C-Codes: Cyclic Lowest-Density MDS Array Codes Constructed Using Starters for RAID 6

The distance-3 cyclic lowest-density MDS array code (called the C-Code) is a good candidate for RAID 6 because of its optimal storage efficiency, optimal update complexity, optimal length, and cyclic symmetry. In this paper, the underlying connections between C-Codes (or quasi-C-Codes) and starters in group theory are revealed. It is shown that each C-Code (or quasi-C-Code) of length $2n$ can be constructed using an even starter (or even multi-starter) in $(Z_{2n},+)$. It is also shown that each C-Code (or quasi-C-Code) has a twin C-Code (or quasi-C-Code). Then, four infinite families (three of which are new) of C-Codes of length $p-1$ are constructed, where $p$ is a prime. Besides the family of length $p-1$, C-Codes for some sporadic even lengths are also presented. Even so, there are still some even lengths (such as 8) for which C-Codes do not exist. To cover this limitation, two infinite families (one of which is new) of quasi-C-Codes of length $2(p-1)$ are constructed for these even lengths.