New self-dual and formally self-dual codes from group ring constructions

Steven T. Dougherty; Joe Gildea; Abidin Kaya; Bahattin Yildiz

doi:10.3934/amc.2020002

Article Contents

2020, Volume 14, Issue 1: 11-22. Doi: 10.3934/amc.2020002

This issue Previous Article Construction and assignment of orthogonal sequences and zero correlation zone sequences for applications in CDMA systems Next Article A construction of bent functions with optimal algebraic degree and large symmetric group

New self-dual and formally self-dual codes from group ring constructions

1.
Department of Mathematics, University of Scranton, Scranton, PA 18510, USA
2.
Department of Mathematics, University of Chester, Chester, UK
3.
Department of Mathematics Education, Sampoerna University, 12780, Jakarta, Indonesia
4.
Department of Mathematics & Statistics, Northern Arizona University, Flagstaff, AZ 86011, USA

^* Corresponding author: Bahattin Yildiz
^* Corresponding author: Bahattin Yildiz

Received: January 2018

Revised: February 2019

Published: August 2019

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Abstract

In this work, we study construction methods for self-dual and formally self-dual codes from group rings, arising from the cyclic group, the dihedral group, the dicyclic group and the semi-dihedral group. Using these constructions over the rings $ \mathbb{F}_2+u \mathbb{F}_2 $ and $ \mathbb{F}_4+u \mathbb{F}_4 $, we obtain 9 new extremal binary self-dual codes of length 68 and 25 even formally self-dual codes with parameters $ [72,36,14] $.

Keywords:

Mathematics Subject Classification: Primary: 94B05; Secondary: 16S34.

Citation:

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Table 1. $ \left[ 64,32,12\right] _{2} $ codes via $ C_{mn} $ with $ m = 4,n = 2 $ over $ \mathbb{F}_{4}+u\mathbb{F}_{4} $

$ \mathcal{C}_{64,i} $	$ r_{A_{1}} $	$ r_{A_{2}} $	$ \|Aut(\mathcal{C}_{i})\| $	$ \beta $ in $ W_{64,2} $
$ \mathcal{C}_{64,1} $	$ \left( B,4,6,2\right) $	$ \left( E,9,7,0\right) $	$ 2^{6} $	0
$ \mathcal{C}_{64,2} $	$ \left( B,6,6,0\right) $	$ \left( E,3,5,8\right) $	$ 2^{5} $	4
$ \mathcal{C}_{64,3} $	$ \left( 9,E,E,0\right) $	$ \left( 6,9,3,A\right) $	$ 2^{5} $	12
$ \mathcal{C}_{64,4} $	$ \left( B,4,C,A\right) $	$ \left( 6,B,D,0\right) $	$ 2^{5} $	16
$ \mathcal{C}_{64,5} $	$ \left( 3,6,E,0\right) $	$ \left( E,B,F,A\right) $	$ 2^{5} $	20
$ \mathcal{C}_{64,6} $	$ \left( 1,E,6,0\right) $	$ \left( C,9,D,8\right) $	$ 2^{5} $	36
$ \mathcal{C}_{64,7} $	$ \left( 9,C,E,2\right) $	$ \left( C,9,D,8\right) $	$ 2^{5} $	48
$ \mathcal{C}_{64,8} $	$ \left( 3,6,4,8\right) $	$ \left( E,B,F,A\right) $	$ 2^{5} $	52

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Table 2. $ \left[ 64,32,12\right] _{2} $ codes via $ C_{mn} $ $ \ $with $ m = 2,n = 4 $ over $ \mathbb{F}_{4}+u\mathbb{F}_{4} $

