A New Design of Binary MDS Array Codes with Asymptotically Weak-Optimal Repair

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1 IEEE TRANSACTIONS ON INFORMATION THEORY 1 A New Design of Binay MDS Aay Codes with Asymptotically Weak-Optimal Repai Hanxu Hou, Membe, IEEE, Yunghsiang S. Han, Fellow, IEEE, Patick P. C. Lee, Senio Membe, IEEE, Yuchong Hu, Membe, IEEE, and Hui Li, Membe, IEEE axiv: v2 [cs.it] 27 Feb 2018 Abstact Binay maximum distance sepaable (MDS) aay codes ae a special class of easue codes fo distibuted stoage that not only povide fault toleance with minimum stoage edundancy but also achieve low computational complexity. They ae constucted by encoding k infomation columns into paity columns, in which each element in a column is a bit, such that any k out of the k + columns suffice to ecove all infomation bits. In addition to poviding fault toleance, it is citical to impove epai pefomance in pactical applications. Specifically, if a single column fails, ou goal is to minimize the epai bandwidth by downloading the least amount of bits fom d healthy columns, whee k d k + 1. If one column of an MDS code is failed, it is known that we need to download at least 1/(d k + 1) faction of the data stoed in each of d healthy columns. If this lowe bound is achieved fo the epai of the failue column fom accessing abitay d healthy columns, we say that the MDS code has optimal epai. Howeve, if such lowe bound is only achieved by d specific healthy columns, then we say the MDS code has weak-optimal epai. Existing binay MDS aay codes that achieve high data ate (i.e., k/(k + ) > 1/2) and optimal epai of infomation column only suppot double fault toleance (i.e., = 2), which is insufficient fo failue-pone distibuted stoage envionments in pactice. This pape fills the void by poposing two explicit constuctions of binay MDS aay codes with moe paity columns (i.e., 3) that achieve asymptotically weak-optimal epai, whee k + 1 d k + ( 1)/2. Codes in the fist constuction have odd numbe of paity This pape was pesented in pat in [1] at the IEEE Intenational Symposium on Infomation Theoy, Aachen, Gemany, June H. Hou is with the School of Electical Engineeing & Intelligentization, Dongguan Univesity of Technology and the Shenzhen Gaduate School, Peking Univesity ( houhanxu@163.com). Y. S. Han is with the School of Electical Engineeing & Intelligentization, Dongguan Univesity of Technology ( yunghsiangh@gmail.com). P. P. C. Lee is with Depatment of Compute Science and Engineeing, The Chinese Univesity of Hong Kong ( pclee@cse.cuhk.edu.hk). Y. Hu is with the School of Compute Science and Technology, Huazhong Univesity of Science and Technology ( yuchonghu@hust.edu.cn). H. Li is with the Shenzhen Gaduate School, Peking Univesity ( lih64@pkusz.edu.cn). This wok was patially suppoted by the National Natual Science Foundation of China (No , , ) and National Keystone R&D Pogam of China (No. 2017YFB , 2016YFB ). Febuay 28, 2018

2 2 IEEE TRANSACTIONS ON INFORMATION THEORY columns and asymptotically weak-optimal epai fo any one infomation failue, while codes in the second constuction have even numbe of paity columns and asymptotically weak-optimal epai fo any one column failue. Index Tems MDS code, binay MDS aay code, optimal epai bandwidth. I. INTRODUCTION Moden distibuted stoage systems deploy easue codes to maintain data availability against failues of stoage nodes. Binay maximum distance sepaable (MDS) aay codes ae a special class of easue codes that achieve fault toleance with minimum stoage edundancy and low computational complexity. Specifically, a binay aay code is composed of k + columns with L bits in each column. Among the k + columns, k infomation columns stoe infomation bits and paity columns stoe paity bits. The L bits in each column ae stoed in the same stoage node. We efe to a disk as a column o a stoage node intechangeably, and an enty in the aay as a bit. When a node fails, the coesponding column of the aay code is consideed as an easue. A code is said to be MDS if any k out of the k + columns suffice to econstuct all k infomation columns. Hence, an MDS code can toleate any failed columns. Examples of binay MDS aay codes include double-fault toleance codes (i.e., = 2) such as X-code [2], RDP codes [3] and EVENODD codes [4], and tiple-fault toleance codes (i.e., = 3) such as STAR codes [5], genealized RDP codes [6], and TIP codes [7]. When a node fails in a distibuted stoage system, one should epai the failed node by downloading bits fom d healthy (helpe) nodes, whee k d k + 1. Minimizing the epai bandwidth, defined as the amount of bits downloaded in the epai pocess, is citical to speed up the epai opeation and minimize the window of vulneability, especially in distibuted stoage in which netwok tansfe is the bottleneck. The epai poblem was fist fomulated by Dimakis et al. [8] based on the concept of infomation flow gaph. It is shown in [8] that the minimum epai bandwidth subject to the minimum stoage edundancy, also known as the minimum stoage egeneating (MSR) point, is given by: dl d k + 1. (1) If the lowe bound in (1) is achieved fo the epai of the failue node fom accessing abitay d healthy nodes, we say that the MDS code has optimal epai. When such lowe bound is Febuay 28, 2018

3 SUBMITTED PAPER 3 only achieved by accessing d specific healthy nodes instead of abitay d healthy nodes, we then say the MDS code has weak-optimal epai. Note that the above epai is exact epai, i.e., the content stoed in the failed node is exactly epoduced in the new node. Anothe vesion of epai is functional epai, i.e., the ecoveed content and the failed content may be diffeent. In functional epai, it is inteesting to note that thee may exist a helpe node selection that can impove the lowe bound in (1) fo some specific paametes n, k, d [9]. Although the minimum epai bandwidth is achievable [8], [10], [11] ove a sufficiently lage finite field, how to constuct binay MDS aay codes that achieve the minimum epai bandwidth emains a challenge. A conventional appoach fo epaiing a failed node is to download all the bits fom any k healthy columns to egeneate the bits in the failue column. As a esult, the total amount of bits downloaded to epai a failue column is k times of the failue bits. Thee have been many studies on educing the epai bandwidth fo a single failed column in binay MDS aay codes. Some appoaches minimize disk eads fo RDP codes [12], EVENODD codes [13] and X-code [14] with d = k + 1; howeve, thei epai bandwidth is sub-optimal and 50% lage than the lowe bpund in (1) when d = k + 1. MDR codes [15], [16] and ButteFly codes [17], [18] ae binay MDS aay codes that achieve optimal epai; howeve, they only povide double-fault toleance (i.e., = 2). How to constuct binay MDS aay codes with both optimal epai and bette fault toleance (i.e., >2) is still an open poblem. In this pape, we fill the void by poposing two explicit constuctions of binay MDS aay codes with moe paity columns (i.e., 3) that achieve asymptotically weak-optimal epai. The main contibutions of this pape ae as follows. 1) We exploit a quotient ing with cyclic stuctue and popose a new appoach of designing binay MDS aay codes with 3 paity columns based on the quotient ing. 2) Two explicit constuctions of binay MDS aay codes with 3 paity columns ae pesented based on the poposed appoach. Codes in the fist constuction have odd numbe of paity columns that is fomed by designing an encoding matix and codes in the second constuction have even numbe of paity columns that is fomed by designing a check matix. 3) Ou constuctions minimize the epai bandwidth by exploiting the poposed quotient ing and choosing the well-designed encoding matix (paity matix), such that the bits accessed in a epai opeation intesect as much as possible. 4) We show that the fist constuction of the poposed binay MDS aay codes has asymptot- Febuay 28, 2018

