Convolutional Codes with Maximum Column Sum Rank for Network Streaming

Transcription

1 1 Convolutional Codes with Maximum Column Sum Rank for Network Streaming Rafid Mahmood, Ahmed Badr, and Ashish Khisti School of Electrical and Computer Engineering University of Toronto Toronto, ON, M5S G4, Canada {rmahmood, abadr, Abstract The column Hamming distance of a convolutional code determines the error correction capability when streaming over a class of packet erasure channels. We show that the column sum rank parallels column Hamming distance when streaming over a network with link failures. We prove rank analogues of several known column Hamming distance properties and introduce a new family of convolutional codes that maximize the column sum rank up to the code memory. Our construction involves finding a class of super-regular matrices that preserve this property after multiplication with non-singular block diagonal matrices in the ground field. Index Terms Column distance, maximum rank distance (MRD) codes, network coding, super-regular matrices, maximum-distance profile (MDP) codes. I. INTRODUCTION In streaming communication, source packets arrive sequentially at the transmitter and are only useful for playback by the receiver in the same order. Erased packets must be recovered within a given maximum delay or they are considered permanently lost. Streaming codes are designed to recover packets within deadlines and have been studied in depth for singlelink communication [1] []. In [], [4], it was shown that the column Hamming distance is an appropriate metric for channels with a bounded number of erasures in a window. The column Hamming distance of a code determines the maximum number of erasures that can occur in any window of the stream for decoding to reman successful [5]. If there are fewer erasures than the distance in every window, each source symbol is recovered within the deadline. Convolutional codes such as m-mds codes are known to achieve the maximum column Hamming distance. These codes are constructed from block Toeplitz super-regular matrices [4] [6]. Several formulations for such matrices have been proposed in prior work [6], [7]. In network communication, a transmitter streams to several users through intermediate nodes. With linear network codes, the problem of decoding is reduced to inverting the transfer matrix between transmitted and received packets [8], [9]. Unreliable links in the network however, can reduce the rank of the channel, making this infeasible. The topic of correction from rank deficiencies has traditionally been treated with the assumption that the receiver has no knowledge of the transfer matrix, as it was originally motivated by non-coherent multiple-antenna channels [10], [11]. A family of end-to-end matrix codes known as subspace codes can decode over a rank deficient channel and are constructed using rank metric codes. Gabidulin codes are the most well-known rank metric codes and often act as constituents for subspace codes [12], [1]. The rank distance of the constituent code determines the maximum loss in the network from which successful decoding is possible. Subspace coding is performed over a single network use, which is transmitted and received instantaneously. It is also possible to code over multiple independent uses, with what is known as multi-shot coding [14]. A symbol lost at one time instance may then be recovered with a delay after subsequent shots. The sum rank of a constituent code gives the maximum number of link failures that can be corrected within the window of transmission by a multi-shot subspace code. As an alternative to block codes, convolutional rank metric codes were introduced for multi-shot communication in [15]. In this work, we assume the receiver has complete knowledge of the channel. Our motivation is internet streaming over the application layer. The transfer matrix is a function of a linear network code, which is transmitted in real time as header bits of the channel packets [9]. Using a coherent channel implies that rather than subspace codes, rank metric codes are now sufficient. The rank metric possesses several parallel properties to the Hamming metric. We introduce the column sum rank as a counterpart to column Hamming distance and define a class of convolutional codes that achieve the maximum column sum rank up to the memory. These codes are rank metric analogues of m-mds codes used in single-link streaming [5]. Interestingly, there has been little prior work on rank metric convolutional codes. To our knowledge, the only previously studied construction appears in [15], where the authors consider the active column sum rank. Their approach differs from the present work both in the code constructions and the distance metric. This paper is outlined as follows. We review fundamentals of extension fields, rank metric block codes, m-mds codes, and super-regular matrices in Section II. The network streaming problem is introduced in Section III. In Section IV, we define and

