MDS array codes with optimal building

1. A computer method of operating a controller of an array of storage nodes, the method comprising:

receiving configuration data at the controller that indicates operating features of the array; and

determining a parity code for operation of the array according to a permutation, wherein the configuration data specifies the array as comprising nodes defined by A=(a i,j ) with size r m ×k for some integers k,m, and wherein for T={v 0 , . . . , v k−1 } ⊂ Z r m a subset of vectors of size k, where for each v=(v 1 , . . . , v m ) ∈T, gcd(v 1 , . . . , v m ,r), where gcd is the greatest common divisor, such that for any l, 0≦l≦r−1, and v ∈T, the code values are determined by the permutation f v l :[0,r m −1]→[0,r m −1] by f v l (x)=x+lv.

2. The computer method of claim 1 , wherein A specified by the configuration data comprises A=(a i,j ), an array of size 2 m ×k for some integers k, m, and k≦2 m , and wherein for T ∈ F 2 m is a subset of vectors of size k that does not contain the zero vector, and for v ∈T the permutation is given by f v :[0,2 m −1]→[0,2 m −1] by f v (x)=x+v, where x is represented in its binary representation.

3. The computer method of claim 2 , wherein for the permutations f 0 , . . . , f m and for sets X 0 , . . . , X m constructed by the vectors {e i } i+0 m , the set X 0 is modified to be X 0 ={x ∈F 2 m :x·(1,1, . . . , 1)=0}.

4. The computer method of claim 2 , further comprising generating an s-duplication code of the code values, such that generating the s-duplication code comprises assigning α i,j (t) =1 for all i, j, t, such that for odd q, and s≦q−1 and assigning for all t∈[0,s−1]

β

i

,

j

(

t

)

=

{

a

t

+

1

,

if

⁢

⁢

u

j

·

i

=

1

a

t

,

o

.

w

.

where u j =Σ l=0 j e l and for even q (powers of 2), and s≦q−2 and assigning for all t ∈[0,s−1]

β

i

,

j

(

t

)

=

{

a

-

t

-

1

,

if

⁢

⁢

u

j

·

i

=

1

a

t

+

1

,

o

.

w

..

5. The computer method of claim 3 , wherein the code comprises a (k+2, k) MDS (maximum distance separable) array code, and the array has size 2 m ×(k+2), wherein the permutations are defined by f j , j ∈[0, k−1] and for systematic information elements a i,j , and for row and zigzag parity elements r i and z i , respectively, for i ∈[0,2 m −1], j ∈[0, k−1] and for row coefficients given by α i,j =1 for all i,j, and zigzag coefficients given by β i,j in some finite field F, the method further comprising:

for a single erasure,

(1) where one parity node is erased, rebuilding the row parity according to

r

i

=

∑

j

=

0

k

-

1

⁢

⁢

a

i

,

j

,

 and rebuilding the zigzag parity by

z

i

=

∑

j

=

0

k

-

1

⁢

⁢

β

f

j

-

1

⁡

(

i

)

,

j

⁢

a

f

j

-

1

⁡

(

i

)

,

j

.

;

(2) where one information node j is erased, rebuilding the elements in rows X j and those in rows X j by zigzags; and

for two erasures,

(1) where two parity nodes are erased, rebuilding by the one parity erasure rebuilding,

(2) where one parity node and one information node is erased, and if the row parity node is erased, then rebuild by zigzags; otherwise rebuilding by rows;

(3) where two information nodes j 1 and j 2 are erased, then

if f j 1 =f j 2 , for any i ∈[0,2 m −1], computing

x i =r i −Σ j≠j 1, j 2 a i,j

y i =z f j1 (i) −Σ j≠j 1, j 2 β f j −1 f j 1 (i),j a f j −1 f j 1 (i),j ;

solving a i,j 1 , a i,j 2 from the equations

[

1

1

β

i

,

j

1

β

i

,

j

2

]

⁡

[

a

i

,

j

1

a

i

,

j

2

]

=

[

x

i

y

i

]

 else, for any i ∈[0,2 m −1], setting i′=i+f j 1 (0)+f j 2 (0), and computing x i , x i′ , y i , y i′ , according to the two information nodes operation (3) and then solving a i,j 1 , a i,j 2 , a i′,j 1 , a i′,j 2 from equations

[

1

1

0

0

0

0

1

1

β

i

,

j

1

0

0

β

i

′

,

j

2

0

β

i

,

j

2

β

i

′

,

j

1

0

]

⁡

[

a

i

,

j

1

a

i

,

j

2

a

i

′

,

j

1

a

i

′

,

j

2

]

=

[

x

i

x

i

′

y

i

y

i

′

]

.

