Generalization of Rashmi-Shah-Kumar Minimum-Storage-Regenerating Codes

Abstract

In this paper, we propose a generalized version of the Rashmi-Shah-Kumar Minimum-Storage-Regenerating(RSK-MSR) codes based on the product-matrix framework. For any $(n,k,d)$ such that $d\ge 2k-2$ and $d\le n-1$, we can directly construct an $(n,k,d)$ MSR code without constructing a larger MSR code and shortening of the larger MSR code. As a result, the size of a finite field over which the proposed code is defined is smaller than or equal to the size of a finite field over which the RSK-MSR code is defined. In addition, the $\{\ell ,{\ell }^{\prime }\}$ secure codes based on the generalized RSK-MSR codes can be obtained by applying the construction method of $\{\ell ,{\ell }^{\prime }\}$ secure codes proposed by Shah, Rashmi and Kumar. Furthermore, the message matrix of the $(n,k,d)$ generalized RSK-MSR code is derived from that of the RSK-MSR code by using the construction method of the $\{\ell =k,{\ell }^{\prime }=0\}$ secure code.

Submitted to IEEE Trans. Information Theory

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