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Table 1 Summary of the constructions of codes with (r,)-cooperative locality considered in this paper. For the codes based on unbalanced bipartite expander graphs (Section 6.2.1), we assume that the underlying bipartite graphs is bi-regular with h and Δ representing its left and right degrees, respectively. Moreover, the graph exhibits expansions from left to right of any set of at most α n left nodes with expansion ratio h(1−ε). Here, the constituent local codes have distance at least t+1. For codes based on double cover of regular expander graphs (Section 6.2.2), Δ and λ denote the degree and the second largest absolute eigenvalue of the underlying graph, respectively. This construction utilizes smaller code of minimum distance at least δ Δ to define local constraints at the vertices of the double cover

From: Cooperative local repair in distributed storage

Construction Cooperative locality Rate \(\text {rate}(\mathcal {C})\) Minimum distance \(d_{\min }(\mathcal {C})\)
Partition code (Section 4.1) (r,) \(\frac {r}{r + \ell ^{2}}\) \(n - k + 1 - \ell \left (\frac {k\ell }{r} - 1\right)\)
Product code (Section 4.2) (r,) \(\left (\frac {r}{r+1}\right)^{\ell }\) +1
Concatenated code (Section 5) (r,) \(\frac {\ell + 2}{\ell + 4}\frac {r}{r + 2}\) +1
Regular bipartite graph-based code (Section 6.1) (r,) \( \geq \frac {r - \ell }{r + \ell }\) +1
Unbalanced bipartite expander graph-based code (Section 6.2.1) (r,) \(\geq 1 + \frac {h}{\Delta }\frac {r}{\ell } - h\) \(\geq \left (2 - \epsilon -\frac {\epsilon }{t}\right)\alpha n\)
Double cover of regular expander graph based-code (Section 6.2.2) (r,) \(\geq 2\frac {r}{\ell \Delta } - 1\) \(\delta \left (\delta - \frac {\lambda }{\Delta }\right)n\)
Hadamard code (Section 7) (r=+1,), \(\forall 1\le \ell \le \frac {n-1}2\) \(\frac {\log (n+1)}{n}\) \(\frac {n+1}2\)