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

The
$(n,k,d)$
regenerating code is a class of
$(n,k)$
erasure codes with the capability to recover a lost code fragment from other
$d$
existing code fragments. In this paper, we focus on the design of exact regenerating codes at minimum 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 been developed in prior work to avoid arithmetic operations in node repairing process. The first result of this paper presents a new class of repair-by-transfer codes via congruent transformations. As compared with prior works, the advantages of proposed codes include: i) the minimum of field size is significantly reduced from
$n \atopwithdelims ()2$
to
$n$
; ii) the encoding complexity is decreased from
$n^4$
to
$n^3$
. Our simulation results show that the proposed code achieves faster operations than the prior approach does under large
$n$
. The second result of this paper presents a new form of coding matrix for product-matrix exact-MBR codes. The proposed coding matrix includes the following advantages: i) the minimum of finite field size is reduced from
$n-k+d$
to
$n$
; ii) the fast Reed-Solomon erasure coding algorithms can be applied on the proposed exact-MBR codes to reduce the time complexities.