6. A controller of an array of storage nodes, the controller comprising:

a host interface through which the controller communicates with a host computer;

a node interface through which the controller communicates with the array of storage nodes;

a processor that operates a controller application for:

receiving configuration data at the controller that indicates operating features of the array; and

determining a parity code for operation of the array according to a permutation, wherein the configuration data specifies the array as comprising nodes defined by A=(a i,j ) with size r m ×k for some integers k,m, and wherein for T={v 0 , . . . , v k−1 } ⊂ Z r m a subset of vectors of size k, where for each v=(v 1 , . . . , v m )∈T, gcd(v 1 , . . . , v m , r), where gcd is the greatest common divisor, such that for any l, 0≦l≦r−1, and v ∈T, the code values are determined by the permutation f v l :[0, r m −1]→[0,r m −1]by f v l (x)=x+lv.

7. The controller of claim 6 , wherein A specified by the configuration data comprises A=(a i,j ), an array of size 2 m ×k for some integers k, m, and k≦2 m , and wherein for T ⊂ F 2 m is a subset of vectors of size k that does not contain the zero vector, and for v ∈ T the permutation is given by f v :[0,2 m −1]→[0,2 m −1]by f v (x)=x+v, where x is represented in its binary representation.

8. The controller of claim 7 , wherein for the permutations f 0 , . . . , f m and for sets X 0 , . . . , X m constructed by the vectors {e i } i=0 m , the set X 0 is modified to be X 0 ={x ∈F 2 m : x·(1,1, . . . , 1)=0}.

9. The controller of claim 7 , further comprising generating an s-duplication code of the code values, such that generating the s-duplication code comprises assigning α i,j (t) =1 for all i, j, t, such that for odd q, and s≦q−1 and assigning for all t ∈[0,s−1]

β

i

,

j

(

t

)

=

{

a

t

+

1

,

if

⁢

⁢

u

j

·

i

=

1

a

t

,

o

.

w

where u j =Σ l=0 j e l , and for even q (powers of 2), and s≦q−2 and assigning for all t ∈[0,s−1]

β

i

,

j

(

t

)

=

{

a

-

t

-

1

,

if

⁢

⁢

u

j

·

i

=

1

a

t

+

1

,

o

.

w

..

10. The controller of claim 8 , wherein the code comprises a (k+2, k) MDS (maximum distance separable) array code, and the array has size 2 m ×(k+2), wherein the permutations are defined by f j , j ∈[0,k−1]and for systematic information elements α i,j , and for row and zigzag parity elements r i and z i , respectively, for i ∈[0,2 m −1], j ∈[0,k−1] and for row coefficients given by α i,j , =1 for all i, j , and zigzag coefficients given by β i,j in some finite field F, the method further comprising:

for a single erasure,

(1) where one parity node is erased, rebuilding the row parity according to

r

i

=

∑

j

=

0

k

-

1

⁢

⁢

a

i

,

j

,

 and rebuilding the zigzag parity by

z

i

=

∑

j

=

0

k

-

1

⁢

⁢

β

f

j

-

1

⁡

(

i

)

,

j

⁢

a

f

j

-

1

⁡

(

i

)

,

j

.

;

(2) where one information node j is erased, rebuilding the elements in rows X j and those in rows X j by zigzags; and

for two erasures,

(1) where two parity nodes are erased, rebuilding by the one parity erasure rebuilding,

(2) where one parity node and one information node is erased, and if the row parity node is erased, then rebuild by zigzags; otherwise rebuilding by rows;

(3) where two information nodes j 1 and 1 2 are erased, then if f j 1 =f j 2 , for any i ∈[0,2 m −1], computing

x i =r i −Σ j≠j 1, j 2 a i,j

y i =z f j1 (i) −Σ j≠j 1, j 2 β f j −1 f j 1 (i),j a f j −1 f j 1 (i),j ;

solving a i,j 1 ,a i,j 2 from the equations

[

1

1

β

i

,

j

1

β

i

,

j

2

]

⁡

[

a

i

,

j

1

a

i

,

j

2

]

=

[

x

i

y

i

]

 else, for any i ∈[0,2 m −1 ], setting i′=i+f j 1 (0)+f j 2 (0), and computing x i , x i′ , y i , y i′ according to the two information nodes operation (3) and then solving a i,j 1 , a i,j 2 a i′,j 1 , a i′,j 2 from equations

[

1

1

0

0

0

0

1

1

β

i

,

j

1

0

0

β

i

′

,

j

2

0

β

i

,

j

2

β

i

′

,

j

1

0

]

⁡

[

a

i

,

j

1

a

i

,

j

2

a

i

′

,

j

1

a

i

′

,

j

2

]

=

[

x

i

x

i

′

y

i

y

i

′

]

.