$ \mathcal{C}_{64,i} $	$ r_{A_{1}} $	$ r_{A_{2}} $	$ r_{A_{3}} $	$ r_{A_{4}} $	$ \|Aut(\mathcal{C}_{64,i})\| $	$ \beta $ in $ W_{64,2} $
$ \mathcal{C}_{64,9} $	$ \left( D,6\right) $	$ \left( 8,F\right) $	$ \left( B,0\right) $	$ \left( 4,A\right) $	$ 2^{6} $	0
$ \mathcal{C}_{64,10} $	$ \left( D,6\right) $	$ \left( 8,F\right) $	$ \left( B,0\right) $	$ \left( 6,8\right) $	$ 2^{5} $	4
$ \mathcal{C}_{64,11} $	$ \left( 7,4\right) $	$ \left( E,0\right) $	$ \left( 8,B\right) $	$ \left( 1,6\right) $	$ 2^{5} $	12
$ \mathcal{C}_{64,12} $	$ \left( 4,A\right) $	$ \left( 0,B\right) $	$ \left( C,B\right) $	$ \left( 6,D\right) $	$ 2^{5} $	16
$ \mathcal{C}_{64,13} $	$ \left( D,C\right) $	$ \left( 4,2\right) $	$ \left( 8,9\right) $	$ \left( 1,C\right) $	$ 2^{5} $	20
$ \mathcal{C}_{64,14} $	$ \left( D,4\right) $	$ \left( 6,8\right) $	$ \left( A,3\right) $	$ \left( 3,4\right) $	$ 2^{5} $	28
$ \mathcal{C}_{64,15} $	$ \left( F,4\right) $	$ \left( 8,5\right) $	$ \left( 9,2\right) $	$ \left( C,A\right) $	$ 2^{6} $	32
$ \mathcal{C}_{64,16} $	$ \left( 5,6\right) $	$ \left( C,2\right) $	$ \left( A,9\right) $	$ \left( 3,4\right) $	$ 2^{5} $	36
$ \mathcal{C}_{64,17} $	$ \left( 5,E\right) $	$ \left( C,A\right) $	$ \left( 8,3\right) $	$ \left( 3,E\right) $	$ 2^{5} $	44
$ \mathcal{C}_{64,18} $	$ \left( 8,9\right) $	$ \left( D,A\right) $	$ \left( C,D\right) $	$ \left( D,3\right) $	$ 2^{5} $	48
$ \mathcal{C}_{64,19} $	$ \left( D,C\right) $	$ \left( E,0\right) $	$ \left( 8,9\right) $	$ \left( 9,E\right) $	$ 2^{5} $	52

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Table 3. $ \left[ 64,32,12\right] _{2} $ codes via $ D_{8} $ over $ \mathbb{F} _{4}+u\mathbb{F}_{4} $

$ \mathcal{C}_{64,i} $	$ r_{A} $	$ r_{B} $	$ \left\vert Aut\left( \mathcal{C} _{64,i}\right) \right\vert $	$ \beta $ in $ W_{64,2} $
$ \mathcal{C}_{64,20} $	$ \left( 6,D,7,E\right) $	$ \left( 2,2,4,5\right) $	$ 2^{6} $	0
$ \mathcal{C}_{64,21} $	$ \left( C,5,F,4\right) $	$ \left( 2,0,C,D\right) $	$ 2^{5} $	4
$ \mathcal{C}_{64,22} $	$ \left( 0,C,8,8\right) $	$ \left( 9,A,B,D\right) $	$ 2^{5} $	8
$ \mathcal{C}_{64,23} $	$ \left( 4,B,A,4\right) $	$ \left( E,D,5,C\right) $	$ 2^{4} $	12
$ \mathcal{C}_{64,24} $	$ \left( F,8,5,E\right) $	$ \left( A,1,D,9\right) $	$ 2^{4} $	16
$ \mathcal{C}_{64,25} $	$ \left( 6,4,9,2\right) $	$ \left( 7,C,C,F\right) $	$ 2^{4} $	20
$ \mathcal{C}_{64,26} $	$ \left( E,3,B,1\right) $	$ \left( 0,7,8,1\right) $	$ 2^{5} $	24
$ \mathcal{C}_{64,27} $	$ \left( 9,9,8,0\right) $	$ \left( 6,6,1,2\right) $	$ 2^{4} $	28
$ \mathcal{C}_{64,28} $	$ \left( 7,0,A,8\right) $	$ \left( F,8,5,C\right) $	$ 2^{6} $	32
$ \mathcal{C}_{64,29} $	$ \left( 6,7,7,6\right) $	$ \left( A,2,4,5\right) $	$ 2^{4} $	$ 36 $
$ \mathcal{C}_{64,30} $	$ \left( 0,6,8,2\right) $	$ \left( 6,3,1,1\right) $	$ 2^{5} $	40
$ \mathcal{C}_{64,31} $	$ \left( 5,F,E,E\right) $	$ \left( 4,1,0,C\right) $	$ 2^{4}\times 3 $	44
$ \mathcal{C}_{64,32} $	$ \left( F,F,6,6\right) $	$ \left( 4,3,A,4\right) $	$ 2^{5} $	48
$ \mathcal{C}_{64,33} $	$ \left( D,D,4,4\right) $	$ \left( 6,B,0,6\right) $	$ 2^{5} $	52

