Balanced Sparsest Generator Matrices for MDS Codes Son Hoang Dau∗, Wentu Song†, Zheng Dong‡, Chau Yuen§ Singapore University of Technology and Design, Singapore Emails: {∗sonhoang dau, †wentu song, ‡dong zheng, §yuenchau}@sutd.edu.sg Abstract—We show that given n and k, for q sufficiently large, In fact, any error-correcting code can be used in the there always exists an [n, k]q MDS code that has a generator aforementioned scheme for sensor networks. We choose to G matrix satisfying the following two conditions: study MDS codes first because their structure, especially their G − (C1) Sparsest: each row of has Hamming weight n k + 1; weight distribution, is well studied (see, for instance [1, Ch. (C2) Balanced: Hamming weights of the columns of G differ from each other by at most one. 11]). Moreover, they have optimal error-correcting capability, given the length and the dimension. We prove that over a I. INTRODUCTION sufficiently large field, there always exists an MDS code that We study the existence and provide a construction of a has a balanced and sparsest generator matrix, which is ideally sparsest and balanced generator matrix of Maximum Distance suitable for the above encoding scheme for sensor networks. Separable (MDS) codes. A generator matrix is the sparsest Necessary notations and definitions are provided in Sec- if it contains the least number of nonzero entries among all tion II. We state and prove our main result in Section III. generator matrices of the same MDS code. A generator matrix II. PRELIMINARIES is balanced if every column contains approximately the same F number of nonzero entries. More specifically, we require that We denote by q the finite field with q elements. Let the number of nonzero entries in each column differs from [n] denote the set {1, 2,...,n}. The support of a vector u Fn u each other by at most one. = (u1,...,un) ∈ q is defined by supp( ) = {i ∈ [n]: Apart from being of theoretical interest, our study on ui 6= 0}. The (Hamming) weight of u is |supp(u)|. We can balanced sparsest generator matrices for MDS codes was also define weight and support of a row or a column of a motivated by its application in error correction for sensor matrix over some finite field, by regarding them as vectors networks. Suppose n sensors, S1,..., Sn, collectively measure over that field. Apart from Hamming weight, we also use k conditions x1,...,xk, such as temperature, pressure, light other standard notions from coding theory such as minimum intensity, etc. Let x = (x1,...,xk), where xi ∈ Fq for each distance, linear [n, k]q and [n,k,d]q codes, MDS codes, and i =1,...,k (Fq is a finite field of q elements). These sensors generator matrices (for instance, see [1]). G Fk×n G transmit the information they collected to a base station, which For a matrix = (gi,j ) ∈ q , the support matrix of , is a data collector. Furthermore, each sensor performs some denoted supp(G),isa k×n binary matrix M = (mi,j ) where encoding on the information it has, before transmitting the mi,j = 0 if gi,j = 0 and mi,j = 1 if gi,j 6= 0. Let M = information back to the base station in the following way. Let (mi,j ) be a k × n binary matrix. We denote by var(M) = M G be an k×n generator matrix of an [n,k,d]q error-correcting (vi,j ) the matrix obtained from by replacing every nonzero code. Sensor Si transmits the scalar product of x and column entry mi,j =1 by ξi,j , where ξi,j ’s are indeterminates. More arXiv:1301.5108v1 [cs.IT] 22 Jan 2013 i of G to the base station. It is well known in classical formally, vi,j = 0 if mi,j = 0 and vi,j = ξi,j if mi,j = 1. coding theory that this coding scheme allows the base station We also denote by gr(M) the bipartite graph G = (V , E ) x d−1 defined as follows. The vertex set V can be partitioned into to retrieve when at most ⌊ 2 ⌋ sensors transmit wrong information. Moreover, the base station can also identify two parts, namely, the left part L = {ℓ1,...,ℓk}, and the right the malfunctioned sensors. For each sensor Si, only those part R = {r1,...,rn}. The edge set is conditions corresponding to nonzero entries of column i of G E = (ℓi, rj ): i ∈ [k], j ∈ [n], mi,j 6=0 . are involved into