JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY Vol. 51 No. 1 Feb : (2016) DOI: / j. issn

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1 JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY Vo. 51 No. 1 Feb : (2016) DOI: / j. issn !"#(k + 2,k)Hadamard MSR $,, ( ,9: ; ) % &:& ' <=4 >,?@A 23(k + 2,k)Hadamard MSR. B 4CD E 2 D,FG HIJKL. M E,N 2 D O, 4E,O 4P. Q N RS?@A<=4TU$V,WXY<= 4 α D '; α / 2,C 2 D,G k + 1 D<=&C Z[F 1 D. \]A^3 E 4 HIJ_L4 2 D `a,b_c TU <=;^ E HIJ_L4 2 D 4 & & 0 4,b_c TU 1 D <=;^ E HIJ_L4 2 D 4 4 & 1,b_c TU 2 D <=. 3 4<= > A Hadamard MSR 4,_c TU <= 1 D <=. ' :' ; ; # ;MSR ; ;c ;TU!:TN ()$:A A New (k + 2,k)Hadamard Minimum Storage Regenerating Code ZHANG Sina, TANG Xiaohu, LI Jie (Schoo of Information Science and Technoogy,Southwest Jiaotong University,Chengdu ,China) Abstract:To reduce the storage capacity of nodes in distributed storage systems,a new (k + 2,k) Hadamard Minimum Storage Regenerating (MSR)code was constructed. Each coding matrix is reated to two vaues,from which the diagona eements of this coding matrix are seected. These two vaues appear in the coding matrix in a repeating pattern,but with different repeating cyces for different matrices. Based on the structure of the coding matrix,a repair strategy was constructed. The repair strategy divides symbos in the faied node into α / 2 groups with two symbos in each group,then the two symbos are recovered by downoading one symbo from each of the other k + 1 nodes. If the two vaues reated to each coding matrix are unequa,the new Hadamard MSR code can optimay repair systematic nodes. If the sum of two vaues reated to each coding matrix is nonzero and the k vaues are the same,the new Hadamard MSR code can optimay repair the first parity node. If the sum of the inverse of two vaues reated to each coding matrix is one,the new Hadamard MSR code can optimay repair the second parity node. The new code reduces the storage capacity to the bond for Hadamard MSR code. Further,it can optimay repair a systematic nodes and one parity node. Key words:distributed;storage;regenerating code;msr code;high code rate;optima;repair ' ><=[F > 4, ; a =,M a 4, OceanStore [1] Tota Reca [2] DHash + + [3]. &A \ 4_ S _ S,' 4. : : (1985 ),,!"#,!"$%&',E mai:nsz1221@ 163. com :()*(1971 ),+,,-, #./,!"$%&01,E mai:xhtang_scce@ home. swjtu. edu. cn :,()*,. 234(k + 2,k)Hadamard MSR [J].,2016,51(1): ,200.