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Table 4. $ \left[ 64,32,12\right] _{2} $ codes via $ D_{16} $ over $ \mathbb{F} _{2}+u\mathbb{F}_{2} $

$ \mathcal{C}_{64,i} $	$ r_{A} $	$ r_{B} $	$ \left\vert Aut\left( \mathcal{C} _{64,i}\right) \right\vert $	$ \beta $ in $ W_{64,2} $
$ \mathcal{C}_{64,34} $	$ \left( 03331uu0\right) $	$ \left( 0003u013\right) $	$ 2^{5} $	0
$ \mathcal{C}_{64,35} $	$ \left( 3031u110\right) $	$ \left( 0u30100u\right) $	$ 2^{5} $	16
$ \mathcal{C}_{64,36} $	$ \left( 031u13uu\right) $	$ \left( u01001u1\right) $	$ 2^{5} $	32
$ \mathcal{C}_{64,37} $	$ \left( 11013uu3\right) $	$ \left( u003111u\right) $	$ 2^{5} $	48
$ \mathcal{C}_{64,38} $	$ \left( 3u13u130\right) $	$ \left( 0u301u0u\right) $	$ 2^{7} $	$ 80 $

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Table 5. New extremal binary self-dual codes of length $ 68 $

$ \mathcal{C}_{68,i} $	$ \mathcal{C} $	$ c $	$ X $	$ \gamma $	$ \beta $
$ \mathcal{C}_{68,1} $	$ \mathcal{C}_{64,29} $	$ 1+u $	$ \left( uu0u33131u1333130u30uu3130113133\right) $	$ 3 $	$ 135 $
$ \mathcal{C}_{68,2} $	$ \mathcal{C}_{64,29} $	$ 1+u $	$ \left( 00u013331u1131310u30u0111u331313\right) $	$ 3 $	$ 139 $
$ \mathcal{C}_{68,3} $	$ \mathcal{C}_{64,29} $	$ 1+u $	$ \left( uu0u33333u131133003u0u113u131331\right) $	$ 3 $	$ 143 $
$ \mathcal{C}_{68,4} $	$ \mathcal{C}_{64,29} $	$ 1+u $	$ \left( uu0011133u311331uu1u0u3110113111\right) $	$ 3 $	$ 151 $
$ \mathcal{C}_{68,5} $	$ \mathcal{C}_{64,29} $	$ 1+u $	$ \left( u0uu333310311331uu30u0331u331113\right) $	$ 3 $	$ 155 $
$ \mathcal{C}_{68,6} $	$ \mathcal{C}_{64,29} $	$ 1 $	$ \left( u0u011131u1311110u30u03110113111\right) $	$ 3 $	$ 161 $
$ \mathcal{C}_{68,7} $	$ \mathcal{C}_{64,38} $	$ 1+u $	$ \left( 33131u0101333uu103310030uu11uu13\right) $	$ 3 $	$ 186 $
$ \mathcal{C}_{68,8} $	$ \mathcal{C}_{64,38} $	$ 1 $	$ \left( 13131uuu0u0u3033u1u130100310u1u0\right) $	$ 3 $	$ 202 $
$ \mathcal{C}_{68,9} $	$ \mathcal{C}_{64,38} $	$ 1 $	$ \left( 133330u301331uu30311003uu0130011\right) $	$ 3 $	$ 204 $

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Table 6. FSD $ \left[ 72,36,14\right] _{2}^{b-1} $codes by $ C_{mn} $ construction over $ \mathbb{F}_{2}+u\mathbb{F}_{2} $

