Steiner systems $ S(2,4, 2^m) $ supported by a family of extended cyclic codes

Qi Wang

doi:10.3934/amc.2022067

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2022, Volume 16, Issue 4: 1011-1022. Doi: 10.3934/amc.2022067

This issue Previous Article Several new classes of optimal ternary cyclic codes with minimum distance four Next Article On construction of lightweight MDS matrices

Steiner systems $ S(2,4, 2^m) $ supported by a family of extended cyclic codes

Qi Wang^1,

Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China

Received: March 2022

Revised: June 2022

Early access: August 15, 2022

Published online: August 15, 2022

The author is supported by the Shenzhen fundamental research program (Grant No. 20200925154814002) and the National Natural Science Foundation of China (Grant No. 11931005).

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Abstract

In [C. Ding, An infinite family of Steiner systems $ S(2, 4, 2^m) $ from cyclic codes, J. Combin. Des. 26 (2018), no.3, 126–144], Ding constructed a family of Steiner systems $ S(2, 4, 2^m) $ for all $ m \equiv 2 \pmod{4} \ge 6 $ from a family of extended cyclic codes. The objective of this paper is to present a family of Steiner systems $ S(2, 4, 2^m) $ for all $ m \equiv 0 \pmod{4} \ge 4 $ supported by this family of extended cyclic codes. The main result of this paper complements the previous work of Ding, and the results in the two papers will show that there exists a binary extended cyclic code that can support a Steiner system $ S(2, 4, 2^m) $ for all even $ m \geq 4 $. Furthermore, this paper also determines the parameters of other $ 2 $-designs supported by this family of extended cyclic codes.

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Mathematics Subject Classification: Primary: 05B05, 51E10; Secondary: 94B15.

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Table 1. Weight distribution of $ \overline{ {\mathcal{C}}_E}^\perp $ (a)

Weight $ w $	No. of codewords $ A_w $
$ 0 $	$ 1 $
$ 2^{m-1}-2^{m-1-h} $	$ (2^m-1)2^{2h} $
$ 2^{m-1} $	$ (2^m-1)(2^{m+1}-2^{2h+1}+2) $
$ 2^{m-1}+2^{m-1-h} $	$ (2^m-1)2^{2h} $
$ 2^m $	$ 1 $

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Table 2. Weight distribution of $ \overline{ {\mathcal{C}}_E}^\perp $ (b)

Weight $ w $	No. of codewords $ A_w $
$ 0 $	$ 1 $
$ 2^{m-1}-2^{(m-2)/2} $	$ (2^{m/2}-1)2^{m} $
$ 2^{m-1} $	$ 2^{m+1}-2 $
$ 2^{m-1}+2^{(m-2)/2} $	$ (2^{m/2}-1)2^{m} $
$ 2^m $	$ 1 $

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Table 3. Weight distribution of $ \overline{ {\mathcal{C}}_E}^\perp $ (c)

Weight $ w $	No. of codewords $ A_w $
$ 0 $	$ 1 $
$ 2^{m-1}-2^{(m+\ell-2)/2} $	$ 2^{m-\ell} (2^m-1)/(2^{\ell/2}+1) $
$ 2^{m-1}-2^{(m-2)/2} $	$ 2^{(2m+\ell)/2} (2^m-1)/(2^{\ell/2}+1) $
$ 2^{m-1} $	$ 2( (2^{\ell/2} -1)2^{m-\ell} +1 )(2^m - 1) $
$ 2^{m-1}+2^{(m-2)/2} $	$ 2^{(2m+\ell)/2} (2^m-1)/(2^{\ell/2}+1) $
$ 2^{m-1}+2^{(m+\ell-2)/2} $	$ 2^{m-\ell} (2^m-1)/(2^{\ell/2}+1) $
$ 2^m $	$ 1 $

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Table 4. Weight distribution of $ \overline{ {\mathcal{C}}_E}^\perp $

Weight $ w $	No. of codewords $ A_w $
$ 0 $	$ 1 $
$ 2^{m-1}-2^{(m+2)/2} $	$ 2^{m-4} (2^m-1)/5 $
$ 2^{m-1}-2^{(m-2)/2} $	$ 2^{m+2} (2^m-1)/5 $
$ 2^{m-1} $	$ 2( 3 \cdot 2^{m-4} +1 )(2^m - 1) $
$ 2^{m-1}+2^{(m-2)/2} $	$ 2^{m+2} (2^m-1)/5 $
$ 2^{m-1}+2^{(m+2)/2} $	$ 2^{m-4} (2^m-1)/5 $
$ 2^m $	$ 1 $

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