4 4 IEEE TRANSACTIONS ON INFORMATION THEORY ically weak-optimal epai with d = k +( 1)/2 fo any single infomation column failue when d is sufficiently lage. We also show that the second constuction of the poposed binay MDS aay codes has asymptotically weak-optimal epai with d = k + /2 1 fo any column failue when d is lage enough. The epai bandwidth of most existing binay MDS aay codes [2] [6] is sub-optimal. Some constuctions [15] [18] of binay MDS aay codes with optimal epai bandwidth only focus on double-fault toleance (i.e., = 2). As fa as we know, the poposed codes ae the fist binay MDS aay codes with asymptotically weak-optimal epai bandwidth and with moe than double-fault toleance. The key diffeences between the poposed codes and the existing binay MDS aay codes ae as follows. Fist, in contast to existing constuctions such as [2], [3], [5], [6], the paity bits in paity columns (except the fist paity column) of the poposed codes ae geneated by summing the bits that coespond to a specific polygonal lines in the aay. Second, the ow numbe of the aay in the poposed codes is exponential in k. The two popeties ae essential fo educing the epai bandwidth. The diffeence between the double-fault toleance optimal epai constuctions [15] [18] and the poposed constuctions is that a quotient ing with cyclic stuctue is employed in the poposed constuction, while is not in [15] [18]. By exploiting the quotient ing, we can choose the well-designed encoding matix (paity matix) and achieve the weak-optimal epai bandwidth asymptotically with lage fault toleance. Unlike the othe binay MDS aay codes [15] [18] with optimal epai bandwidth, the poposed codes with the fist constuction have asymptotically weak-optimal epai bandwidth fo ecoveing one infomation column failue. Note that the infomation failue column is ecoveed by accessing d specific columns instead of any abitay set of d healthy columns. Similaly, the poposed codes with the second constuction have asymptotically weak-optimal epai bandwidth fo ecoveing any one column failue (including both infomation column and paity column), and the failue column is also ecoveed by accessing d specific healthy columns. Pevious studies [19], [20] also exploit simila techniques (applying quotient ings) to educe computational complexity of egeneating codes. In this wok, we show that when τ (a paamete that will be found late) is lage enough and satisfies some conditions, we can find some constuctions of binay MDS aay codes that can achieve the weak-optimal epai asymptotically. The ing in [19], [20] can be viewed as a special case of the poposed ing with τ = 1. Moeove, the main esults between [19], [20] and this pape ae diffeent. It was shown in [19], [20] that the fundamental tadeoff cuve between the stoage and epai bandwidth of functional-epai Febuay 28, 2018

5 SUBMITTED PAPER 5 egeneating codes is also achievable unde a quotient ing, and the existing poduct-matix constuction of egeneating codes still woks unde the quotient ing with less computational complexity. While in this pape, we use a moe geneal ing to constuct a new class of binay MDS aay codes with asymptotically weak-optimal epai bandwidth by choosing the welldesigned geneato matix o paity matix. Even though the poposed binay MDS aay codes and constuctions of high data ate MSR codes [10], [11], [21] [26] ae all based on constucting geneato matices o paity matices, the poposed codes ae constucted ove binay field and the encoding matices o paity matices ae designed on the ing with a cyclic stuctue. This pape is oganized as follows. An appoach of designing binay aay codes with geneal numbe of paity columns is given in Section II. In Section III, two constuctions of binay MDS aay codes with moe than thee paity columns ae pesented. In Section IV, a sufficient condition of the MDS popety condition fo two constuctions is pesented. We also illustate the MDS popety of = 3 fo the fist constuction and = 4 fo the second constuction in detail. Two efficient epai algoithms ae poposed in Section V fo the poposed two constuctions to ecovey single infomation column easue and single column easue espectively. It is also shown that both two constuctions can achieve the weak-optimal epai bandwidth in (1) asymptotically. We make conclusion and emak in Section VI. II. NEW DESIGN OF BINARY MDS ARRAY CODES Conside the binay MDS aay code with k 2 infomation columns and 3 paity columns. Each column of this code stoes (p 1)τ bits, whee p is a pime numbe such that 2 is a pimitive element in the field Z p and τ will be specified late. Assume that a file of size k(p 1)τ denoted by infomation bits s 0,i, s 1,i,..., s (p 1)τ 1,i F (p 1)τ 2 fo i = 1, 2,..., k, which ae used to geneate (p 1)τ paity bits s 0,j, s 1,j,..., s (p 1)τ 1,j F (p 1)τ 2 fo j = k +1, k +2,..., k +. The infomation bits s 0,i, s 1,i,..., s (p 1)τ 1,i ae stoed in infomation column i (column i) fo i = 1, 2,..., k, and the (p 1)τ paity bits s 0,j, s 1,j,..., s (p 1)τ 1,j ae stoed in paity column j k (column j) fo j = k + 1, k + 2,..., k +. Fo i = 1, 2,..., k and µ = 0, 1,..., τ 1, we define the exta bit s (p 1)τ+µ,i associated with bits s µ,i, s τ+µ,i,..., s (p 2)τ+µ,i as p 2 s (p 1)τ+µ,i s lτ+µ,i. (2) l=0 Febuay 28, 2018

6 6 IEEE TRANSACTIONS ON INFORMATION THEORY Fo example, when p = 3, k = 4 and τ = 4, the exta bit of s µ,i, s 4+µ,i is s 8+µ,i = s µ,i + s 4+µ,i. Fo j = k + 1, k + 2,..., k +, τ exta bits s (p 1)τ,j, s (p 1)τ+1,j,..., s pτ 1,j fo paity column j k ae added duing the encoding pocedue. It will be clea late that the exta bit s (p 1)τ+µ,j of paity column j k also satisfies (2) fo j = k + 1, k + 2,..., k + and µ = 0, 1,..., τ 1, if we eplace i with j in (2). Fo l = 1, 2,..., k+, we epesent (p 1)τ bits s 0,l, s 1,l,..., s (p 1)τ 1,l in column l, togethe with the τ exta bits s (p 1)τ,l, s (p 1)τ+1,l,..., s pτ 1,l, by a polynomial s l (x) of degee at most pτ 1 ove the ing F 2 [x], i.e., s l (x) s 0,l + s 1,l x + s 2,l x s pτ 1,l x pτ 1. (3) The polynomial s i (x), which coesponds to infomation column i fo i = 1, 2,..., k, is called a data polynomial; the polynomial s j (x), which coesponds to paity column j k fo j = k + 1, k + 2,..., k +, is called a coded polynomial. The k data polynomials and coded polynomials can be aanged as a ow vecto (4) can be obtained by taking the poduct [s 1 (x), s 2 (x),, s k+ (x)]. (4) [s 1 (x), s 2 (x),, s k+ (x)] = [s 1 (x), s 2 (x),, s k (x)] G k (k+) (5) with opeations pefomed in F 2 [x]/(1 + x pτ ). The k (k + ) geneato matix G k (k+) is composed of the k k identity matix I k k and a k encoding matix P k as ] G k (k+) = [I k k P k. (6) The poposed code can also be descibed equivalently by a (k + ) check matix H (k+). Given the ow vecto in (4), we have [s 1 (x), s 2 (x),, s k+ (x)] H (k+) = 0. (7) Let R pτ denote the quotient ing F 2 [x]/(1 + x pτ ). An element a(x) in R pτ can be epesented by a polynomial of the fom a(x) = a 0 + a 1 x + + a pτ 1 x pτ 1 with coefficients fom the binay field F 2. Addition is the usual tem-wise addition, and multiplication is pefomed with modulo 1 + x pτ. In R pτ, multiplication by x can be intepeted as a cyclic shift. This is cucial Febuay 28, 2018

