Cyclic codes over local Frobenius rings of order 16

Steven T. Dougherty; Abidin Kaya; Esengül Saltürk

doi:10.3934/amc.2017005

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2017, Volume 11, Issue 1: 99-114. Doi: 10.3934/amc.2017005

This issue Previous Article DNA cyclic codes over rings Next Article Generalized Hamming weights of codes over the $\mathcal{GH}$ curve

Cyclic codes over local Frobenius rings of order 16

1.
Department of Mathematics, University of Scranton, Scranton, PA 18518, USA
2.
Department of Mathematics, Fatih University, Istanbul, 34500, Turkey
3.
Department of Mathematics, University of Scranton, Scranton, PA 18518, USA

^* Corresponding author
^* Corresponding author

Received: June 2015

Published: May 2017

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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Abstract

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.

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

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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}) $	$\left[92, 66, 8\right] _{2}^{\ast }$

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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. $	$\left[92, 78, 4\right] _{2}^{b-1}$

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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) $	$\left(68, 2^{57}, 4\right) _{2}^{\ast }$

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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] $	$% \left(28, 2^{15}, 6\right) _{2}^{\ast }$

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Cyclic codes over local Frobenius rings of order 16

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