MDS array codes with independent parity symbols

Abstract

-A n e w family of MDS a r r a y codes is presented. T h e code a r r a y s contain p information c o l u m n s a n d r i n d e p e n d e n t p a r i t y columns, w h e r e p is a prime. We give necessary a n d sufficient conditions for our codes to be MDS, a n d t h e n prove t h a t if p belongs to a certain class of p r i m e s t h e s e conditions a r e satisfied u p to r 5 8. We also develop efficient decodi n g procedures for t h e case of t w o a n d t h r e e c o l u m n errors, a n d a n y n u m b e r of c o l u m n erasures. Finally, we present u p p e r a n d lower b o u n d s o n t h e average n u m b e r of p a r i t y b i t s which have to be u p d a t e d in a n MDS code over GF(2"), following a n u p d a t e i n a single information bit. We show t h a t t h e u p p e r b o u n d o b t a i n e d f r o m o u r codes is close to t h e lower b o u n d a n d does n o t depend o n t h e size of t h e code symbols.

Improved decoding algorithm for the generalized EVENODD array code

“…If The parameters of the generalized EVENODD array code is ( 8,5,4), the triple information column erasures at position r, s, t where r=0,s=2,t=3. We can obtain the equation which is constructed by the parity symbols as follow: From the equations above we can obtain the symbol c 1,2 directly.…”

“…To the best of our knowledge, there exist only few classes of MDS array codes of distance 4 that are very suitable for storage: the Blaum-Roth code and the generalized EVENODD code [8,9]. But it is unfortunately that the Blaum-Roth code is nonsystematic.…”

On the Minimum Number of Multiplications Necessary for Universal Hash Functions

“…A characterization of MDS matrix is that every square sub-matrix has full rank. There are some known systematic form of MDS code based on Vandermonde matrix [5]. We call a matrix S k×(n−k) d-MDS if every square submatrix of size d has full rank.…”

Parity declustering for fault-tolerant storage systems via t-designs

“…The ingredient balanced (t − 1)-parity group G in Algorithm 3 can be constructed by applying the method in Example III.2 to any MDS horizontal array code that tolerates t − 1 disk failures. Except from the well-known RS codes, some other known MDS horizontal (t−1)-failure tolerant codes (t > 3) were studied by Blomer et al [28], Blaum et al [29], [30], Huang and Xu [9]. More details can be found in the full version [20].…”