2 2 derive several properties for the column sum rank. We prove equivalence between the capability of a convolutional code to recover from rank deficiencies with delay and its column sum rank. Codes maximizing this metric are referred to as Maximum Sum Rank (MSR) codes. Our construction in Section V first presents a new class of matrices that preserve super-regularity after multiplication with block diagonal matrices in the ground field. Such super-regular matrices are used to construct the generator for a MSR code. We conclude with code examples and a discussion on the necessary field size. A. Extension Fields II. BACKGROUND INFORMATION For a prime q, let F q M be an extension field and F q [X] be a polynomial ring of F q. A primitive element α F q M is one whose consecutive powers generate the entire field [16]. The minimal polynomial p(x) F q [X] of a primitive α is the monic polynomial of degree M, for which p(α) = 0. A minimal polynomial is irreducible by definition and if any f(x) F q [X] also has α as a root, then p(x) f(x) [17]. The extension field is isomorphic to the vector space F M q. Let α 0,..., α M 1 F q M be a basis for this vector space. A normal basis is one where each α i = α qi for some α F q M, commonly referred to as a normal element [17]. We will denote α [i] = α qi to be the i-th Frobenius power of α. Every element in F q M can be represented as a linearized polynomial f(x) F q [X] evaluated at a normal α. A polynomial is referred to as linearized when monomial terms only have Frobenius powers. The coefficients of a linearized polynomial map to the entries of a vector f = (f 0,..., f M 1 ) T F M q, giving us an easy isomorphism between the extension field and vector space. f(α) = M 1 f i α [i] f = (f 0,..., f M 1 ) T (1) The q-degree (denoted deg q ) determines the largest Frobenius power in the polynomial. In this paper, we will frequently treat elements in the extension field as linearized polynomials evaluated at α. For every finite extension of a prime field, there exists at least one element that is both normal and primitive [18]. We refer to such an element as a primitive normal. B. Rank Metric Codes A vector space over the extension field is isomorphic to a matrix space over the ground field. We use the bijection φ : F n q M Fq M n to transform vectors of linearized polynomials to matrices via (1). The column rank of a vector x in the extension field refers to the rank of φ(x). The rank distance between two vectors x, ˆx F n q was defined in [12], where it was also revealed M to be a metric. d R (x, ˆx) = rank (φ(x) φ(ˆx)) For a linear block code C[n, k] over F q M, the minimum rank is defined similarly to minimum Hamming distance, and must satisfy a Singleton-like bound given by d R (C) min { } 1, M n (n k)+1 [12]. Maximum Rank Distance (MRD) codes achieve this bound with equality. They also have the following property. Theorem 1 (Gabidulin, [12]). Let G F k n q M matrix A F n k q satisfies rank GA = k. be the generator matrix of a MRD code. The product of G with any full-rank A complementary theorem was proven in [12] for the parity-check matrix of a MRD code. We use the equivalent property for the generator matrix, which arises from the fact that the dual of a MRD code is also a MRD code [12]. We will assume M n throughout this paper; a MRD code is then also MDS. Gabidulin codes are the most well known family of such block codes [12]. To construct a Gabidulin code, let g 0,..., g F q M be a set of linearly independent elements over F q. In practice, g i are often drawn from a subset of a normal basis. The generator matrix for a Gabidulin code C[n, k] is given as follows. G = g 0 g 1... g g [1] 0 g [1] 1... g [1] g [k 1] 0 g [k 1] 1... g [k 1]

3 C. m-mds Codes Let C[n, k, m] be a linear time-invariant convolutional code with memory m over F q M. For a source s [0,j] = (s 0,..., s j ), the channel packets x [0,j] = s [0,j] G EX j are found using the extended form generator matrix given below G 0 G 1... G j G EX G 0... G j 1 j =.... (2). G 0 where G j F k n and G q M j = 0 for j > m [19]. In this paper, we assume that G 0 has full row rank. This is necessary in order to guarantee that G EX j also has full row rank. The Hamming weight of x [0,j] is measured by summing the Hamming weight of each x t. The j-th column distance of a code determines the minimum Hamming weight amongst channel packets whose initial source s 0 is non-zero [5]. d H (j, C) = min wt H(x [0,j] ) x [0,j] C,s 0 0 We will simplify the notation when C is obvious. The column distance has several properties that were treated in [4], [5]. We will be proving rank metric analogues of two properties. 1) Assuming all prior packets have been recovered, if there are at most d H (j) 1 symbol erasures when a receiver observes x [0,j], then the packet s t is recoverable by time t + j. 2) The j-th column distance is upper bounded d H (j) (n k)(j + 1) + 1. If d H (j) achieves this bound, then d H (i) does as well for all i j. A family of codes known as m-mds codes achieve the upper bound for d H (m). The extended generator for a m-mds code is constructed by carefully taking a sub-matrix of k(m + 1) rows from a block Toeplitz super-regular matrix [5]. We will next define super-regularity and give a previous construction of such a matrix with a block Toeplitz structure. D. Super-regular Matrices In this paper, the rows and columns of a matrix are indexed starting from the 0-th. For r N, let σ be a permutation of the set {0,..., r 1} and let S r denote the symmetric group, or the set of all possible permutations. The sign function sgn σ indicates 1 or 1 respectively when σ is constructed from an even or odd number of transpositions. The Leibniz formula calculates the determinant of a matrix D = [D i,j ] of order r. det D = r 1 sgn σ D i,σ(i) () σ S r The product of entries for a single σ is referred to as term in the summation. A trivial determinant is one where each term in the Leibniz formula is equal to 0. A super-regular matrix is defined as a matrix for which every square sub-matrix with a non-trivial determinant is non-singular. From (2), the generator for a convolutional code is a block Toeplitz matrix and so we focus on super-regular matrices with the same structure. In [5], the authors provided a construction for a Toeplitz super-regular matrix that exists for every prime field F q. A second block Toeplitz matrix, which we outline below, was proposed in [7]. For n, m N, let M = q n(m+2) 1. Let α F q M be a primitive element and a root of the minimal polynomial p(x). For j = 0,..., m, we define T j F n n below. q M α [nj] α [nj+1]... α [n(j+1) 1] α [nj+1] α [nj+2]... α [n(j+1)] T j = (4) α [n(j+1) 1] α [n(j+1)]... α [n(j+2) 2] We then construct the block Toeplitz matrix T = [ T i,j ]. T 0 T 0 T 1 T = T 0... T m 1 T m Each non-zero entry is a linearized monomial Ti,j (X) evaluated at α. The q-degrees of these monomials increase as one (5)

