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The $(n,k,d)$ regenerating code is a class of $(n,k)$ erasure codes with thecapability to recover a lost code fragment from other $d$ existing codefragments. This paper concentrates on the design of exact regenerating codes atMinimum Bandwidth Regenerating (MBR) points. For $d=n-1$, a class of$(n,k,d=n-1)$ Exact-MBR codes, termed as repair-by-transfer codes, have beendeveloped in prior work to avoid arithmetic operations in node repairingprocess. The first result of this paper presents a new class ofrepair-by-transfer codes via congruent transformations. As compared with theprior works, the advantages of the proposed codes include: i) The minimum ofthe finite field size is significantly reduced from $n \choose 2$ to $n$. ii)The encoding complexity is decreased from $n^4$ to $n^3$. As shown insimulations, the proposed repair-by-transfer codes have lower computationaloverhead when $n$ is greater than a specific constant. The second result ofthis paper presents a new form of coding matrix for product-matrix Exact-MBRcodes. The proposed coding matrix includes a number of advantages: i). Theminimum of the finite field size is reduced from $n-k+d$ to $n$. ii). The fastReed-Solomon erasure coding algorithms can be applied on the Exact-MBR codes toreduce the time complexities.

Novel Repair-by-Transfer Codes and Systematic Exact-MBR Codes with Lower Complexities and Smaller Field Sizes