Regenerating codes represent a class of block codes applicable for distributed storage systems. The [n, k, d] regenerating code has data recovery capability while possessing arbitrary k out of n code fragments, and supports the capability for code fragment regeneration through the use of other arbitrary d fragments, for k ≤ d ≤ n - 1. Minimum storage regenerating (MSR) codes are a subset of regenerating codes containing the minimal size of each code fragment. The first explicit construction of MSR codes that can perform exact regeneration (named exact-MSR codes) for d ≥ 2k - 2 has been presented via a product-matrix framework. This paper addresses some of the practical issues on the construction of exact-MSR codes. The major contributions of this paper include as follows. A new product-matrix framework is proposed to directly include all feasible exact-MSR codes for d ≥ 2k - 2. The mechanism for a systematic version of exact-MSR code is proposed to minimize the computational complexities for the process of message-symbol remapping. Two practical forms of encoding matrices are presented to reduce the size of the finite field.
Sian-Jheng Lin, Wei-Ho Chung, Han, Y. S., & Al-Naffouri, T. Y. (2015). A Unified Form of Exact-MSR Codes via Product-Matrix Frameworks. IEEE Transactions on Information Theory, 61(2), 873–886. doi:10.1109/tit.2014.2378255
This work was supported by the Ministry of Science and Technology, Taiwan, under Grants NSC 102-2221-E-001-006-MY2, MOST 103-3113-E-110-002, and NSC 101-2221-E-011-069-MY3. Part of this work was presented at the 2013 IEEE 24th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC 2013).