4 4 moves further down and to the right inside T. deg q Ti,j(X) + 1 deg q Ti,j (X), if i < i deg q Ti,j (X) + 1 deg q Ti,j (X), if j < j We use the variable X when discussing properties of polynomials and evaluate at α specifically when calculating the determinant of a matrix. To show that T is super-regular, let D F r r be any square sub-matrix. If D has a non-trivial determinant, q M then det D must be non-zero. Each non-trivial term in the Leibniz formula is now a product of linearized monomials, which we denote D σ (α) = r 1 D i,σ(i)(α). The determinant det D becomes a polynomial D(α) = σ S r sgn σd σ (α). We bound the degree of D(X) using (6), which is preserved in all sub-matrices of T. Lemma 1 (Almeida et al, [7]). For T defined in (5), let D be any sub-matrix with a non-trivial determinant. Let D(X) be the polynomial which evaluates its determinant. The degree of D(X) is bounded as follows. 1 deg D(X) < q n(m+2) 1 In [7], the matrices in (4) contained entries with double exponents α 2i rather than Frobenius powers α [i]. As a result, the Lemma 1 included in the prior work requires q = 2. A direct extension reveals that the Frobenius power can be used for (4) and the corresponding q in the bounds. For completeness, we prove the extension for the upper bound in Appendix A. The lower bound is derived by algorithmically finding a unique σ = arg max σ deg D σ (X), which generates the highest degree monomial term. The algorithm does not change in the extension, so we refer the reader to [7] for details. The polynomials D(X) and D σ (X) share the same degree and therefore, D(X) is not the zero polynomial. Given the upper bound, deg D(X) < deg p(x) and consequently, p(x) D(X) [17]. As a result, α is not a root of D(X) and any D with a non-trivial determinant is also non-singular. (6a) (6b) III. NETWORK STREAMING PROBLEM The problem is defined in three steps: encoding, the network model, and decoding. A. Encoder At each time instance t 0, a source packet s t F k q arrives causally at a transmitter node. A channel packet x M t F n q is M a function of the previous source packets, i.e. x t = γ t (s 0,..., s t ). We consider the class of linear time invariant encoders for our scenario and will use convolutional codes with the structure in (2). A rate R = k n encoder with memory m generates the channel packet in the following manner. m x t = s t i G i B. Network Model Channel packets are instantaneously transmitted and received over a network. A receiver sequentially observes y t = x t A t, where A t F n n q is the channel matrix at time t [14], [20]. A single transmission is referred to as a network shot. Each shot is independent of all others. Communication over a window [t, t + W 1] of W shots is therefore given by y [t,t+w 1] = x [t,t+w 1] A [t,t+w 1], where A [t,t+w 1] = diag (A t,..., A t+w 1 ) is the block diagonal channel transfer matrix [14]. Let ρ t = rank (A t ). Clearly, t+w 1 i=t ρ i = rank (A [t,t+w 1] ). When the network is operating perfectly, ρ t = n, but unreliable connections cause a rank deficient channel matrix. In [21], it is shown that each failing link can reduce the rank of A t by at most 1. To facilitate the extension from the point-to-point streaming problem, we introduce a network variation of the sliding window model by using rank deficiencies in place of symbol erasures. Definition 1. Consider a network where the receiver observes y t = x t A t, with rank (A t ) = ρ t. A Rank Deficient Sliding Window Channel C R (N, W ) has the property that in any sliding window of length W, the block diagonal channel rank does not decrease by more than N, i.e. t+w 1 i=t ρ i nw N for all t 0. In analysis, we disregard the linearly dependent columns of A t and the associated received symbols. At each time instance, the receiver effectively observes y t = x ta t, where A t Fq n ρt is the reduced channel matrix containing only the linearly independent columns. C. Decoder Let T be the maximum delay permissible by the decoder. A packet received at time t must be recovered by t + T, i.e. there should exist a sequence of functions that solves ŝ t = η t (y 0,..., y t+t ). If ŝ t = s t, then the source packet is perfectly recovered by the deadline. Otherwise, the source packet is declared lost. In the linear model, the decoding problem is reduced to inverting