$ n $	$ m $	$ r_{1},\ldots ,r_{n} $	$ A_{14} $	$ A_{16} $	$ A_{18} $
$ 2 $	$ 9 $	$ 13u10u000,03u100011 $	$ 8820 $	$ 123039 $	$ 1210564 $
$ 2 $	$ 9 $	$ 30u0031uu,10u1u3u3u $	$ 8856 $	$ 122850 $	$ 1210492 $
$ 2 $	$ 9 $	$ u00u31uu1,01uu3u013 $	$ 8784 $	$ 123417 $	$ 1207344 $
$ 2 $	$ 9 $	$ 3031uu10u,300333u30 $	$ 8928 $	$ 122436 $	$ 1210776 $
$ 2 $	$ 9 $	$ uu0003103,300u1303u $	$ 9288 $	$ 120690 $	$ 1208328 $
$ 2 $	$ 9 $	$ 1333u1313,11u3u31uu $	$ 9360 $	$ 119583 $	$ 1216936 $
$ 3 $	$ 6 $	$ 11010u,30u1u0,103u11 $	$ 8820 $	$ 123327 $	$ 1207092 $
$ 3 $	$ 6 $	$ u30u33,333130,0301u3 $	$ 9180 $	$ 121194 $	$ 1209304 $
$ 3 $	$ 6 $	$ 33u101,0u0311,13u1u3 $	$ 9504 $	$ 119151 $	$ 1212760 $
$ 3 $	$ 6 $	$ 3uuu0u,u31u03,10uu13 $	$ 9648 $	$ 118170 $	$ 1215172 $

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Table 7. FSD $ \left[ 72,36,14\right] _{2}^{b-1} $codes by $ C_{mn} $ over $ \mathbb{F}_{2} $

$ n $	$ m $	$ r_{1},\ldots ,r_{n} $	$ A_{14} $	$ A_{16} $	$ A_{18} $
$ 3 $	$ 12 $	$ 000100000010,110001110110,000010011010 $	$ 8496 $	$ 124911 $	$ 1209160 $
$ 3 $	$ 12 $	$ 011110101111,010110100010,101011000100 $	$ 8568 $	$ 124362 $	$ 1211068 $
$ 3 $	$ 12 $	$ 111100000101,100100101100,111000111110 $	$ 9072 $	$ 121653 $	$ 1210816 $
$ 3 $	$ 12 $	$ 100111000011,011000001011,010001011011 $	$ 9144 $	$ 121221 $	$ 1211328 $
$ 3 $	$ 12 $	$ 001110100000,000000111000,110111001000 $	$ 9468 $	$ 119601 $	$ 1209700 $
$ 4 $	$ 9 $	$ 001011011,010011110,001100010,101000011 $	$ 8388 $	$ 125730 $	$ 1206348 $
$ 4 $	$ 9 $	$ 010110011,100110010,101100111,000101011 $	$ 8712 $	$ 123741 $	$ 1209160 $
$ 4 $	$ 9 $	$ 001111011,100100100,010101100,010111000 $	$ 8820 $	$ 123039 $	$ 1210564 $
$ 4 $	$ 9 $	$ 111011010,101110101,111000101,110001001 $	$ 8928 $	$ 122328 $	$ 1212076 $
$ 4 $	$ 9 $	$ 000010001,111001101,101101110,110011100 $	$ 8928 $	$ 122769 $	$ 1206784 $
$ 4 $	$ 9 $	$ 110001100,101110111,001100010,100110110 $	$ 9036 $	$ 121761 $	$ 1211868 $
$ 4 $	$ 9 $	$ 101100101,101110111,010011000,010010110 $	$ 9036 $	$ 121977 $	$ 1208276 $
$ 6 $	$ 6 $	$ 000100,110100,010111,101000,100000,010111 $	$ 8388 $	$ 125973 $	$ 1203436 $
$ 6 $	$ 6 $	$ 001010,011001,110010,111011,010100,110101 $	$ 8784 $	$ 123570 $	$ 1206532 $
$ 6 $	$ 6 $	$ 101001,001110,110110,101000,000110,000100 $	$ 9360 $	$ 120114 $	$ 1210564 $

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