2 1,: (k + 2,k)Hadamard MSR 189 4,G. $ V 4 ' Facebook 4 Hadoop Googe Coossus Microsoft Azure [4]. Q,W D )& M = kα 4 ; )4 n ( )& α), k _ U, MDS (maximum distance separabe)s. M,<= 4 & >,, D<=XY,&TUG ( ) & α), G k D<= G,,& T U M / k D, M D., [5][ Ac) #(minimum storage regenerating,msr),b MDS S, `. B W<= 4 & >,MTUXY<=4 M / k D, G d D(d k)_ <= G ' [6], 4 >( &TU )& dm / k(d - k + 1)< M. (n = k + 2,k) MSR : (1),^KL <=, ; (2) G MSR `, c) [7 8]. 4(n = k + 2,k) MSR [9 15]. G, [9][ 4 Hadamard MSR 4 <= <= 4 ; D > MDS. N RS Hadamard MSR _ Q 4', <= 4, <= 4, c 3S. [16]\]A(k + 2,k)Hadamard MSR 4<= > α 4 & 2 k,, [9][ 4 Hadamard MSR 4<= > α = 2 k + 1, 4 A D 3 4(k + 2,k)Hadamard MSR. 3 4<= > α A 2 k, c 3S,_c TU <=, c TU D <=, D S4 ), <= 2,GXY <=, <=`, <=4X Y, &. 4 X,! "#$014 %. 1 (k + 2,k)Hadamard MSR $# *+,-./0 M& F q ', (k + 2,k) MSR D )& M = kα 4,#; k + 2 ) & α 4. G,k 4, 2 4(S). N k + 2 '* k + 2 D <=. 4<= & <=,G <= & <=. ^W<= i (1 i k + 2) 4 D α +, % > c i )-, c i =(c i,1,c i,2,,c i,α) T, X.S, <= k _)-& <= 4, c k + 1 = c 1 + c c k, <= k _)-& <= 4(S /, c k + 2 = A 1 c 1 + A 2 c A k c k, G,A i (1 i k) α α 4E, & <= i 4 E. ^ B MSR K (k + 2,k) Hadamard MSR, E A 1,A 2,,A k & HE,G E A i (1 i k)_)-& A i = diag(a i,1,a i,2,,a i,α), G,a i, F q,1 α. (k + 2,k)Hadamard MSR 40? ) ) 1 _,M(k + 2,k)Hadamard MSR, <= 4 <= 4 `, G 4.,M <= 4, <= 4,,(k + 2,k)HadamardMSR 1 1 (k + 2,k)Hadamard MSR $#*+ Tab. 1 Structure of (k + 2,k)Hadamard MSR code <= 1 <= 2 <= k <= k + 2 <= k + 2 c i,1 c k+2,1 = k a i,1 c i,1 1 c 1,1 c 2,1 c k,1 c k + 1,1 = k 2 c 1,2 c 2,2 c k,2 c k+1,2 = k α c 1,α c 2,α c k,α c k+1,α = k c i,2 c k+2,2 = k c i,α c k+2,α = k a i,2 c i,2 a i,α c i,α

3 190 c 3 S. 1 ) 1 _,M C, (c 1,,,c k,,c k + 1,,c k + 2,) D(k + 2,k) > MDS,1 α,,(k + 2,k)Hadamard MSR _ α > MDS 4 /.,^(k + 2, k)hadamard MSR MDS S, E 4 HIJ 2 a i, 0 a i, - a j, 0,G, 1 i j k,1 α. M(k + 2,k)Hadamard MSR,XY<=4 c TU ' `34 [17]. M 4 k + 1 D<=453,XY<= ( )& α)4 TU '; α / 2 6 4,C 78 2 D,^XY<=4 s ( s)4 '& TU,& N 2 D,C D53<=[F4 ( )& 1)9 1Z:4 s #;4. 2! " #(k + 2,k)Hadamard MSR $ [9][ 4(k + 2,k)Hadamard MSR 4<= > α & 2 k + 1, [16]; & <= >, G A,?@A 234(k + 2,k)Hadamard MSR. 3 4<= > α & 2 k, A. (k + 2,k)MSR 1 E A 1,A 2,,A k ;, (k + 2,k)Hadamard MSR 4 E & H. M3(k + 2,k)Hadamard MSR, E A i (1 i k) H('4IJ a i,(1 α = 2 k + 1 )L & μ i, + 1 ) = 1(mod 2), a i, = ν i, + 1 ) = 0(mod 2, E 51! { ), (1) ) A i = diag (μ i 1,ν i 1,,μ i 1,ν i 1 2 k - i + 1 D G :μ i,ν i F q,q 2k + 3;1 2 i 2 i +4 1 %>. &$ 4\],W3 E H IJ4L < 4= )-. 