7 SUBMITTED PAPER 7 to educe epai bandwidth fo one column failue. Note that we do not need to stoe the exta bits on disk, they ae pesented only fo notational convenience. Conside the sub-ing C pτ of R pτ which consists of polynomials in R pτ with 1 + x τ being a facto, C pτ {a(x)(1 + x τ ) mod (1 + x pτ ) a(x) R pτ }. (8) In fact, C pτ is an ideal, because c(x) R pτ, s(x) C pτ, c(x)s(x) C pτ. One can veify that the poduct of h(x) 1 + x τ + + x (p 1)τ and any polynomial in C pτ is zeo. The polynomial h(x) is called the check polynomial of C pτ. The multiplication identity of C pτ is e(x) 1 + h(x) = x τ + x 2τ + + x (p 1)τ = (1 + x τ )(x τ + x 3τ + + x (p 2)τ ), (9) as b(x) C pτ, e(x)b(x) = (1 + h(x))b(x) = b(x) mod (1 + x pτ ). (10) Theoem 1. The coefficients of polynomial s i (x) satisfy (2) if and only if s i (x) is in C pτ. Poof. Suppose that the coefficients of s i (x) satisfy (2). By efomulating s i (x), we have p 2 p 2 s i (x) = s 0,i + s 1,i x + + s (p 1)τ 1,i x (p 1)τ 1 + x (p 1)τ s lτ,i + + x pτ 1 p 2 = s 0,i + s τ,i x τ + + s (p 2)τ,i x (p 2)τ + x (p 1)τ s lτ,i + l=0 l=0 l=0 p 2 s 1,i x + s τ+1,i x τ s (p 2)τ+1,i x (p 2)τ+1 + x (p 1)τ+1 s lτ+1,i + + p 2 s τ 1,i x τ 1 + s 2τ 1,i x 2τ s (p 1)τ 1,i x (p 1)τ 1 + x pτ 1 l=0 l=0 s (l+1)τ 1,i s lτ+τ 1,i = s 0,i (1 + x (p 1)τ ) + s τ,i (x τ + x (p 1)τ ) + + s (p 2)τ,i (x (p 2)τ + x (p 1)τ ) + s 1,i (x + x (p 1)τ+1 ) + s τ+1,i (x τ+1 + x (p 1)τ+1 ) + + s (p 2)τ+1,i (x (p 2)τ+1 + x (p 1)τ+1 ) + + s τ 1,i (x τ 1 + x pτ 1 ) + s 2τ 1,i (x 2τ 1 + x pτ 1 ) + + s (p 1)τ 1,i (x (p 1)τ 1 + x pτ 1 ). Febuay 28, 2018

8 8 IEEE TRANSACTIONS ON INFORMATION THEORY This is educed to show that x iτ+j + x (p 1)τ+j is a multiple of 1 + x τ fo i = 0, 1,..., p 2 and j = 0, 1,..., τ 1. It is tue fom the fact x iτ+j + x (p 1)τ+j = x iτ+j (1 + x (p i 1)τ ) = x iτ+j (1 + x τ )(1 + x τ + x 2τ + + x (p i 2)τ ). This veifies that the polynomial s i (x) is in C pτ. Convesely, suppose that s i (x) = pτ 1 l=0 s l,ix l is in C pτ. By (8), s i (x) can be witten as s i (x) a(x)(1 + x τ ) mod (1 + x pτ ) = (a 0 + a (p 1)τ ) + (a 1 + a (p 1)τ+1 )x + + (a τ + a 0 )x τ + + (a pτ 1 + a (p 1)τ 1 )x pτ 1. Theefoe, we obtain s µ,i = a µ + a (p 1)τ+µ, s τ+µ,i = a τ+µ + a µ,, s (p 1)τ+µ,i = a (p 1)τ+µ + a (p 2)τ+µ, fo µ = 0, 1,..., τ 1. We can check that s µ,i + s τ+µ,i + + s (p 2)τ+µ,i = (a µ + a (p 1)τ+µ ) + (a τ+µ + a µ ) + + (a (p 2)τ+µ + a (p 3)τ+µ ) Theefoe, the coefficients of s i (x) satisfy (2). Since the equation = a (p 1)τ+µ + a (p 2)τ+µ = s (p 1)τ+µ,i. (1 + x τ )(x τ + x 3τ + + x (p 2)τ ) + 1 h(x) = 1 holds ove F 2 [x], 1 + x pτ can be factoized as a poduct of two co-pime factos 1 + x τ and h(x). We show in the next lemma that R pτ is isomophic to F 2 [x]/(1 + x τ ) F 2 [x]/(h(x)). Lemma 2. The ing R pτ is isomophic to the poduct ing F 2 [x]/(1 + x τ ) F 2 [x]/(h(x)). Poof. We need to find an isomophism between R pτ and F 2 [x]/(1+x τ ) F 2 [x]/(h(x)). Indeed, we can set up an isomophism θ : R pτ F 2 [x]/(1 + x τ ) F 2 [x]/(h(x)) by defining θ(f(x)) (f(x) mod 1 + x τ, f(x) mod h(x)). The mapping θ is a ing homomophism and a bijection, because it has an invese function φ((a(x), b(x))) given by φ((a(x), b(x))) [a(x) h(x) + b(x) e(x)] mod 1 + x pτ. Febuay 28, 2018

9 SUBMITTED PAPER 9 In the following, we show that the composition φ θ is the identity map of R pτ. Fo any polynomial f(x) R pτ, thee exist two polynomials g 1 (x), g 2 (x) F 2 [x] such that f(x) = g 1 (x)(1 + x τ ) + f(x) mod (1 + x τ ), f(x) = g 2 (x)h(x) + f(x) mod h(x). Then we have φ(θ(f(x))) = [h(x)(f(x) mod (1 + x τ )) + e(x)(f(x) mod h(x))] mod 1 + x pτ = [h(x)(f(x) g 1 (x)(1 + x τ )) + (1 + h(x))(f(x) g 2 (x)h(x))] mod 1 + x pτ = [h(x)f(x) h(x)g 1 (x)(1 + x τ ) + f(x) + h(x)f(x) e(x)g 2 (x)h(x)] mod 1 + x pτ = [f(x) h(x)g 1 (x)(1 + x τ ) e(x)g 2 (x)h(x)] mod 1 + x pτ = [ f(x) (1 + x τ )(x τ + x 3τ + + x (p 2)τ )g 2 (x)h(x) ] mod 1 + x pτ = f(x). The composition φ θ is thus the identity mapping of R pτ and the lemma is poved. By Lemma 2, we have that C pτ is isomophic to F 2 [x]/(h(x)) in the next lemma. Lemma 3. The ing C pτ is isomophic to F 2 [x]/(h(x)). Futhemoe, the isomophism θ : C pτ F 2 [x]/(h(x)) can be defined as θ(f(x)) f(x) mod h(x). Fo example, when p = 5 and τ = 2, C 10 is isomophic to the ing F 2 [x]/(1+x 2 +x 4 +x 6 +x 8 ) and the element 1 + x 8 in C 10 is mapped to 1 + x 8 mod (1 + x 2 + x 4 + x 6 + x 8 ) = x 2 + x 4 + x 6. If we apply the function φ to x 2 + x 4 + x 6, we can ecove φ(0, x 2 + x 4 + x 6 ) = (x 2 + x 4 + x 6 )(x 2 + x 4 + x 6 + x 8 ) = 1 + x 8 mod (1 + x 10 ). When τ = 1, the ing C p is discussed in [19], [20] and used in a new class of egeneating codes with low computational complexity. Note that C pτ is isomophic to a finite field F 2 (p 1)τ if and only if 2 is a pimitive element in Z p and τ = p i fo some non-negative intege i [27]. We need the following definition about e(x)-invese befoe intoducing the explicit constuctions of the poposed aay codes. Definition 1. A polynomial f(x) R pτ is called e(x)-invetible if we can find a polynomial f(x) R pτ such that f(x) f(x) = e(x), whee e(x) is given in (9). The polynomial f(x) is called e(x)-invese of f(x). Febuay 28, 2018

