Constructing self-dual codes from group rings and reverse circulant matrices

Joe Gildea; Adrian Korban; Abidin Kaya; Bahattin Yildiz

doi:10.3934/amc.2020077

Article Contents

2021, Volume 15, Issue 3: 471-485. Doi: 10.3934/amc.2020077

This issue Previous Article Finding small solutions of the equation

$ \mathit{{Bx-Ay = z}} $

and its applications to cryptanalysis of the RSA cryptosystem Next Article New optimal error-correcting codes for crosstalk avoidance in on-chip data buses

Constructing self-dual codes from group rings and reverse circulant matrices

1.
University of Chester, Department of Mathematical and Physical Sciences, Thornton Science Park, Pool Ln, Chester CH2 4NU, England
2.
Sampoerna University, Department of Engineering Fundamentals, 12780, Jakarta, Indonesia
3.
Northern Arizona University, Department of Mathematics & Statistics, Flagstaff, AZ 86001, USA

^* Corresponding author: Adrian Korban
^* Corresponding author: Adrian Korban

Received: September 30, 2019

Revised: December 31, 2019

Early access: April 2020

Published: August 2021

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Abstract

In this work, we describe a construction for self-dual codes in which we employ group rings and reverse circulant matrices. By applying the construction directly over different alphabets, and by employing the well known extension and neighbor methods we were able to obtain extremal binary self-dual codes of different lengths of which some have parameters that were not known in the literature before. In particular, we constructed three new codes of length 64, twenty-two new codes of length 68, twelve new codes of length 80 and four new codes of length 92.

Keywords:

Mathematics Subject Classification: 94B05, 16S34.

Citation:

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Table 1. Self-dual codes over $\mathbb{F}_{4}+u\mathbb{F}_{4}$ of length $64$ from $C_{2, 2}$

$\mathcal{C}_{i}$	$r_{\sigma(v_1)}$	$r_{\sigma(v_2)}$	$r_C$	$\|Aut(\mathcal{C}_i)\|$	$\beta$
$1$	$(0, 9, 2, 1)$	$(0, 0, A, 4)$	$(3, C, 3, 3)$	$2^{4}$	$0$
$2$	$(0, 9, 4, F)$	$(0, 0, 0, 6)$	$(2, 5, 2, 2)$	$2^{5}$	$0$

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Table 2. Extremal Self-dual codes of length $68$ from Theorem 2.1

$\mathcal{M}_{68, i}$	$\mathcal{C}_i$	$c$	$X$	$\gamma$	$\beta$	$\|Aut(\mathcal{M}_{68, i})\|$
$1$	$1$	$1$	$(3u0u01u103103030u01u0u0301u10013)$	$\bf{0}$	$\bf{40}$	$2$
$2$	$2$	$u+1$	$(33313311u3uu13110u1030u1u31u31u3)$	$\bf{3}$	$\bf{77}$	$2$

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Table 3. Self-dual codes over $ \mathbb{F}_{2}+u\mathbb{F}_{2} $ of length $ 64 $ from $ C_{4,2} $

$ \mathcal{E}_{i} $	$ r_{\sigma(v_1)} $	$ r_{\sigma(v_2)} $	$ r_C $	$ \|Aut(\mathcal{E}_i)\| $	$ \beta $
$ 1 $	$ (u,0,1,1,u,u,u,u) $	$ (u,u,1,1,0,u,u,3) $	$ (0,1,0,1,0,1,0,1) $	$ 2^5 $	$ 0 $
$ 2 $	$ (u,u,1,3,u,u,u,u) $	$ (u,u,1,1,u,u,0,3) $	$ (u,3,u,3,u,3,u,3) $	$ 2^6 $	$ 0 $
$ 3 $	$ (u,0,u,u,0,0,1,3) $	$ (u,u,0,1,u,0,3,1) $	$ (3,3,3,3,3,3,3,3) $	$ 2^7 $	$ 80 $

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Table 4. New codes of length 64 as neighbors

$ \mathcal{L}_{64,i} $	$ \mathcal{E}_{i} $	$ (x_{33},...,x_{64}) $	$ W_{64,i} $	$ \beta $	$ \|Aut(\mathcal{L}_{64,i})\| $
$ 1 $	$ 3 $	$ (01110001001001101000011001011111) $	$ 1 $	$ \bf{58} $	$ 2^2 $
$ 2 $	$ 3 $	$ (11011001001101110110010110011010) $	$ 2 $	$ \bf{54} $	$ 2^3 $
$ 3 $	$ 3 $	$ (11111100101011111001111001010010) $	$ 2 $	$ \bf{62} $	$ 2 $

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Table 5. Extremal Self-dual codes of length $ 68 $ from Theorem 2.1

