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