Diagonal code[1] 

Member of an explicit family of high-rate \([n,k,d,\alpha, \beta = \frac{\alpha}{d-k+1}, M=k\alpha]\) MSR codes for any \(r\) and \(n\). Such codes can optimally repair any \(f\) failed nodes from any \(d\) helper nodes for all \(d\), \(1 \le f \le r\) and \(k \le d \le n-f\) simultaneously. These codes can be constructed over any base field \(GF(q)\) as long as \(|GF(q)| \ge sn\), where \(s = \text{lcm}(1,2,\cdots,r)\).

Let \(C \in GF(q)^{\alpha \times n}\) be a codeword, with \(C_i\) being the \(i\)-th coordinate. Then, the code is defined as \begin{equation} \mathsf{C} = \{(C_1,C_2,\cdots,C_n) \sum_{i=1}^nA_i^{t-1}C_i = 0, t=1,2,\cdots,r\}~, \tag*{(1)}\end{equation} where the matrices \(A_i\) are diagonal \(\alpha \times \alpha\) matrices.