Song–Cai–Yue rate conjecture for sequentially recoverable locally recoverable codes

Let $\mathcal{C}$ denote an $(n,k,r,t)_{\text{seq}}$ code over a finite field $\mathbb{F}_q$. Let

$$m=\lceil \log_r(k) \rceil.$$

Song–Cai–Yue conjecture. An achievable upper bound on the rate $k/n$ is

$$\frac{k}{n} \leq \frac{1}{1+\sum_{i=1}^{m}\frac{a_i}{r^i}}, \qquad a_i\geq 0,\quad a_i\in\mathbb{Z},\quad \sum_{i=1}^{m}a_i=t.$$

This conjecture concerns the maximum achievable rate of sequentially recoverable locally recoverable codes from any number of multiple erasures. The paper presents it as a conjectured tight rate bound; its resolution is not established in the supplied text.