1 a i, = a i, +,a j, a j, + DL μ i, DL ν i,g,1 i j k, [2 i s + 1, 2 i s + ],0 s 2 k -. > (1),< 1 a?, + 2 ) + 1 ent 2 ( + 1 ) + j - 1 1(mod 2), j = i, ) + 1 ) (mod 2), j i. i = j, + 2 ) ( + 1 = ent + 2 ) = + 1 ) + 1. i > j, + 2 ) ( + 1 =ent + 1 ) + 2 i - j + 1 ) (mod 2). i < j, X.S,@ 2s = 2 j - i φ +, 0 φ 2 k - j + 1-1,0 2 j - i - 2, b ( + 1) + m - + m 2, j - 1 b [2 i s + 1,2 i s + ], ent ent ( ( + 1 ) = φ + 1, ) = ( + 1)2 φ + + m + 1 ) = φ a i, a i,2 k DL μ i, DL ν i, G,1 i k, [1,2 k - 1 ]. b X.S,@ = s + m, 1 m,0 s 2 k -, + 1 ) = s + 1, 2 k ) ent 2 ( ) m s 2 ) s(mod 2). > (1),a i, a i,2 k - +1 DL μ i, DL ν i # M3(k + 2,k)Hadamard MSR, <=

4 1,: (k + 2,k)Hadamard MSR 191 i(1 i k)4tu W ( [2 i s + 1,2 i s + ],0 s 2 k - ) + '&,& TUB,G <= <=[F4 N 2 4. A\] E 4 HIJ4L μ i ν i (1 i k) 2B C,3(k + 2,k)Hadamard MSR _c TU <=. 7 1 ^ μ i ν i (1 i k),3(k + 2,k) Hadamard MSR _c TU <=. <= i(1 i k) 4 & c i,1, c i,2 1,,c i,2 k,& c i, c i, + ( [2 i s + 1, 2 i s + ],0 s 2 1, k - ), G <= c j, + c j, j k,j i; (2) i - <= k + 1 c k + 1, + c 1, & k + 1, + 2 i - (c i, + c j, +2 i-1)+ k j = 1,j i (c j, + c j, +2 i-1); (3) <= k + 2 c k + 2, + c k + 2, +, & (a i,c i, + a i, +2i -1c i, +2i -1)+ k j = 1,j i (a j,c j, + a j, +2i -1c j, +2i -1). (4) 1< 1 _D,a j, = a j, + 2,, (3) (4)4c E _1 (2), (3)4 1 E (4)4 1 E ; c i, c i, + 2 4$, a i, a j, + 2,B$ _. 1< 1 D,a i, a j, + 2 DL μ i, DL ν i,, μ i ν i,b_ c i, c i, + 2.,^ μ i ν i (1 i k),3(k + 2,k) Hadamard MSR _c TU <= #34-. 3(k + 2,k)Hadamard MSR \ 2 D <= _c TU, KL D c TU. ^_c TU <= k + 1,GTU W ( [1,2 k - 1 ]) 2 k '&. &T UB, <=[F4 N 2, <= k + 2 [F4b N 2 F. A\] E HIJ4L μ i ν i (1 i k) 2 B C,3 (k + 2,k) Hadamard MSR _c TU <= k ^ μ 1 + ν 1 = μ 2 + ν 2 = = μ k + ν k 0, 3(k + 2,k)Hadamard MSR _c TU < = k + 1. < = k & c k + 1,1, c k + 1,2 1,,c k + 1,2 k,& c k + 1, c k + 1,2 k - + 1( [1, 2 k - 1 ]), k D <= c i, + c i,2 k - + 1, 1 i k, (5) W (5)4 E`G, c k + 1, + c k + 1,2 k - + 1, (6) <= k + 2 c k + 2, - c k + 2,2 k - + 1, & (a 1,c k +1, - a 1,2 k - +1c k +1,2 k - +1)+ k [(a j, - a 1,)c j, + j = 1 (a 1,2 k a j,2 k - +1)c j,2 k - +1]. (7) (6) (7) ; c k + 1, c k + 1, $,^ $ _, a 1, 6 (7)4c E_, + a 1,2 k a j, + a j,2 k = a 1, + a 1,2 k - + 1, 2 j k. 1< 2 _D,a i, a i,2 k DL μ i, DL ν i, a i, + a i,2 k = μ i + ν i, HI, μ 1 + ν 1 = μ 2 + ν 2 = = μ k + ν k 0, b_ c k + 1, c k + 1, ,^ μ 1 + ν 1 = μ 2 + ν 2 = = μ k + ν k 0, 3(k + 2,k)Hadamard MSR _c TU < = k + 1. ^3(k + 2,k)Hadamard MSR _c TU <= k + 2,G' TU <= k + 1 `, ( [1,2 k - 1 ]) 2 k '&. &TUB, <=[F4 N 2 4, <= k + 1 [F4b N 2 4 F. A\] E 4 HIJ4L μ i ν i (1 i k) 2B C,3(k + 2,k)Hadamard MSR _c TU <= k ^ (1 i k),3(k + 2, k)hadamard MSR _c TU <= k + 2. < = k & c k + 2,1, c 1,,c k,& k + 2,2 k + 2,2 c k + 2, c k + 2,2 k - + 1( [1, 2 k - 1 ]), k D <= a i,c i, + a i,2 k - + 1c i,2 k - + 1, 1 i k, (8) W (8)4 E`G, c k + 2, + c k + 2,2 k - + 1, (9) <= k + 1 c k + 1, - c k + 1,2 k - + 1, & c k +2, - k [(a i, - 1)c i, + c i,2 k - +1]. (10) (9) (10) ; c k + 2, c k + 2,2 k

