Product-matrix (PM) code[1]

Description

Code constructed using two explicit constructions, with each construction corresponding to one of the two extreme points of the storage-bandwidth trade-off curve [2].

For the MBR point, the parameters satisfy \(n-1 \ge d \ge k\), \(\alpha=d\beta\), and \begin{align} M=\left(kd-\binom{k}{2}\right)\beta~. \tag*{(1)}\end{align} For the MSR point, the parameters satisfy \(d \ge 2k-2\), \(\alpha=(d-k+1)\beta\), and \(M=k\alpha\).

PM codes are the first explicit constructions for all values of the system parameters \([n,k,d]\) at the MBR point, and for all parameters satisfying \([n,k,d \ge 2k-2]\) at the MSR point. Both constructions are based on a common product-matrix framework, which makes them easy to implement.