5 5 the transfer matrix between s t and y 0,..., y t+t. A linear code C is declared feasible over C R (N, W ) if there exists an encoder and decoder for the code, which completely recovers every source packet with delay T. We assume that T = W 1 for the remainder of this paper and will design convolutional codes with memory m = T in application. Coding for channels where T W 1 will be briefly addressed near the end. Our codes aim to maximize the sum rank metric, which is introduced in the next section. IV. MAXIMUM SUM RANK CODES The sum rank distance between channel packets x [0,j] and ˆx [0,j] is defined as the sum of the rank distance between each x t and ˆx t. Using the bijection φ( ) from (1) to map vectors in the extension field to matrices in the ground field, we introduce the j-th column sum rank of a code as an analogue of column Hamming distance. d R (j) = min x [0,j] C,s 0 0 t=0 j rank (φ(x t )) The authors in [15] proposed an alternative metric for their code construction: the active column sum rank. Using the state trellis, active column rank is the minimum sum rank amongst channel packets that exit the zero state at time 0 and do not re-enter the zero state between time 1 and j 1. In contrast, the column sum rank includes packets that return back to the zero state before time j. Our metric, rather than the active rank, is necessary and sufficient for network streaming over a window of length W. Theorem 2. Let C be a convolutional code used to stream over the window [0, W 1]. For t = 0,..., W 1, let A t F n ρt q be full column rank matrices. In addition, let A = diag (A 0,..., A W 1) be the channel matrix. If d R (W 1) > nw rank (A ), then s 0 is recoverable by time W 1. Conversely, if d R (W 1) nw rank (A ), then there exists at least one s 0 which cannot be recovered within the window. Proof: Consider two source packets s = (s 0,..., s j ) and ŝ = (ŝ 0,..., ŝ j ), where s 0 ŝ 0. Suppose they respectively generate channel packets x and ˆx, where both xa = y and ˆxA = y. Then (x ˆx)A = 0, where the difference x ˆx is a hypothetical channel packet whose sum rank is at least d R (W 1). However, (x t ˆx t )A t = 0 implies rank (φ(x t ) φ(ˆx t )) n rank (A t ) for t W 1. We arrive at the following contradiction on the sum rank of the channel packet by summing each of the inequalities. W 1 t=0 rank (φ(x t ) φ(ˆx t )) nw rank A < d R (W 1) For the converse, let s = (s 0,..., s W 1 ), where s 0 0, generate x with sum rank equal to d R (W 1). For t = 0,..., W 1, let ρ t = n rank (φ(x t )). Then there exist A t F n ρt q such that x t A t = 0. From summing all of the ρ t, the matrix A = diag (A 0,..., A W 1) must have rank equal to nw d R (W 1) and s is indistinguishable from the all-zero source over this channel. A time-invariant encoder implies this theorem is sufficient to show that all source packets are recoverable over the sliding window channel. Assuming all prior packets are decoded, we recover s t using the window [t, t + W 1]. The contributions of s 0,..., s t 1 can simply be negated from the received packets. Theorem 2 is a rank metric analogue to Property 1 in Section II-C, which guarantees the necessity and sufficiency of column Hamming distance in single-link streaming [4]. We have replaced symbol erasures with rank deficiency and the column Hamming distance with column sum rank. We next parallel Property 2 in Section II-C. Firstly, d R (j) (n k)(j + 1) + 1. The sum rank of a packet cannot exceed its Hamming weight, meaning that the upper bound on column Hamming distance is inherited by the column sum rank. Furthermore, if the j-th column sum rank is maximized, all prior ones are as well. Lemma 2. If d R (j) = (n k)(j + 1) + 1, then d R (i) = (n k)(i + 1) + 1 for all i j. Proof: It suffices to prove for i = j 1. Let C be a code for which d R (j 1) is at most (n k)j, but d R (j) achieves the maximum. Consider the source packets s [0,j 1] which generate x [0,j 1] = s [0,j 1] G EX j 1 with sum rank equal to d R (j 1). We next determine s j to obtain x j = j 1 t=0 s tg j t + s j G 0. The first summation produces a vector whose Hamming weight is at most n. Because rank (G 0 ) = k, s j can be selected specifically in order to negate up to k non-zero entries of the first summation. This implies that wt H (x j ) n k and consequently, rank (x j ) n k. Therefore, the sum rank of x [0,j] is upper bounded by d R (j 1) + n k (n k)(j + 1), which is a contradiction. Codes achieving maximum d R (j) up to the memory are referred to as Maximum Sum Rank (MSR) codes. They directly parallel m-mds convolutional codes, which maximize the m-th column Hamming distance [5]. By Theorem 2, a MSR code with memory W 1 can recover each source packet with delay W 1, provided that the rank of the channel matrix is greater than kw in every sliding window. We will show the existence of MSR codes in the next section. The following theorem

6 6 provides further insight on decoding with delay j. The theorem also serves as an extension of Theorem 1 to convolutional codes transmitted over independent network uses. Theorem. For t = 0,..., j, let A t F n ρt q be a set of full column rank matrices. Let ρ t satisfy the following condition t ρ i k(t + 1) (7) for all t j and with equality for t = j. We construct A = diag (A 0,..., A j ). The following statements are equivalent for any convolutional code. 1) d R (j) = (n k)(j + 1) + 1 2) G c ja is non-singular. Proof: We first prove 1 2. Consider a code where d R (j) = (n k)(j + 1) + 1 and suppose there exists an A satisfying (7), for which G c ja is singular. Then there exists channel packets x [0,j], where x [0,j] A = 0. The first packet x 0 is not necessarily non-zero, meaning there is no guarantee on the sum rank of x [0,j]. We let l = arg min t x t 0 and consider the vector x [l,j], whose sum rank is at least d R (l j). Because x t A t = 0 for t = l,..., j, we bound rank (φ(x t )) n ρ t for these instances. The sum rank of x [l,j] is bounded below. j rank (φ(x t )) n(j l + 1) t=l j t=l (n k)(j l + 1) The second line follows from j t=l ρ t k(j l + 1), which can be derived when (7) is met with equality for t = j. However, d R (j l) = (n k)(j l + 1) + 1 due to Lemma 2. The sum rank of x [l,j] is less than this, which