10 10 IEEE TRANSACTIONS ON INFORMATION THEORY We show in the next lemma that 1 + x b is e(x)-invetible in R pτ. Lemma 4. Let b be an intege with 1 b < pτ and the geatest common diviso (GCD) of b and p is gcd(b, p) = 1, and let gcd(b, τ) = a. The e(x)-invese of 1 + x b in R pτ is 2τ/a 1 i=τ/a x ib + 4τ/a 1 i=3τ/a x ib + + (p 1)τ/a 1 i=(p 2)τ/a x ib. (11) Poof. We can check that, in R pτ, ( (1 + x b ) (x τ a b + x ( τ a +1)b + + x (2 τ a 1)b )+ ) (x 3 τ a b + x (3 τ a +1)b + + x (4 τ a 1)b ) + + (x (p 2) τ a b + x ((p 2) τ a +1)b + + x ((p 1) τ a 1)b ) =(x τ a b + x 2 τ a b + x 3 τ a b + + x (p 2) τ a b + x (p 1) τ a b ). It is sufficient to show that the above equation is equal to e(x), i.e., x τ a b + x 2 τ a b + x 3 τ a b + + x (p 2) τ a b + x (p 1) τ a b e(x) mod (1 + x pτ ). (12) Conside the ing of integes modulo pτ, which is denoted Z pτ. In Z pτ, thee is a set τz pτ (0, τ, 2τ,..., (p 1)τ). Now we conside (ib/a) mod pτ Z pτ fo i {1, 2,..., p 1}. Theefoe, (τib/a) mod pτ τz pτ. Next, we want to show that, fo i j {1, 2,..., p 1}, iτb/a jτb/a mod pτ. Assume that iτb/a mod pτ = jτb/a mod pτ. Then thee exists an intege l such that iτb/a = lpτ + jτb/a. The above equation can be futhe educed to (i j)b/a = lp. Since gcd(b, p) = 1, gcd(b/a, p) = 1. Hence, we have p (i j). Howeve, this is impossible due to the fact that 1 j < i p 1. Similaly, we can pove that, fo 1 i p 1, iτb/a mod pτ 0. Febuay 28, 2018

11 SUBMITTED PAPER 11 Hence, we can obtain that (τb/a, 2τb/a,..., (p 1)τb/a) (τ, 2τ,..., (p 1)τ) mod pτ. Theefoe, (12) holds. By Lemma 1, we have s i (x) C pτ fo i = 1, 2,..., k. Let f(x) be any enty of the geneato matix o check matix. If f(x) / C pτ, one may eplace the enty by (f(x)e(x) mod (1+x pτ )) C pτ without changing the esults. This is due to the fact that s i (x)e(x) = s i (x) mod (1 + x pτ ), fo 1 i k. Hence, afte eplacing all f(x) of the geneato matix o check matix with (f(x)e(x) mod (1 + x pτ )), we have an equivalent geneato matix o check matix such that the coded polynomials in (4) can be computed ove the ing C pτ via (5) o (7). The encoding pocedue can be descibed in tems of polynomial opeations as follows. Given k(p 1)τ infomation bits, by (3), one appends τ exta bits fo each of (p 1)τ infomation bits and foms k data polynomials that belong to C pτ. Afte obtaining the vecto in (4) by choosing some specific encoding matix o check matix, one stoes the coefficients in the polynomials of degees 0 to (p 1)τ 1 and dops the coefficients in highe degees. The poposed aay code can be consideed as punctued systematic linea code ove C pτ. III. TWO EXPLICIT CONSTRUCTIONS OF BINARY MDS ARRAY CODES The pupose of this pape is to find suitable encoding matices P k o check matices H (k+) such that the coesponding codes ae MDS codes and the epai bandwidth of one single failue is asymptotically weak-optimal. In the section, we will give two explicit constuctions of binay MDS aay codes, whee the fist constuction is fomed by an encoding matix and the second constuction is fomed by a check matix. The epai bandwidth of the fist constuction is asymptotically weak-optimal fo any one single infomation failue, while the epai bandwidth of the second constuction is asymptotically weak-optimal fo any one single infomation o paity failue. Note that, fo codes constucted fom both constuctions, not all paametes exist fo them to be MDS codes. A sufficient condition will be deived fo them to be MDS codes when p is lage enough. Some constucted codes with small p ae also poved to be MDS codes. Febuay 28, 2018

12 12 IEEE TRANSACTIONS ON INFORMATION THEORY A. The Fist Constuction: Encoding Matix The constucted code is denoted by C 1 (k,, d, p) with P k given as follows: 1 x x 2 x d k x η x 2η x (d k)η x (d k)ηk 2 x (d k 1)ηk 2 x ηk 2 1 x η2 x 2η2 x (d k)η2 x (d k)ηk 3 x (d k 1)ηk 3 x ηk 3 P k x ηk 3 x 2ηk 3 x (d k)ηk 3 x (d k)η2 x (d k 1)η2 x η2 1 x ηk 2 x 2ηk 2 x (d k)ηk 2 x (d k)η x (d k 1)η x η x d k x d k 1 x, (13) whee η = d k + 1, 1 k 4, 3 is an odd numbe, d = k + ( 1)/2, τ = (d k + 1) k 2, and p > d k. Since evey data polynomial is in C pτ and C pτ is an ideal, we have the following lemma. Lemma 5. Fo j = k + 1, k + 2,..., k +, each coded polynomial s j (x) in C 1 (k,, d, p) belongs to C pτ. By Lemma 1 and Lemma 5, the coefficients of the coded polynomials s j (x) satisfy (2) if we eplace i with j in (2). Let (i : j) = {i, i + 1,..., j} and P k (i : j) be the sub-matix of P k with column index detemined by (i : j). In P k, the sub-matix P k (η + 1 : 2η 1) is a clockwise otation of the sub-matix P k (2 : η) by 180 degees. The last ow of P k (2 : η) is an all one vecto, and the exponent of the enty in ow i and column j of P k (2 : η) is η i 1 times of that in the fist ow and column j fo i = 2, 3,..., k 1 and j = 1, 2,..., d k. Example 1. Conside k = 4, p = 3, = 3. Hence, d = = 5 and τ = 4. The 32 infomation bits ae epesented by s 0,i, s 1,i,..., s 7,i, fo i = 1, 2, 3, 4. The encoding matix is 1 x 1 1 x 2 x 4 P 4 3 =. 1 x 4 x x The Example 1 is illustated in Fig. 1. Note that the exta bits calculated fom the infomation bits do not need to be stoed and they ae only used to calculate the paity bits. 1 η and d k + 1 will be intechangeably used in the wok. Febuay 28, 2018

13 SUBMITTED PAPER 13 Infomation Columns Paity Columns s 0,1 s 0,2 s 0,3 s 0,4 s 0,1 +s 0,2 +s 0,3 +s 0,4 s 11,1 +s 10,2 +s 8,3 +s 0,4 s 0,1 +s 8,2 +s 10,3 +s 11,4 s 1,1 s 1,2 s 1,3 s 1,4 s 1,1 +s 1,2 +s 1,3 +s 1,4 s 0,1 +s 11,2 +s 9,3 +s 1,4 s 1,1 +s 9,2 +s 11,3 +s 0,4 s 2,1 s 2,2 s 2,3 s 2,4 s 2,1 +s 2,2 +s 2,3 +s 2,4 s 1,1 +s 0,2 +s 10,3 +s 2,4 s 2,1 +s 10,2 +s 0,3 +s 1,4 s 3,1 s 3,2 s 3,3 s 3,4 s 3,1 +s 3,2 +s 3,3 +s 3,4 s 2,1 +s 1,2 +s 11,3 +s 3,4 s 3,1 +s 11,2 +s 1,3 +s 2,4 s 4,1 s 4,2 s 4,3 s 4,4 s 4,1 +s 4,2 +s 4,3 +s 4,4 s 3,1 +s 2,2 +s 0,3 +s 4,4 s 4,1 +s 0,2 +s 2,3 +s 3,4 s 5,1 s 5,2 s 5,3 s 5,4 s 5,1 +s 5,2 +s 5,3 +s 5,4 s 4,1 +s 3,2 +s 1,3 +s 5,4 s 5,1 +s 1,2 +s 3,3 +s 4,4 s 6,1 s 6,2 s 6,3 s 6,4 s 6,1 +s 6,2 +s 6,3 +s 6,4 s 5,1 +s 4,2 +s 2,3 +s 6,4 s 6,1 +s 2,2 +s 4,3 +s 5,4 s 7,1 s 7,2 s 7,3 s 7,4 s 7,1 +s 7,2 +s 7,3 +s 7,4 s 6,1 +s 5,2 +s 3,3 +s 7,4 s 7,1 +s 3,2 +s 5,3 +s 6,4 Fig. 1: The infomation and paity columns in Example 1. When infomation column 1 fails, the bits in the solid line box ae downloaded to epai the infomation bits s 0,1, s 2,1, s 4,1, s 6,1 and the bits in the dashed box ae used to epai the infomation bits s 1,1, s 3,1, s 5,1, s 7,1. B. The Second Constuction: Check Matix The second constuction is denoted by C 2 (k,, d, p) with the check matix defined as 1 x x 2 x x 2 x 2 2 x ( 2 1) x ( 2 ) 2 1 x 2( 2 ) 2 1 x ( 2 1)( 2 ) x ( 2 )d 1 x 2( 2 )d 1 x ( 2 1)( 2 )d 1 x ( 2 1)( 2 )d 1 x ( 2 )d 1 x (d 1)( 2 )d 1, x ( 2 1)( 2 ) 2 1 x ( 2 ) 2 1 x ( 2 1)( 2 )d x ( 2 1) 2 x 2 x ( 2 )d 1 x ( 2 1) x 1 (14) whee k 4, 4 is an even numbe, the matix 0 is the 2 2 zeo matix, d = k + 2 1, τ = ( 2 )d 1, and p >. Unlike the definition given in (4), it will be shown late that, fo 2 Febuay 28, 2018