5 192 51! $,^ $ _, (10)4c E, a - 1 i, + a - 1 i,2 k = 1. 1< 2 _D,a i, a i,2 k DL μ i, DL ν i, a - 1 i, + a - 1 i,2 k = HI, b_ c k + 2, c k + 2,2 k , i,,^, i i k, 3(k + 2,k)Hadamard MSR _c TU < = k "$# MDS 0: A\] E 4 HIJ4L μ i ν i (1 i k) 2B C,3(k + 2,k)Hadamard MSR 2 MDS S. 7 4 ^ μ i 0,ν i 0,μ i ν i,μ i μ j,ν i ν j (1 i j k),3(k + 2,k)Hadamard MSR MDS S. ^(k + 2,k)Hadamard MSR MDS S, E 4 HIJ 2 a i, 0 a i, - a j, 0(1 i j k,1 α). 1 (1) D,3 4 a i, L μ i J ν i,b a i, - a j, 4L 4,'*& μ i - μ j,ν i - ν j,μ i - μ j,ν i - ν j HI, ^ a i, 0 a i, - a j, 0 ;K,b μ i 0, ν i 0,μ i ν j,μ i μ j,ν i ν j.,^ μ i 0,ν i 0,μ i ν j,μ i μ j,ν i ν j (1 i j k),3(k +2,k)Hadamard MSR MDS S. AL 3(k + 2,k)Hadamard MSR M 4&. N' O,3 ^_c TU <= MDS S, μ i ν i,μ i 0,ν i 0,μ i ν j,μ i μ j,ν i ν j, 1 i j k, μ i,ν i (1 i k)& 2k D ` 4. μ 1 + ν 1 = μ 2 + ν 2 = = μ k + ν k 0 (_c TU <= k 'C ) 1 + ν = 2 + ν = = k + ν - 1 k = 1 (_c TU <=k 'C ) μ i ν i (1 i k) k ` & 0 4.,3(k + 2,k)Hadamard MSR & F q RP q 2k + 3. M& F q (q 2k + 3)',^3 KQc TU <= k + 1,μ i ν i (1 i k)_l μ i = i,ν i = q - ; ^KQc TU <= k + 2,μ i ν i (1 i k) _L μ i = 2 + t ν +(t + 1) - 1, G,1 t q * ; ; A 2 3 4(k + 2,k)Hadamard MSR,G<= > A Hadamard MSR 4. ; A<=4' TU$V,Q BTU $V,\]A3 _c TU <= <= 4 'C. : [1] RHEA S,WELLS C,EATON P,et a. Maintenance free goba data storage[j]. IEEE Internet Computing, 2001,5(5): [2] BHAGWAN R,TATI K,CHENG Y C,et a. Tota reca: System support for automated avaiabiity management[c] Symposium Networked Systems Design and Impementation. San Francisco: ACM, 2004: [3] DABEK F,LI Jinyang,SIT E,et a. Designing a DHT for ow atency and high throughput[c] Symposium Networked Systems Design and Impementation (NSDI). San Francisco:ACM,2004:85 98 [4] HUANG Cheng,SIMITCI H,XU Yi,et a. Erasure coding in windows azure storage[c] Usenix annua Technica Conference. Boston:ACM,2012: [5] DIMAKIS A G,GODFREY P B,WU Yunnan,et a. Network coding for distributed storage systems[j]. IEEE Transactions on Information Theory, 2010, 56(9): [6] R S,TUV. Q WXY E 4UY <= $ [J].,2014, 49(2): FAN Weni,LIU Zhigang. Ranking method for node importance based on efficiency matrix[j]. Journa of Southwest Jiaotong University,2014,49(2): [7] DIMAKIS A G,RAMCHANDRAN K,WU Yunnan,et a. A survey on network codes for distributed storage[j]. Proceedings of the IEEE,2011,99(3): ("# 200 $)

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