is a contradiction. A is singular. Let m = arg min i d R (i) (n k)(i + 1) be the first instance where the column sum rank is not maximum. Consider x [0,m] whose sum rank is equal to d R (m). We assume m > 0 and will discuss the case for m = 0 at the end. The sum rank of x [0,m 1] is equal to (n k)m k 1 for some k 1 0 and the sum rank of x [0,m] is equal to (n k)(m + 1) k 2 for some k 2 0. We prove 2 1 by using a code with d R (j) (n k)(j + 1) and constructing an A for which G EX j We will show that there exist a set of matrices A t F n ρt q we will construct A [0,m] to have rank (m + 1)k. ρ t satisfying both (7) and x t A t = 0 for t = 0,..., m. In addition, Let ρ t = n rank (φ(x t )) for t = 0,..., m 1. Clearly, there exist A t for which x t A t = 0. Summing ρ t confirms that these matrices satisfy (7) for t m 1. t t ρ i = n(t + 1) rank (φ(x i )) k(t + 1) 1 The second line follows from the fact that the sum rank of x [0,t] is at least (n k)(t + 1) + 1. In fact, the exact summation for t = m 1 is known to be m 1 ρ i = mk 1 k 1. It remains to choose an appropriate ρ m. The rank of our x m is n k 1 k 1 k 2, so there should exist an A m with rank ρ m = k k 1 that satisfies x m A m = 0. To confirm this is possible, we will check whether ρ m n. The sum rank of our x [0,m 1] cannot exceed the sum rank of x [0,m], which is bounded by (n k)(m + 1). Using the sum rank of x [0,m 1], we bound k 1 n k 1. Inputting the inequality into the equation for ρ m guarantees that ρ m n and that A m possesses full column rank. Alternatively if m = 0, then rank (φ(x 0 )) n k, and we simply use an A 0 with rank ρ 0 = k. The remaining A m+1,..., A j can be any full-rank n k matrices, thus satisfying (7) for all t j. The product G c ja is given below. ( ) ( G EX m X A ) ( [0,m] G EX Y A = m A [0,m] XA ) [m+1,j] [m+1,j] YA [m+1,j] X and Y denote the remaining blocks that comprise G EX j. Note that G EX m A [0,m] is a square singular matrix. Therefore, det G EX j A is also zero. The constraints in (7) ensure that [0, j] is a feasible decoding window. For every t j, if t ρ i < k(t + 1), then the decoder does not possess sufficient information to decode. If the bound is achieved for some t, then we can invert G EX t A [0,t] in order to recover s 0 with delay t. In the next section, we will construct an extended generator matrix. This theorem will be useful afterwards to verify that the generator produces a MSR code.

7 7 A. Preservation of Super-regularity V. CODE CONSTRUCTION Because MSR codes are also m-mds, it is natural to assume that their generators can be constructed using super-regular matrices. MRD block codes however, have an additional property over MDS block codes, given by Theorem 1. Theorem extends this for the convolutional counterparts. In this section, we connect Theorem to super-regularity. Consider the case when the element α generating the matrix in (4) is primitive normal. T remains super-regular, but now each of the blocks T j resemble generator matrices for rate R = 1 Gabidulin codes. Let A t F n n q be a non-singular matrix in the ground field and construct A = diag (A t,..., A t ) from m + 1 copies. The product F = TA has the following structure. F 0 F 0 F 1 F = F 0... F m 1 F m We have let F j = T j A t for j = 0,..., m. It can be shown that each F j has the structure f [nj] 0 f [nj] 1... f [nj] f [nj+1] F j = 0 f [nj+1] 1... f [nj+1] f [n(j+1) 1] 0 f [n(j+1) 1] 1... f [n(j+1) 1] where f = (f 0,..., f ) = (α [0],..., α [] )A t is a linearly independent set over F q. [12]. In addition, each element f i is a linearized polynomial f i (X) evaluated at α, with coefficients from A t = [A k,i ]. We can then write every non-zero entry of F as follows. f [j] i = A k,i α [k+j] (9) k=0 The polynomials on any given column have the same set of coefficients, but the degrees of each monomial term increase as one moves downwards along F, i.e. f [j] i (X) = f i (X [j] ). The degree of each linearized polynomial is bounded below. j deg q f [j] i (X) n 1 + j (10) These bounds depend primarily on the Frobenius power j. Consequently, the polynomial entries on any fixed row of F i all share the same bound. Although A was constructed by repeating a single A t, similar results are reached when letting A = diag (A 0,..., A m ) be constructed from different matrices. The non-zero entries remain linearized polynomials with the same bounds, but each column block of F is now generated using a different A t. Consequently, there are a different set of linearly independent f 0,..., f for different column blocks. Overall, the structure of F remains similar to T, but with polynomials of varying degrees rather than monomials with fixed degrees. Consequently, we propose a weakened version of (6). Lemma. For t = 0,..., m, let A t F n n q be non-singular matrices. We construct A = diag (A 0,..., A m ). Let T be be the super-regular matrix in (5). The product F = TA satisfies the following. 1) deg q F i,j(x) + 1 deg q F i,j (X), if i < i 2) deg q F i,j (X) + 1 deg q F i,j (X), if F i,j is an entry of a different column block to the left of that from which F i,j is drawn. Statement 1 in Lemma () is identical to (6a) and follows directly from (10). Statement 2 is a weakened variation of (6b) that only holds when the two entries are drawn from different column blocks of F. The above lemma is used to show that F is also a super-regular matrix when α is primitive normal. Theorem 4. For t = 0,..., m, let A t F n n q be any non-singular matrices. We construct A = diag (A 0,..., A m ). Let T F q M be the super-regular matrix in (5). If M q n(m+2) 1 and α is primitive normal, then F = TA is super-regular. Proof: We show that F is super-regular in three parts, moving from specific to increasingly general cases. The problem in subsequent cases can be converted to the first case, which we will prove super-regular. Furthermore, we assume without loss of generality that A 0 = A 1 = = A m = A t. This simplifies notation and allows us to freely use the previous polynomial structures and bounds. Case 1: Consider when the degrees of the polynomials f i (X) are strictly increasing, i.e. deg f 0 (X) < < deg f (X). (8)