14 14 IEEE TRANSACTIONS ON INFORMATION THEORY C 2 (k,, d, p), k data polynomials can be placed in any k positions of the codewods. To compute the coded polynomials, we should solve a linea equation system with the encoding coefficients being a sub-matix of (14). In the following, we fist give a constuction fo = 4. The cases fo > 4 will be given late. A C 2 (k, 4, d, p) contains k + 4 polynomials s 1 (x), s 2 (x),..., s k+4 (x), whee we select s 3 (x), s 4 (x),..., s k+2 (x) as data polynomials and s 1 (x), s 2 (x), s k+3 (x), s k+4 (x) as coded polynomials. The check matix H (k+4) 4 is given as follows: x x 2 x 4 x 8 x 2k H (k+4) x 2k x 8 x 4 x 2 x x k 2k x 3 2k x 2 2k x 2k 1 Fom the fist two columns of H (k+4) 4, we have T. (15) s 1 (x) + s 2 (x) + + s k+1 (x) + s k+2 (x) = 0, (16) xs 1 (x) + x 2 s 2 (x) + + x 2k s k+1 (x) + s k+2 (x) = 0. (17) Fist, we can compute the summation of cyclic-shifted vesion of the data polynomials as p 1 (x) s 3 (x) + s 4 (x) + + s k+1 (x) + s k+2 (x), p 2 (x) x 4 s 3 (x) + x 8 s 4 (x) + + x 2k s k+1 (x) + s k+2 (x). Substituting (16) into p 1 (x) and (17) into p 2 (x), we have xs 2 (x) = xp 1(x) + p 2 (x). 1 + x Note that we may solve fo s 2 (x) by left cyclic shifting of xs 2 (x). It is easy to see that p 1 (x), p 2 (x) C pτ, and theefoe xp 1 (x) + p 2 (x) C pτ. By Lemma 4, 1 + x b is e(x)-invetible and we can compute g(x) fom (1 + x b )g(x) = f(x) as g(x) = f(x) ( 2τ/a 1 i=τ/a x ib + 4τ/a 1 i=3τ/a x ib + + (p 1)τ/a 1 i=(p 2)τ/a x ib ), (18) whee f(x) C pτ and gcd(b, τ) = a. By letting b = 1 and f(x) = xp 1 (x) + p 2 (x), we can solve g(x) = xs 2 (x) by (18) and then s 1 (x) by (16). As f(x) C pτ, the esulting polynomial g(x) in (18) is also in C pτ. Theefoe, the coded polynomials s 1 (x), s 2 (x) ae in C pτ. The othe two coded polynomials s k+3 (x), s k+4 (x) can be computed similaly and ae also in C pτ. Febuay 28, 2018

15 SUBMITTED PAPER 15 Next, we demonstate how to efficiently compute g(x) via (18). We can compute the coefficients g j fo j = 0, 1,..., a 1 by g j = 2τ/a 1 i=τ/a f (j ib) mod pτ + 4τ/a 1 i=3τ/a f (j ib) mod pτ + + (p 1)τ/a 1 i=(p 2)τ/a f (j ib) mod pτ. (19) Since (1+x b )g(x) = f(x), once g 0, g 1,..., g a 1 ae known, we can compute the othe coefficients of g(x) iteatively by g bl+j = f bl+j + g b(l 1)+j (20) with the index l unning fom 1 to pτ/a 1 and 0 j a 1. It can be shown that bl 1 + j mod pτ bl 2 + j mod pτ fo l 1 l 2 {1, 2,..., pτ/a 1}. Theefoe, we can compute all the othe coefficients of g(x) by (20). We can count that thee ae a( p 1 2 τ 3pτ τ 4a 1) + (pτ a) = a 2 XORs involved in solving g(x) fom (1 + x b )g(x) = f(x). Example 2. We give a constuction with k = 4, p = 3, = 4, and then d = 5 and τ = 16. Each column stoes (p 1)( 2 )d 1 = 32 bits and thee ae k + = 8 columns. We have 8 polynomials s i (x) = 47 l=0 s l,ix l fo i = 1, 2,..., 8. Suppose that s 3 (x), s 4 (x), s 5 (x), s 6 (x) ae fou data polynomials. Fist, we compute p 1 (x) = 47 l=0 p l,1x l, p 2 (x) = 47 l=0 p l,2x l as p 1 (x) = s 3 (x) + s 4 (x) + s 5 (x) + s 6 (x), p 2 (x) = x 4 s 3 (x) + x 8 s 4 (x) + x 16 s 5 (x) + s 6 (x). Then, we can compute xs 2 (x) via xs 2 (x) = xp 1(x) + p 2 (x) 1 + x by (19) and (20), and obtain s 2 (x) by left cyclic shifting of xs 2 (x), i.e., l j=0 s l,2 = (p j,1 + p j+1,2 ) + 31 i=16 (p 47 i,1 + p 48 i,2 ) fo l = 0, 1,..., 46; 31 i=16 (p 47 i,1 + p 48 i,2 ) fo l = 47. We can then compute s 1 (x) as s 1 (x) = s 2 (x) + p 1 (x), i.e., p l,1 + l j=0 s l,1 = (p j,1 + p j+1,2 ) + 31 i=16 (p 47 i,1 + p 48 i,2 ) fo l = 0, 1,..., 46; p l, i=16 (p 47 i,1 + p 48 i,2 ) fo l = 47. We can also compute columns 7 and 8 in a simila way. Febuay 28, 2018

