{2}^{\ast }$
     | Show Table
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