8 j X X X X X X X X X X X X 0 X X 0 X X j + n 1 X X X 0 0 X Fig. 1: The result of the Gaussian elimination applied on a single row of D i. Non-zero entries are denoted by X. The inverse isomorphism of the result generates a vector of polynomials with increasing degrees. Because the polynomials are linearized, the degrees are forced to be the following. deg f 0 (X) = 1, deg f 1 (X) = q,..., deg f (X) = q (11) In this case, F fully satisfies (6), as opposed to only Lemma. Every polynomial entry in F has the same degree as the monomial at the same position in T. If we let D be a sub-matrix of F, then we can construct D from T using the same row and column indices. The monomial entries of D share the same degrees as the polynomials in the corresponding positions of D. The Leibniz formula for the determinant yields deg D(X) = deg D(X). Lemma 1 holds for D(X) and it follows that D is non-singular. Therefore, F is super-regular. It is clear that F or its sub-matrices need only fully satisfy (6) in order for Lemma 1 to apply. In the remaining cases, we will modify the matrices to return to Case 1. Case 2: Consider a more general scenario where the degrees of f i (X) are different but not necessarily in increasing order. As a result, (6b) does not always hold but Lemma permits (6a) to be true. Permuting the columns of F allows us to re-arrange each column block to produce a matrix ˆF that satisfies (11). The set of sub-matrices of F and ˆF are identical up to column permutations and therefore, every sub-matrix of F is non-singular. Case : We now consider the scenario with no restrictions on the degrees of f i (X). Naturally, there may exist multiple polynomials sharing the same degree. Column permutations alone cannot transform F to satisfy (11). As a result, we will show how by using elementary column operations, each sub-matrix of F with a non-trivial determinant can be transformed to one with an increasing degree distribution. The matrix is then interpreted as a sub-matrix of a super-regular ˆF that satisfies (11). Let D be a sub-matrix of F with a non-trivial determinant. Lemma is clearly preserved. The authors in [7] revealed that D takes the following shape. D = O 1 O 2.. D 1 O h... D h Each O i is a zero matrix and each D i is a matrix containing non-zero polynomials drawn from a single column block of F. Let k i be the number of columns in each D i. The polynomials on each row of D i share the same bounds on degrees and are linearly independent amongst themselves. We apply elementary column operations on each D i separately in order to ensure that D satisfies (6b). Using the isomorphism in (1), each row of D i maps to a matrix in F M ki q. This matrix may possess non-zero entries only between the j-th to +jth rows. Because the polynomials are linearly independent, the matrix has full column rank. Using Gaussian elimination, we transform it to reduced column echelon form. An example of the desired structure is provided in Fig. 1. Applying the inverse isomorphism on the result matrix gives a vector of polynomials with strictly increasing degrees. By (9), the column operations that modify one row will modify all other rows to the same degree differences. The operations imply there exists a matrix M i F ki ki q that ensures D i M i satisfy the conditions of (6b). By constructing M = diag (M h,..., M 0 ), we produce ˆD = DM, for which (6) is completely satisfied. ˆD can be seen as a sub-matrix of a super-regular matrix meeting (11). Then det ˆD = det D det M implies that D is non-singular and F is super-regular. For this proof, we had let A 0 = A 1 = = A m. Without this assumption, the product F uses a different set of f i (X) in each column block. Because column operations are performed for each D i independently, the polynomial degrees can always be transformed in order to satisfy (11). The key technique in this proof is the elementary column operation matrix M. An example is provided in Appendix B. It is interesting to note that in general, F can be transformed using elementary column operations from the onset to a D 0 (12)

9 9 super-regular matrix whose polynomials satisfy (11). However, it is not obvious whether this preserves super-regularity. While column permutations will not change the set of sub-matrices, column addition operations on F produces a matrix with an entirely different set of sub-matrices. As a result, our proof opts to directly show that each sub-matrix is indeed non-singular. B. Encoder We permute the rows of T to resemble the extended generator matrix in (2). This re-arranged matrix is also super-regular. T 0 T 1... T m T 0... T m 1 T =... (1). T 0 G EX m [5]. is constructed as a sub-matrix of k(m + 1) rows from T. This process parallels the construction of m-mds generators Theorem 5. Let T be the super-regular matrix generated over F q M using Theorem 4. Let 0 i 1 < < i k < n and construct a (m + 1)k (m + 1)n sub-matrix G EX m of T from rows indexed jn + i 1,..., jn + i k for j = 0,..., m. This matrix is the extended generator of a MSR convolutional code C[n, k, m]. Proof: We( will) show that G EX m