16 16 IEEE TRANSACTIONS ON INFORMATION THEORY The encoding pocedue with > 4 is descibed as follows. Since we will show that C 2 (k,, d, p) satisfy the MDS condition in Theoem 7 (in Section IV), the encoding pocedue can be implemented as a special case of decoding pocedue. Thee ae k+ polynomials s 1 (x), s 2 (x),..., s k+ (x), and assume that k data polynomials ae s 2 +1, s 2 +2,..., s 2 +k. We fist eplace each enty f(x) of H (k+) with f(x)e(x) mod (1 + x pτ ), and then solve the linea equations fo coded polynomials accoding to the modified check matix ove C pτ. As C 2 (k,, d, p) satisfy the MDS condition, we can always compute the coded polynomials. IV. THE MDS PROPERTY Let Ḡ k (k+) and H (k+) be the matices by eplacing each enty f(x) of G k (k+) and H (k+) with f(x)e(x) mod (1+x pτ ) espectively. The codes satisfy MDS popety if and only if the deteminant of any k k sub-matix of Ḡk (k+) o sub-matix of H (k+) is e(x)-invetible. Recall that C pτ is isomophic to F 2 [x]/(h(x)) by Lemma 3, the necessay and sufficient MDS condition is equivalent to that the deteminant of any k k sub-matix of Ḡk (k+) o sub-matix of H (k+), afte educing modulo h(x), is invetible in F 2 [x]/(h(x)). Note that the deteminant of any k k sub-matix of Ḡk (k+) o sub-matix of H (k+) afte educing modulo h(x) can be computed by fist educing each enty of the squae matix by h(x), and then computing the deteminant afte educing modulo h(x). Fo any integes i, j, we have (x ipτ+j mod (1 + x pτ )) mod h(x) = x j mod h(x) and (f(x)e(x) mod (1 + x pτ )) mod h(x) = (f(x)e(x) mod h(x)) mod 1 + x pτ = [f(x)(1 + h(x)) mod h(x)] mod 1 + x pτ = (f(x) mod 1 + x pτ ) mod h(x) = f(x) mod h(x). It is sufficient to show that the deteminant of any k k sub-matix of G k (k+) (o equivalently any squae sub-matix of (13) by Coollay 3 and Theoem 8 in [28]) o sub-matix of H (k+), afte modulo h(x), is invetible ove F 2 [x]/(h(x)). Theoem 6. Let h(x) be factoized as a poduct of powes of ieducible polynomials ove F 2 : h(x) (f 1 (x)) l1 (f 2 (x)) l2 (f t (x)) lt, (21) Febuay 28, 2018

17 SUBMITTED PAPER 17 whee l i 1 fo i = 1, 2,..., t. C 1 (k,, d, p) (o C 2 (k,, d, p)) is an MDS code if and only if the deteminant of any l l sub-matix of (13) fo l = 1, 2,..., min{k, } (o the deteminant of any sub-matix of (14)) is a non-zeo polynomial in F 2 [x]/(f i (x)) fo i = 1, 2,..., t. Poof. By Chinese emainde theoem, the ing F 2 [x]/h(x) is isomophic to the diect sum of t ings F 2 [x]/(f 1 (x)) l 1, F 2 [x]/(f 2 (x)) l 2,..., F 2 [x]/(f t (x)) lt, and the mapping θ is defined by θ(a(x)) (a(x) mod (f 1 (x)) l 1,..., a(x) mod (f t (x)) lt ), whee a(x) F 2 [x]/h(x). The invese of θ is given by θ 1 (a 1 (x),..., a t (x)) t a i (x) h i (x) < h i (x) 1 > fi (x) l i i=1 modh(x), whee a i (x) F 2 [x]/(f i (x)) l i and h i (x) = h(x)/(f i (x)) l i fo i = 1, 2,..., t, < h i (x) 1 > fi (x) l i denotes the multiplicative invese of h i (x) mod f i (x) l i and 1 i t. It can be checked that θ(θ 1 (a 1 (x),..., a t (x))) = (a 1 (x),..., a t (x)). A moe geneal vesion of the polynomial Chinese emainde theoem is pesented in [29, Theoem 1]. The code is MDS if and only if the deteminants of all the sub-matices ae invetible in F 2 [x]/h(x). Suppose that a deteminant a(x) is invetible, i.e., thee exists a polynomial g(x) such that g(x)a(x) = 1 mod h(x). If we apply the mapping θ to g(x)a(x) mod h(x), then we have θ(g(x)a(x) mod h(x)) = θ(1) = (1 mod (f 1 (x)) l 1,..., 1 mod (f t (x)) lt ). Theefoe, a(x) is invetible in F 2 [x]/(f i (x)) l i and is a non-zeo polynomial in F 2 [x]/f i (x) fo i = 1, 2,..., t. Convesely, suppose that a deteminant a(x) mod f i (x) is a non-zeo polynomial in F 2 [x]/f i (x) fo i = 1, 2,..., t. As f i (x) is ieducible polynomial, we have gcd(f i (x), a(x)) = 1. Hence, gcd(f i (x) l i, a(x)) = 1. Theefoe, thee exists a polynomial g i (x) such that g i (x)a(x) = 1 mod f i (x) l i. By Chinese emainde theoem, thee exists a unique polynomial f(x) F 2 [x]/h(x) such that θ(f(x)a(x) mod h(x)) = (g 1 (x)a(x) mod f 1 (x) l 1,, g t (x)a(x) mod f t (x) lt ) = (1,, 1). Febuay 28, 2018

18 18 IEEE TRANSACTIONS ON INFORMATION THEORY By applying the invese mapping θ 1 to θ(f(x)a(x) mod h(x)), we obtain that θ 1 (θ(f(x)a(x) mod h(x))) = f(x)a(x) mod h(x) = θ 1 (g 1 (x)a(x) mod (f 1 (x)) l 1,..., g t (x)a(x) mod (f t (x)) lt ) t = θ 1 (1,..., 1) = h i (x) < h i (x) 1 > fi (x) l i mod h(x). i=1 As gcd((f i (x)) l i, (f j (x)) l j ) = 1 fo i j, we have gcd(h 1 (x), h 2 (x),, h t (x)) = 1. By Bézout s identity [30], thee exist polynomials b 1 (x), b 2 (x),..., b t (x) in F 2 [x]/h(x) such that t h i (x) b i (x) mod h(x) = 1 i=1 and b i (x) can be computed as b i (x) =< h i (x) 1 > fi (x) l i. Theefoe, we obtain that f(x)a(x) mod h(x) = t h i (x) b i (x) mod h(x) = 1 mod h(x), and a(x) is invetible in F 2 [x]/h(x). This completes the poof. A sufficient MDS condition is given in the next theoem. i=1 Theoem 7. Let h(x) be factoized as in (21), whee deg(f 1 (x)) deg(f 2 (x)) deg(f t (x)). If deg(f 1 (x)) is lage than ( 1)( (η 1)((d k)ηk 1 η k 1 ( 1)/2 ) η k 1 + η k ( 1)/2 (η 1) 2 ), (22) then C 1 (k,, d, p) is an MDS code fo k. Similaly, C 2 (k,, d, p) is an MDS code fo k, if deg(f 1 (x)) is lage than ( 2 1)( 2 )d 1 ( ( 2 )d 2 2 )d 1 ( 2 )d (23) 2 Poof. It is easy to see that (22) is lage than d k and (23) is lage than. By Theoem 6, 2 we should show that the deteminants of all sub-matices ae invetible in F 2 [x]/(f i (x)) fo i = 1, 2,..., t. If the maximum degee of the non-zeo deteminant is less than deg(f 1 (x)), then the deteminant is invetible in F 2 [x]/(f i (x)). Since any l l, 1 l, sub-matix of (13) is contained in an sub-matix, the maximum degee among all deteminants of l l sub-matices is no lage than that of all deteminants Febuay 28, 2018