$ \mathcal{N}_{68,i} $	$ \mathcal{E}_i $	$ c $	$ X $	$ \gamma $	$ \beta $	$ \|Aut(\mathcal{N}_{68,i})\| $
$ 1 $	$ 1 $	$ 3 $	$ (01330u3131uuu3330uuuu000333u1u1u) $	$ \bf{0} $	$ \bf{39} $	$ 2 $
$ 2 $	$ 2 $	$ 1 $	$ (0013u1111uu1u0uuuu101u1333330130) $	$ \bf{3} $	$ \bf{79} $	$ 2 $
$ 3 $	$ 2 $	$ 1 $	$ (u30u1u03u10uu113uuu01131u111u030) $	$ \bf{3} $	$ \bf{85} $	$ 2 $

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Table 6. $ [80,40,14] $ Self-dual codes over $ \mathbb{F}_{4}+u \mathbb{F}_{4} $ from $ C_{5} $

$ \mathcal{D}_{i} $	$ r_{\sigma(v_1)} $	$ r_{\sigma(v_2)} $	$ r_C $	$ \|Aut(\mathcal{D}_i)\| $	$ (\beta,\alpha) $
$ 1 $	$ (A,A,A,1,3) $	$ (0,2,1,3,E) $	$ (7,7,7,7,7) $	$ 2^3 \cdot 5 $	$ (0,-120) $
$ 2 $	$ (0,A,2,6,F) $	$ (2,1,E,2,1) $	$ (6,6,6,6,6) $	$ 2^2 \cdot 5 $	$ (0,-125) $
$ 3 $	$ (A,A,0,4,F) $	$ (2,A,6,2,F) $	$ (1,1,1,1,1) $	$ 2^2 \cdot 5 $	$ (0,-150) $
$ 4 $	$ (0,A,A,4,5) $	$ (0,3,6,A,B) $	$ (E,E,E,E,E) $	$ 2^2 \cdot 5 $	$ (0,-155) $
$ 5 $	$ (2,0,A,4,5) $	$ (2,A,4,0,5) $	$ (B,B,B,B,B) $	$ 2^2 \cdot 5 $	$ (0,-180) $
$ 6 $	$ (0,A,B,B,E) $	$ (0,2,1,3,1) $	$ (6,6,6,6,6) $	$ 2^2 \cdot 5 $	$ (0,-190) $
$ 7 $	$ (0,A,2,6,F) $	$ (2,2,6,2,7) $	$ (B,B,B,B,B) $	$ 2^2 \cdot 5 $	$ (0,-200) $
$ 8 $	$ (0,0,A,6,F) $	$ (2,1,E,0,3) $	$ (6,6,6,6,6) $	$ 2^2 \cdot 5 $	$ (0,-215) $
$ 9 $	$ (A,0,1,4,7) $	$ (0,3,E,2,7) $	$ (B,B,B,B,B) $	$ 2^2 \cdot 5 $	$ (0,-230) $
$ 10 $	$ (A,2,A,1,4) $	$ (0,0,7,1,F) $	$ (7,7,7,7,7) $	$ 2^2 \cdot 5 $	$ (0,-250) $
$ 11 $	$ (A,A,3,B,4) $	$ (0,A,4,0,7) $	$ (4,4,4,4,4) $	$ 2^2 \cdot 5 $	$ (0,-275) $
$ 12 $	$ (0,2,B,1,E) $	$ (0,0,1,1,3) $	$ (4,4,4,4,4) $	$ 2^2 \cdot 5 $	$ (10,-370) $

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Table 7. Self-dual codes over $ \mathbb{F}_{2} $ of length $ 68 $ from $ C_{17} $

$ \mathcal{C}_{i} $	$ r_{\sigma(v_1)} $	$ r_{\sigma(v_2)} $	$ \gamma $	$ \beta $	$ \|Aut(\mathcal{C}_i)\| $
$ 1 $	$ (0,0,0,0,0,0,0,1,1,0,1,1,0,1,1,1,1) $	$ (0,0,0,1,0,0,0,1,1,1,0,0,1,0,1,1,1) $	$ 0 $	$ 255 $	$ 2 \cdot 17 $
$ 2 $	$ (0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,1,1) $	$ (0,0,0,1,0,1,1,1,1,0,1,1,0,1,1,1,1) $	$ 0 $	$ 272 $	$ 2^2 \cdot 17 $

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Table 8. New codes of length 68 as neighbors