satisfies Theorem. Assume without loss of generality that i 1 = 0,..., i k = k 1. Each T i Gi is divided into, where G i F k n are the blocks of the extended generator matrix. For t = 0,..., m, let A q M t F n n q be T i non-singular matrices. We similarly divide A t = ( A t A t), where the two blocks A t F n ρt q and A t F n (n ρt) q represent the reduced channel matrix and some remaining matrix respectively. Let A = diag (A 0,..., A m ). The product can be written as T 0 A 0 T 1 A 1... T m A m T 0 A 1... T m 1 A m TA =..... T 0 A m where ( Gi A j G T i A j = i A ) j T ia j T ia j The sub-matrix of TA containing only the rows and columns involving G i A j is equal to the product G EX m A. If the ranks ρ t satisfy the conditions in (7), then G EX m A has a non-trivial determinant [7]. By Theorem 4, this matrix is non-singular. Therefore, G EX m satisfies Theorem and C[n, k, m] achieves d R (m) = (n k)(m + 1) + 1. The above technique can be used to construct convolutional codes of any parameter that achieve the maximum column sum rank up to the code memory. We can now directly use an MSR code for network streaming over a sliding window channel. A MSR code C[n, k, T ] can feasibly recover every packet over C R (N, W ) with delay T = W 1 if N < d R (W 1). In practice, the decoding deadline T is not always exactly equal to W 1. If the decoder relaxes the delay constraint, i.e. T W, the same code C[n, k, T ] achieves perfect recovery over every sliding window channel that has N < d R (W 1). However if T < W 1, then the maximum N achievable is d R (T ) 1. C. Example Although the bound on the field sizegiven in Theorem 4 is large, it is only a sufficiency constraint required for the proof. We present examples of MSR codes over fields with smaller sizes below. Example 1. Let α F 2 7 be a primitive normal element. Let T be the super-regular matrix from (1) generated over F 2 7, but with dimensions given by n = and m = 2. We pick the rows i 1 = 0, i 2 = 2 and let G j be defined as follows. ( ) α [jn] α G j = [jn+1] α [jn+2] α [jn+2] α [jn+] α [jn+4] Numerically, G EX 2 can be shown to satisfy Theorem, therefore making it the generator matrix for a MSR code C[, 2, 2]. Example 2. Let β F 2 10 be a primitive normal element. Let T be the super-regular matrix from (1) generated over F 2 10, but with dimensions given by n = 4 and m = 1. We pick the rows i 1 = 0, i 2 = and let G j be defined as below. ( ) β [jn] β G j = [jn+1] β [jn+2] β [jn+] β [jn+] β [jn+4] β [jn+5] β [jn+6]

10 10 G EX 1 is the generator matrix for a MSR code C[4, 2, 1]. In both cases, Theorem 4 guarantees the construction only if M = 2 11, i.e. α and β are primitive normal elements of F VI. CONCLUSION In this paper, we map the relationship between column sum rank and network streaming. We prove several properties of the column sum rank that parallel analogous properties of column Hamming distance. A convolutional code construction maximizing this metric up to the code memory is proposed. This MSR code is a rank metric counterpart to the m-mds convolutional code. We use matrices over extension fields that preserve super-regularity after multiplication with block diagonal matrices in the ground field. The proof requires large field sizes but we numerically show that codes can be constructed for smaller fields. Future work involves pursuing a more detailed study on the field requirements. Moreover, we have only considered a specific class of rank deficient sliding window channels. In single-link streaming over burst erasure or mixed erasure channels, structured constructions using m-mds codes as constituents have been revealed as more powerful alternatives []. MSR codes may be similarly useful constituents for analogous codes over networks having burst or mixed link failures. As MSR codes directly extend m-mds codes, it is likely that rank metric parallels exist for convolutional codes maximizing other Hamming distance metrics. APPENDIX A PROOF OF LEMMA 1 Proof: The bounds were proven in [7] for q = 2 in order to show that the matrix is super-regular. We will prove the upper bound for more general q, but refer the reader to the original work for the lower bound. The degree of D(X) is bounded by considering the degree of the polynomial terms. When constructing each D σ (X), up to two entries from the last row and column can be used. Choosing D r 1,r 1 prevents selecting a second entry, but due to (6), D r 1,r 1 (X) has a greater degree than the sum of any other two entries. In addition, deg D r 1,r 1 (X) q n(m+2) 2. We next consider the second last row and column. However, the degrees of entries here are bounded by deg D r 1,r 1 (X) q n(m+2) 4. This argument is used recursively to show r 1 r 1 deg D i,σ(i) (X) k=0 q n(m+2) 2 2k r 1 = q n(m+2) 2 k=0 q 2k < q n(m+2) q 2 < q n(m+2) 1 APPENDIX B EXAMPLE OF THE COLUMN TRANSFORMATION IN THEOREM 4 Example. Let n = 4, m = 1 and T be the super-regular matrix of Theorem 4 generated from a primitive and normal element α F Let A 0 and A 1 be the two non-singular square matrices given below A 0 = , A 1 = We will use A = diag (A 0, A 1 ) as the block-diagonal matrix and generate the product TA. The entries f 0 = α [1] + α [2], f 1 = α [0], f 2 = α [0] + α [2], f = α [] are linearly independent polynomials. Similarly, the entries g 0 = α [0] + α [1], g 1 = α [1] + α [2] + α [], g 2 = α [1], g = α [0] + α [2] are also respectively linearly independent. Note that the q-degree of these polynomials are upper and lower bounded between and 0. Furthermore, for any fixed row of TA, the polynomials derived from f i (X) always have a lower degree than those from g i (X).

