Constructions of Partial MDS Codes Over Small Fields

Abstract

Partial MDS (PMDS) codes are a class of erasure-correcting array codes that combine local correction of the rows with global correction of the array. An $\boldsymbol {m}\times \boldsymbol {n}$ array code is called an $(\boldsymbol {r};\boldsymbol {s})$ PMDS code if each row belongs to an ${[}\boldsymbol {n},\boldsymbol {n}-\boldsymbol {r}, \boldsymbol {r}+\textbf {1}{]}$ MDS code and the code can correct erasure patterns consisting of $\boldsymbol {r}$ erasures in each row together with $\boldsymbol {s}$ more erasures anywhere in the array. While a recent construction by Calis and Koyluoglu generates $(\boldsymbol {r};\boldsymbol {s})$ PMDS codes for all $\boldsymbol {r}$ and $\boldsymbol {s}$ , its field size is exponentially large. In this paper, a family of PMDS codes with field size ${\mathcal{ O}}\left ({\max \{\boldsymbol {m},\boldsymbol {n}^{\boldsymbol {r}+\boldsymbol {s}}\}^{\boldsymbol {s}} }\right)$ is presented for the case where $\boldsymbol {r}= {\mathcal{ O}}(1), \boldsymbol {s}= {\mathcal{ O}}(1)$ .