19 SUBMITTED PAPER 19 of sub-matices. It is sufficient to calculate the maximum degee of the deteminants of all sub-matices of (13) fo C 1 (k,, d, p). Note that the size of the matix (13) is k, we need to fist choose ows fom the k ows to fom an sub-matix and then calculate the maximum exponent of the deteminant of the sub-matix. The deteminant is computed as the summation (with plus o minus signs) of all possible multiplications of enties that ae in diffeent ows and diffeent columns. Denote the ow index and column index of l-th enty that is involved in computing the deteminant with maximum degee among all the deteminants as i l and j l, espectively, whee l = 1, 2,...,, 1 i 1 < < i k and (j 1, j 2,..., j ) is a pemutation of (1, 2,..., ). Thee exists an intege t fo 1 t such that j t = d k + 1. If the maximum degee of all the deteminants does not contain the enty x (d k)ηk 2 in column d k + 1 ow k 1, i.e., i l k 1 and j l d k + 1 fo all l, then we have that the exponent of the multiplication of enties with ow indices {i 1, i 2,..., i t 1, i t+1,..., i, k 1} and column indices {j 1, j 2,..., j t 1, j t+1,..., j, d k + 1} is lage than the exponent of the deteminant with ow indices i 1, i 2,..., i and columns indices j 1, j 2,..., j, as (d k)η k 2 is the lagest exponent of all enties in (13) and is lage than the exponent of all the enties in column d k + 1 except x (d k)ηk 2. This contadicts to the assumption that the deteminant with ow indices i 1, i 2,..., i and columns indices j 1, j 2,..., j has the maximum degee among all the deteminants. Theefoe, the maximum degee of all the deteminants should contain the enty x (d k)ηk 2 in column d k + 1 and ow k 1. Then, it is educed to find 1 enties that ae in diffeent columns and ows with each othe in the k matix (13) except ow k 1 and column d k+1. By the same agument, we can obtain that the maximum degee of all the deteminants contains the enty x (d k)ηk 2 in column d k+2 and ow 2. Repeating the above pocedue, we can obtain that the maximum degee of all deteminants contains the enties in column d k + 2 l and ow k l fo l = 1, 2,..., ( 1)/2 + 1, and enties in column d k l and ow 1 + l fo l = 1, 2,..., ( 1)/2. It can be computed that the maximum degee is 2((d k)η k 2 + (d k 1)η k η k ( 1)/2 1 ), which is (22). By epeating the same agument fo (14), the maximum degee among all the deteminants is Febuay 28, 2018

20 20 IEEE TRANSACTIONS ON INFORMATION THEORY achieved when the ow indices of the sub-matix ae d 2 4 to d and the maximum degee is as given in (23). By Theoem 7, we can choose p with deg(f 1 (x)) is lage than (22) and (23) to ensue the MDS popety fo C 1 (k,, d, p) and C 2 (k,, d, p) espectively. Although this lowe bound given in Theoem 7 is exponentially inceasing on k, d, and, we will show that the lowe bound can be geatly educed when the paametes ae specified in the following. If 2 is a pimitive element in Z p and τ is a powe of p, h(x) is ieducible and F 2 [x]/(h(x)) is the finite field F 2 (p 1)τ [27]. If d k + 1 = p fo C 1 (k,, d, p) o 2 = p fo C 2(k,, d, p), then τ is a powe of p. Accoding to Theoem 6, the MDS condition is educed to that the deteminant of each sub-matix is non-zeo in F 2 [x]/(h(x)). It can be shown, by compute seach, that C 1 (k, 5, k + 2, 3) is an MDS code fo k = 3, 4,..., 12. If τ is a powe of 2, we have h(x) = 1 + x τ + x 2τ + + x (p 1)τ = (1 + x + + x p 1 ) τ. (24) As 1 + x + + x p 1 is ieducible [31], we can diectly have the next theoem by Theoem 6. Theoem 8. If τ is a powe of 2, and the deteminant of any l l sub-matix of (13) fo l = 1, 2,..., min{k, } (o any sub-matix of (14)) is invetible in F 2 [x]/(1+x+ +x p 1 ), then C 1 (k,, d, p) (o C 2 (k,, d, p)) satisfy the MDS popety. Next, we chaacteize the detailed MDS condition fo codes with some specific paametes. The MDS condition of C 1 (k, 3, d, p) given in [1] is a special case and is summaized in the next theoem. Theoem 9. [1, Theoem 2] Let k 4. If p 2k 1 is a pime such that 2 is a pimitive element in Z p, then the code C 1 (k, 3, d, p) satisfies the MDS popety. Theoem 9 is a sufficient condition, the codes with paametes that do not satisfy the condition in Theoem 9 may also satisfy the MDS popety. Fo example, the code in Fig. 1 is MDS, as the deteminant of each l l sub-matix is not divisible by 1 + x + x 2 + x 3 + x 4 fo l = 1, 2, 3. Next, we conside the MDS condition of C 2 (k, 4, d, p). When = 4, we have τ = 2 k. By Theoem 8, we need to pove that the deteminant of each 4 4 sub-matix of H (k+4) 4 (15) is not divisible by 1 + x + + x p 1. Fo any two positive integes i, j such that i < j, we have x i + x j = x i (1 + x j i ) = x i (1 + x)(1 + x + + x j i 1 ). Febuay 28, 2018

21 SUBMITTED PAPER 21 A polynomial with even numbe of tems can be witten as multiple pais of x i + x j, we thus have that a polynomial with even numbe of tems must have a facto of 1 + x. It is easy to check that any deteminant f(x) of the 4 4 sub-matix has even numbe of tems and f(x) can be witten as f(x) = (1 + x)g(x). Suppose that the deteminant f(x) is not divisible by 1 + x + + x p 1, then g(x) is not divisible by 1 + x + + x p 1 because 1 + x + + x p 1 is ieducible polynomial and is not a multiple of 1 + x. Recall that the polynomial 1 + x p can be factoized as 1 + x p = (1 + x)(1 + x + + x p 1 ), so f(x) is not divisible by 1 + x p. Convesely, if f(x) is not divisible by 1+x p, then we can diectly have that f(x) is not divisible by 1 + x + + x p 1. Theefoe, it is sufficient to show that the deteminant is not divisible by 1 + x p. An uppe bound of p fo which C 2 (k, 4, d, p) is MDS is summaized in the next theoem. Theoem 10. Let p be a pime such that 2 is a pimitive element in Z p. If p is lage than (k 1) 2 k + 17, (25) then C 2 (k, 4, d, p) is MDS fo k 6. Poof. As = 4, we have 2 = 2 that is stictly less than the value in (25). Note that each enty in (15) is a polynomial with at most one non-zeo tem. Fo each 4 4 sub-matix of (15), each enty is also a polynomial with at most one non-zeo tem. By expanding the deteminant of 4 4 sub-matix, the deteminant is a polynomial ove F 2 [x] with at most 24 non-zeo tems and can be witten as x e 1 + x e x et = x e 1 (1 + x e 2 e x et e 1 ), whee t is a positive even numbe with t 24, and e 1 < < e t. If e t e 1 < p 1, then e i e 1 < p 1 fo i = 2,..., t and the deteminant is a non-zeo polynomial in F 2 [x]/(1+x+ +x p 1 ). If e 1+t/2 < p 1, then e i < p 1 fo i = 1, 2,..., 1+t/2 and thee exists at least one i such that e i and e j ae not conguent modulo p fo j = 1,..., i, i + 1,..., t. This is due to the fact that the numbe of e i that can be chosen fom is lage than the numbe of e j fo j = 2 + t/2,..., t, and e j < p 1 fo j = 1, 2,..., i, i + 1,..., 1 + t/2. Hence, the deteminant is not divisible by 1 + x p. Thus, e t e 1 p 1, e 1+t/2 p 1 is the emaining case to be poved. Febuay 28, 2018

22 22 IEEE TRANSACTIONS ON INFORMATION THEORY Let H 1 and H 2 be the sub-matices of H (k+4) 4 given in (15) with the ow indices being 1, 2, 3, k + 2, k + 3, k + 4 and 4 to k + 1 espectively. Hence, we have 1 x x x H 1 (26) 1 1 x 4 x 2 2k 0 0 x 2 x 2k 0 0 x 1 and 1 x 8 x 2k x k 2k 1 x 16 x 2k 1 x (k 1) 2k H 2. (27) x 2k x 8 x 3 2k Fo k 6, it is sufficient to show that the deteminant of the matix consisting of any g ows of (26) and any 4 g ows of (27) is not divisible by 1 + x p fo g = 0, 1, 2, 3, 4. When g = 0, the 4 4 sub-matix with ow indices being i, j, l, m is 1 x 2i+2 x 2k i+1 x (k i+1) 2k 1 x 2j+2 x 2k j+1 x (k j+1) 2k, 1 x 2l+2 x 2k l+1 x (k l+1) 2k 1 x 2m+2 x 2k m+1 x (k m+1) 2k whee 1 i < j < l < m k 2. The deteminant of it is x 2l k l (k l 3 +1)2 k, (28) l 1 l 2 l 3 {i,j,l,m} whee the numbe of non-zeo tem is t. Since the maximum degee and the minimum degee of tems in (28) is espectively 2 m k j+1 + (k i + 1)2 k and 2 i k l+1 + (k m + 1)2 k, e t e 1 = (m i)2 k + 2 m k j i k l+1. Clealy, e t e 1 is maximal when m = k 2, l = k 3, i = 1, j = 2. Theefoe, e t e 1 (k 2)2 k + 2 k 1 24 and, accoding to (25), it is less than p 1. Febuay 28, 2018