$ \mathcal{N}_{68,i} $	$ \mathcal{C}_{i} $	$ (x_{35},x_{36},...,x_{68}) $	$ \gamma $	$ \beta $	$ \mathcal{N}_{68,i} $	$ \mathcal{C}_{i} $	$ (x_{35},x_{36},...,x_{68}) $	$ \gamma $	$ \beta $
$ \mathcal{N}_{68,1} $	$ \mathcal{C}_{1} $	$ (1001110010101010011000001000111011) $	$ \boldsymbol{0} $	$ \boldsymbol{183} $	$ \mathcal{N}_{68,2} $	$ \mathcal{C}_{1} $	$ (1101000000010001110101011010100001) $	$ \boldsymbol{0} $	$ \boldsymbol{185} $
$ \mathcal{N}_{68,3} $	$ \mathcal{C}_{1} $	$ (0001000010011000110101100010101000) $	$ \boldsymbol{0} $	$ \boldsymbol{189} $	$ \mathcal{N}_{68,4} $	$ \mathcal{C}_{1} $	$ (0100000001110110100011110011101111) $	$ \boldsymbol{0} $	$ \boldsymbol{191} $
$ \mathcal{N}_{68,5} $	$ \mathcal{C}_{1} $	$ (0110110001001000110010110111100001) $	$ \boldsymbol{0} $	$ \boldsymbol{193} $	$ \mathcal{N}_{68,6} $	$ \mathcal{C}_{2} $	$ (0000001110101000111001011000001101) $	$ \boldsymbol{0} $	$ \boldsymbol{195} $
$ \mathcal{N}_{68,7} $	$ \mathcal{C}_{2} $	$ (1001000111000100110010000111111111) $	$ \boldsymbol{0} $	$ \boldsymbol{197} $	$ \mathcal{N}_{68,8} $	$ \mathcal{C}_{2} $	$ (0110100100000001010000101001011100) $	$ \boldsymbol{0} $	$ \boldsymbol{199} $
$ \mathcal{N}_{68,9} $	$ \mathcal{C}_{2} $	$ (1010111001110010001010100100011010) $	$ \boldsymbol{0} $	$ \boldsymbol{200} $	$ \mathcal{N}_{68,10} $	$ \mathcal{C}_{2} $	$ (0000000100000111100111110000110110) $	$ \boldsymbol{0} $	$ \boldsymbol{203} $
$ \mathcal{N}_{68,11} $	$ \mathcal{C}_{1} $	$ (1001010000011000011101100011101101) $	$ \boldsymbol{1} $	$ \boldsymbol{189} $	$ \mathcal{N}_{68,12} $	$ \mathcal{C}_{1} $	$ (0110100111000110000001001001100011) $	$ \boldsymbol{1} $	$ \boldsymbol{201} $
$ \mathcal{N}_{68,13} $	$ \mathcal{C}_{1} $	$ (1010011111110001111001110111001110) $	$ \boldsymbol{1} $	$ \boldsymbol{203} $	$ \mathcal{N}_{68,14} $	$ \mathcal{C}_{1} $	$ (1111011111101101100101100000010101) $	$ \boldsymbol{1} $	$ \boldsymbol{205} $
$ \mathcal{N}_{68,15} $	$ \mathcal{C}_{1} $	$ (1011110111111110101101111111101111) $	$ \boldsymbol{1} $	$ \boldsymbol{213} $	$ \mathcal{N}_{68,16} $	$ \mathcal{C}_{2} $	$ (1010001111110100000010100011101001) $	$ \boldsymbol{1} $	$ \boldsymbol{216} $
$ \mathcal{N}_{68,17} $	$ \mathcal{C}_{1} $	$ (1011110011111011001101111100111101) $	$ \boldsymbol{1} $	$ \boldsymbol{217} $	$ \mathcal{N}_{68,18} $	$ \mathcal{C}_{2} $	$ (0000010011001100100101011101110101) $	$ \boldsymbol{1} $	$ \boldsymbol{233} $

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Table 9. Self-dual codes over $ \mathbb{F}_{2} $ of length $ 92 $ from $ C_{23} $

$ \mathcal{C}_{i} $	$ r_{\sigma(v_1)} $	$ r_{\sigma(v_2)} $	$ \gamma $	$ \beta $	$ \|Aut(\mathcal{C}_i)\| $	Type
$ 1 $	$ (0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,0,1,1,1) $	$ (0,0,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,0,1,1,1) $	$ 0 $	$ \textbf{759} $	$ 2 \cdot 23 $	$ W_{92,1} $
$ 3 $	$ (0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,1,0,1,1) $	$ (0,0,0,0,1,1,0,0,0,1,1,0,1,0,0,1,1,0,1,0,1,1,1) $	$ 0 $	$ \textbf{1012} $	$ 2 \cdot 23 $	$ W_{92,1} $
$ 13 $	$ (0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,1,1,1) $	$ (0,0,0,0,1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,1) $	$ -46 $	$ \textbf{1564} $	$ 2^2 \cdot 23 $	$ W_{92,1} $
$ 16 $	$ (0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,0,1,1) $	$ (0,0,0,0,0,0,1,0,0,0,0,1,1,0,1,0,1,1,1,0,0,0,1) $	$ -46 $	$ \textbf{1978} $	$ 2 \cdot 23 $	$ W_{92,1} $

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