encoding. So it is sufficient for Si to measure only such conditions. Thus, if G is sparse then in average, For any k × n matrix N, we define f(N) = P det(P ), n P each sensor only needs to measure a few among k conditions where the product is taken over all k submatrices of order in order to achieve the desired error correction capability. On k of N. Q top of that, if columns of G have approximately the same III. MAIN RESULT number of nonzero entries then the sensors are required to measure approximately the same number of conditions. This A sparsest generator matrix of an [n, k]q MDS code would balance guarantees an even distribution of workload among have precisely n−k+1 nonzero entries in every row. Moreover, k(n−k+1) sensors, which is an important criterion for sensor networks if it is balanced, then each column contains either ⌊ n ⌋ k(n−k+1) where energy saving is a critical issue. or ⌈ n ⌉ nonzero entries. Hereafter, we often use Ri, i ∈ [k], and Cj , j ∈ [n], to denote the supports of row i and ∪i∈I Ri ≥ n − k + |I|, for every subset ∅ 6= I ⊆ [k]. (2) column j, respectively, of a k ×n binary matrix M. Note that Proof: Suppose that (1) holds and that there exists a Ri ⊆ [n] and Cj ⊆ [k]. nonempty set I ⊆ [k] satisfying Lemma 1. Let M = (mi,j ) be a k×n binary matrix. Suppose ∪i∈I Ri ≤ n − k + |I|− 1. (3) that each row of M has weight n − k +1. Then M is the We aim to obtain a contradiction. The condition (3) is equiv- support matrix of a generator matrix of some [n, k]q MDS n−1 alent to code over a sufficiently large field Fq (q> ) if and only k−1 ∩i I Ri ≥ k − |I| +1. (4) if f(var(M)) 6≡ 0. ∈ M G G Hence there exists a set J of k − |I| +1 columns of M that Proof: Suppose = supp( ), where = (gi,j ) satisfies ∩j∈J Cj ≥ |I|. Equivalently we have is a generator matrix of some [n, k]q MDS code. Due to a well-known property of MDS codes (see [1, p. 319]), ∪j∈J C j ≤ k − |I| < k − |I| +1= |J|. (5) every submatrix of order k of G has nonzero determinant. We obtain a contradiction between (1) and (5). The “only if” Therefore, f(G) 6= 0. Note that f(var(M)) can be regarded direction can be proved in a similar manner. as a multivariable polynomial in Fq[...,ξi,j ,...]. Moreover, since M = supp(G), we deduce that f(G) can be obtained Lemma 5. Let M = (mi,j ) be a k×n binary matrix. Suppose from f(var(M)) by substituting ξi,j by gi,j for all i, j where that each row of M has weight n − k +1. Then M is the gi,j 6=0. As f(G) 6=0, we conclude that f(var(M)) 6≡ 0. support matrix of a generator matrix of some [n, k]q MDS M F n−1 Now suppose that f(var( )) 6≡ 0. Note that each column code over a sufficiently large field q (q> k−1 ) if and only of var(M) belongs to precisely n−1 submatrices of order k if (2) holds. k−1 of var(M). Hence the exponent of each ξi,j in f(var(M)) n−1 M Proof: The proof follows from Lemma 1-4. is at most k−1 . Since f(var( )) 6≡ 0, by [2, Lemma n−1 F We present below our main result. 4], if q > k−1 then there exist gi,j ∈ q (for i, j M where mi,j = 1) so that f(var( ))(...,gi,j,...) 6= 0. Let Theorem 6 (Main Theorem). Suppose 1 ≤ k ≤ n and q > G n−1 = (gi,j ) (for i, j where mi,j = 0 we set gi,j = 0). Since . Then there always exists an [n, k] MDS code that has G M k−1 q f( ) = f(var( ))(...,gi,j,...) 6= 0, again by [1, p. 319], a generator matrix G satisfying the following two conditions. G we deduce that is a generator matrix of an [n, k]q MDS (C1) Sparsest: each row of G has weight n − k +1. G code. Therefore, each row of has weight at least n − k +1, (C2) Balanced: column weights of G differ from each other due to the Singleton Bound (see [1, p. 33]). Since each row by at most one. of M also has weight n − k +1, we deduce that gi,j 6= 0 whenever mi,j =1. Therefore, M = supp(G). By Lemma 5, to prove Theorem 6, we need to show that there always exists a k × n binary matrix M satisfying the M Lemma 2. Let = (mi,j ) be a k × n binary matrix. following properties Then var M if and only if every bipartite subgraph f( ( )) 6≡ 0 (P1) each row of M has weight n − k +1, induced by the k left-vertices and some k right-vertices in (P2) column weights of M differ from each other by at most M gr( ) has a perfect matching.