23 SUBMITTED PAPER 23 Now conside g = 1. When the 4 4 sub-matix is consisted of the fouth ow of H 1 in (26) and ows 1, 2, 3 of H 2 in (27), i.e., 1 1 x 4 x 2 2k 1 x 8 x 2k x k 2k, 1 x 16 x 2k 1 x (k 1) 2k 1 x 32 x 2k 2 x (k 2) 2k the deteminant of the above matix is x 2 2k +2 k x 2 2k +2 k x 2 2k +2 k x 2 2k +2 k x 2 2k +2 k x 2 2k +2 k x (k 2) 2k x (k 2) 2k x (k 2) 2k +2 k 1 + x (k 2) 2k +2 k x (k 2) 2k +2 k + x (k 1) 2k x (k 2) 2k +2 k x (k 1) 2k x (k 1) 2k +2 k 2 + x (k 1) 2k +2 k x (k 1) 2k +2 k + x k 2k x (k 1) 2k +2 k x k 2k x k 2k +2 k 2 + x k 2k +2 k x k 2k +2 k 1 + x k 2k +2 k If k 8, then the above polynomial has 24 tems and the exponents in the polynomial ae in ascending ode. Note that e 13 = (k 1) 2 k + 16, which is less than p 1. If k = 6, the above polynomial becomes x x x x x x x x x x x x x x x x x x x x 448. The lowe bound of p 1 in (25) is 337 when k = 6. Hence, e 11 = 320 in the above equation is less than p 1. When k = 7, the deteminant becomes x x x x x x x x x x x x x x x x x x x x x x 992. The lowe bound of p 1 in (25) is 785 when k = 7. Theefoe, e 12 = 780 is less than p 1. Afte expanding the deteminant fo all the othe 4 4 sub-matices when g = 1, we can also detemine that e 1+t/2 < p 1. When g = 2, we fist conside the 4 4 sub-matix consisting of the thid ow, the fouth ow of (26) and ows 1, 2 of (27), i.e., 1 x x 4 x 2 2k. 1 x 8 x 2k x k 2k 1 x 16 x 2k 1 x (k 1) 2k Febuay 28, 2018

24 24 IEEE TRANSACTIONS ON INFORMATION THEORY The deteminant of this sub-matix is x 12 + x 20 + x 2k 1 + x 2k x 2k +4 + x 2k x 2 2k +8 + x 2 2k x 2 2k +2 k x 2 2k +2 k x 2 2k +2 k +4 + x 2 2k +2 k x (k 1) 2k + x (k 1) 2k x (k 1) 2k +2 k + x k 2k +8 + x k 2k x k 2k x k 2k +2 k 1 + x k 2k +2 k When k 6, we have t = 20 and e 11 = 3 2 k + 4 is less than p 1. Similaly, we can pove e 1+t/2 < p 1 fo the othe cases when g = 2. When g = 3, 4, it is easy to check that eithe e t e 1 o e 1+t/2 is less than p 1. Fom the above discussion, the deteminants of all 4 4 sub-matices in (15) ae not divisible by 1 + x p. The codes C 2 (k, 4, d, p) thus satisfy MDS popety fo k 6. TABLE I: All values of p that C 2 (k, 4, d, p) ae MDS codes fo = 4 and k = 2, 3,..., 13. k p 11 11, 19 19, 37 19, 29, 61 19, 29, 53, 37, 61, , 107 k p 53, 53, 67, 107, 67,101,131,149, 67,101,131,149, ,101,131, 179, 211, 61, 59, 61, 139, 163, 491, 173, 491, 491, 509, 613, 347, 491, 509, 613, 653, 709, 107, 163, 107, 139, 509, 613, 509, 613, 709, 653, 709, 1741, 2027, 1741, 1949, 1973, 2027, 4093, 491, 163, 491, 709, 1741, 1741, 2027, 4093, 6827, 8171, 6827, 8171, 16363, 16381, , , , 39937, Indeed, the uppe bound of p in Theoem 10 is exponential in k. Howeve, since we ae inteested in small k, we can fist compute each 4 4 deteminant that can be viewed as a polynomial g(x) in F 2 [x]. Then we can check by compute seach whethe polynomial g(x) is a multiple of 1 + x p o not. By using this pocedue, all values of p fo which the codes C 2 (k, 4, d, p) ae MDS codes ae found and summaized in Table I fo k = 2, 3,..., 13. V. WEAK-OPTIMAL REPAIR PROCEDURE FOR ONE COLUMN FAILURE In this section, we demonstate how to ecove the bits stoed in any infomation column fo the fist constuction when an infomation column is failed. We also pesent an efficient epai algoithm fo the second constuction when a single column is failed. We pove that the poposed pocedues ae with asymptotically weak-optimal epai bandwidth. Febuay 28, 2018

25 SUBMITTED PAPER 25 A. Repai Pocedue of the Fist Constuction In this subsection, we assume that the infomation column f is eased, whee 1 f k. We want to ecove the bits s 0,f, s 1,f,..., s (p 1)τ 1,f stoed in the infomation column f by accessing bits fom k 1 othe infomation columns and d k + 1 paity columns. Recall that we can compute the exta bits by (2). Fo notational convenience, we efe the bits of column i as the pτ bits s 0,i, s 1,i,..., s pτ 1,i in this section. as and Fo j = 1, 2,..., and 0 l pτ 1, we define the l-th paity set of the j-th paity column P l,j = {s l (j 1)η 0,1, s l (j 1)η 1,2,..., s l (j 1)η k 2,k 1, s l,k }, fo 1 j d k + 1, P l,j = {s l,1, s l (2η j)η k 2,2,..., s l (2η j)η 1,k 1, s l (2η j)η 0,k}, fo d k + 2 j. Note that all the indices of the elements in P l,j ae taken modulo pτ. It is clea that paity set P l,j consists of infomation bits which ae used to geneate the paity bit s l,k+j. That is, s l,k + k 1 i=1 s l,k+j = s l (j 1)η i 1,i 0 l pτ 1, 1 j d k + 1; s l,1 + k i=2 s l (2η j)η k i,i 0 l pτ 1, d k + 2 j. When we say an infomation bit is epaied by a paity column, it means that we access the paity bits of the paity column, and all the infomation bits, excluding the eased bits, in this paity set. Assume that the infomation column f has failed. When 1 j d k + 1, we can epai the l-th bit in this failed column by the j-th paity column: s l+(j 1)η s l,f = f 1,k+j + s l+(j 1)η f 1,k + k 1 i=1,i f s l+(j 1)η f 1 (j 1)η i 1,i 1 f k 1; s l,k+j + k 1 i=1 s l (j 1)η i 1,i f = k. (29) When d k + 2 j, we can epai the bit s l,f by the j-th paity column: s l,k+j + k i=2 s l,f = s l (2η j)η k i,i f = 1; s l+(2η j)η k f,k+j + s l+(2η j)η k f,1 + k i=2,i f s l+(2η j)η k f (2η j)η k i,i 2 f k. (30) The epai algoithm is stated in Algoithm 1. In the algoithm, we divide the k infomation columns into two pats. The fist pat has k/2 columns and the second pat has k k/2 columns. If a column in the fist pat fails, we epai the failue column by the fist d k + 1 Febuay 28, 2018