11 11 ( ) T TA = 0 A 1 = T 0 A 0 T 1 A 1 g 0 g 1 g 2 g g [1] 0 g [1] 1 g [1] 2 g [1] g [2] 0 g [2] 1 g [2] 2 g [2] g [] 0 g [] 1 g [] 2 g [] 2 g [4] 0 g [5] 1 g [5] 2 g [5] f 0 f 1 f 2 f g [4] 0 g [4] 1 g [4] f [1] 0 f [1] 1 f [1] 2 f [1] g [5] f [2] 0 f [2] 1 f [2] 2 f [2] g [6] 0 g [6] 1 g [6] 2 g [6] f [] 0 f [] 1 f [] 2 f [] g [7] 0 g [7] 1 g [7] 2 g [7] Now consider the sub-matrix of the product formed from rows R i, i {1, 2, 4, 5} and columns C j, j {, 4, 5, 6}. We denote this matrix D. g [1] 0 g [1] 1 g [1] 2 g [2] D = 0 g [2] 1 g [2] 2 f g [4] 0 g [4] 1 g [4] 2 f [1] g [5] 0 g [6] 1 g [7] 2 This matrix does not satisfy (6b). The degree of the polynomials in C 4 and C 6 are the same and lower than the degree of the polynomials in C 5. We apply the following elementary column operations on D to construct a new matrix ˆD. 1) C 5 C 6 2) C 4 C 5 C 4 For the new matrix, the entries in C 4 are generated from Frobenius powers of ĝ 0 = g 0 g 2 = α [0]. ˆD now completely satisfies (6) and it follows that it is non-singular. REFERENCES [1] E. Martinian and C.-E. W. Sundberg, Burst erasure correction codes with low decoding delay, IEEE Trans. on Inf. Theory, vol. 50, no. 10, pp , [2] D. Leong, A. Qureshi, and T. Ho, On coding for real-time streaming under packet erasures, in IEEE Int. Symp. on Inf. Theory (ISIT), 201, pp [] A. Badr, P. Patil, A. Khisti, W. Tan, and J. Apostolopoulos, Layered constructions for low-delay streaming codes, to appear in IEEE Trans. on Inf. Theory. [4] V. Tomas, J. Rosenthal, and R. Smarandache, Decoding of convolutional codes over the erasure channel, IEEE Trans. on Inf. Theory, vol. 58, no. 1, pp , [5] H. Gluesing-Luerssen, J. Rosenthal, and R. Smarandache, Strongly-MDS convolutional codes, IEEE Trans. on Inf. Theory, vol. 52, no. 2, pp , [6] R. Hutchinson, R. Smarandache, and J. Trumpf, Superregular matrices and the construction of convolutional codes having a maximum distance profile, Lin. Algebra and Its App., vol. 428, pp , [7] P. Almeida, D. Napp, and R. Pinto, A new class of super regular matrices and MDP convolutional codes, Lin. Algebra and Its App., vol. 49, pp , 201. [8] R. Ahlswede, N. Cai, S. Li, and R. Yeung, Network information flow, IEEE Trans. on Inf. Theory, vol. 46, no. 4, pp , [9] T. Ho, M. Medard, R. Kotter, D. R. Karger, M. Effros, J. Shi, and B. Leong, A random linear network coding approach to multicast, IEEE Trans. on Inf. Theory, vol. 52, no. 10, pp , [10] R. Kotter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. on Inf. Theory, vol. 54, no. 8, pp , [11] D. Silva, F. R. Kschischang, and R. Kotter, A rank-metric approach to error control in random network coding, IEEE Trans. on Inf. Theory, vol. 54, no. 9, pp , [12] E. M. Gabidulin, Theory of codes with maximum rank distance, Problems Inf. Transm., vol. 21, no. 1, pp. 1 12, [1] R. Roth, Maximum-rank array codes and their application to crisscross error correction, IEEE Trans. on Inf. Theory, vol. 7, no. 2, pp. 28 6, [14] R. W. Nobrega and B. F. Uchoa-Filho, Multishot codes for network coding using rank-metric codes, in IEEE Wireless Network Coding Conf. (WiNC), 2010, pp [15] A. Wachter-Zeh, M. Stinner, and V. Sidorenko, Convolutional codes in rank metric with application to random network coding, to appear in IEEE Trans. on Inf. Theory. [16] R. Lidl and H. Niederreiter, Introduction to finite fields and their applications. Cambridge University Press, [17] P. J. McWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes. North Holland Mathematical Library, [18] H. W. Lenstra and R. J. Schoof, Primitive normal bases for finite fields, Math. of Comp., vol. 48, no. 177, pp , [19] R. Johannesson and K. S. Zigangirov, Fundamentals of Convolutional Coding. IEEE Press, [20] A. Khisti., D. Silva, and F. R. Kschischang, Secure-broadcast codes over linear-deterministic channels, in IEEE Int. Symp. on Inf. Theory (ISIT), 2010, pp [21] S. Jalali, M. Effros, and T. Ho, On the impact of a single edge on the network coding capacity, in Inf. Theory and App. (ITA), 2011, pp. 1 5.