Construction of Optimal Locally Repairable Codes of Triangular Association Schemes

doi:10.16183/j.cnki.jsjtu.2023.151

Abstract

Abstract:

As a new erasure code for distributed storage systems, locally repairable codes (LRCs) can effectively realize the reliable and efficient storage of massive data. The construction of locally repairable codes with (r,t) locality has become a research hotspot recently. Therefore, the construction methods of locally repairable codes based on triangular association schemes are proposed, which can construct optimal binary locally repairable codes with arbitrary (r,t) locality. Performance analyses show that the LRCs constructed with availability t=2 reach the optimal code rate bound, the LRCs constructed with arbitrary locality r>2 and availability t>2 reach the optimal minimum distance bound. The LRC constructed in this paper performs better in terms of code rate and more flexible parameter selection than those constructed based on near-regular graphs and direct product codes, etc.

Key words: distributed storage system, locally repairable code (LRC), triangular association schemes, minimum distance

CLC Number:

TN911.2

WANG Jing, LI Jinghui, YANG Jiarong, WANG E. Construction of Optimal Locally Repairable Codes of Triangular Association Schemes[J]. Journal of Shanghai Jiao Tong University, 2024, 58(10): 1596-1605.

Figures/Tables 8

Fig.1

Tab.1

Comparative analysis of parameters of Construction 1

构造方式	构造参数	最小距离	相对距离
构造1构造的BLRCs	n= $(r + 1) (r + 2) 2$ , k= $r (r + 1) 2$ , r,t=2	d_min=3	$6 (r + 1) (r + 2)$
基于近正则图构造的BLRCs^[18]	$n=k+\left\lceil\frac{2 k}{r}\right\rceil,\left\lceil\frac{2 k}{r}\right\rceil \geqslant r+2, t=2$	d_min=3	$3 k + 2 k r$
基于单位矩阵变换构造的BLRCs^[21]	n=r²+2r, k=r², r,t=2	d_min=3	$3 r (r + 2)$
基于直积码构造的BLRCs^[12](t=2)	n=(r+1)², k=r², r,t=2	d_min=4	$4 (r + 1) 2$

Tab.1

Fig.2

Tab.2

Fig.3

Tab.3

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Fig.5

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构造方式	构造参数	最小距离	相对距离
推论1构造的BLRCs	n= $(t + 1) (t + 2) 2$ , k=t+1, r=2,t	d_min=t+1	$2 t + 2$
基于单纯形码构造的BLRCs^[22]	n=2^m-1, k=m, r=2, t=2^m^-1-1	d_min=t+1	$t + 1 2 t + 1$
基于直积码构造的BLRCs^[12] (r = 2)	n=3^t, k=2^t, r=2,t	d_min=2^t	$23 t$

构造方式	n	k	r	t	R
构造2构造的BLRCs	$(r + 1) (r + 2) … (r + t) t!$	$r (r + 1) … (r + t - 1) t!$	r>2	t>2	$r r + t$
基于迹函数构造的BLRCs^[13]	2^m-1	2^m^-1-1	m-1	m	$2 r - 1 2 r + 1 - 1$
基于超图构造的BLRCs^[16]	v+ $1 r$ vt	v	v≥t(r-1)+1, r\|vt		$r r + t$
基于阵列LDPC码构造的BLRCs^[14]	r²+rt+1	r²	r(奇素数)	t(偶数)	$r 2 r 2 + r t + 1$
基于直积码构造的BLRCs^[12]	(r+1)^t	r^t	r	t	$r r + 1 t$