Localized dimension growth in random network coding: A convolutional approach

Abstract

We propose an efficient Adaptive Random Convolutional Network Coding (ARCNC) algorithm to address the issue of field size in random network coding. ARCNC operates as a convolutional code, with the coefficients of local encoding kernels chosen randomly over a small finite field. The lengths of local encoding kernels increase with time until the global encoding kernel matrices at related sink nodes all have full rank. Instead of estimating the necessary field size a priori, ARCNC operates in a small finite field. It adapts to unknown network topologies without prior knowledge, by locally incrementing the dimensionality of the convolutional code. Because convolutional codes of different constraint lengths can coexist in different portions of the network, reductions in decoding delay and memory overheads can be achieved with ARCNC. We show through analysis that this method performs no worse than random linear network codes in general networks, and can provide significant gains in terms of average decoding delay in combination networks.

Localized Dimension Growth: A Convolutional Random Network Coding Approach to Managing Memory and Decoding Delay

“…Assume s generates a source message per unit time, consisting of a fixed number of m source symbols represented by a size m row vector x t = (x 1,t , x 2,t , · · · , x m,t ), x i,t ∈ F q . Time t is indexed from 0, where the (t + 1)-th message generated at time t, consistent with our previous work [1], [10]. Source messages are collectively represented by a power series x(z) = t≥0 x t z t , where x t is the message generated at t and z denotes a unit-time delay.…”

“…2 illustrates its topology. Assuming unit capacity links, the min-cut to each sink is m. In combination networks, routing is insufficient and network coding is needed to achieve the multicast capacity m. Here coding is performed only at s, since each intermediate node has only one incoming edge but s. For a general n m combination network, we showed in [1] that the expected average first decoding time can be significantly improved by ARCNC when compared to BNC. At time t − 1, for a sink r, F r (z) is a size m × m matrix of polynomials of degree t − 1.…”

“…This variable was called the stopping time in [1]. For a sink r decodable at time T r , the decoding delay is T r + 1 since time is indexed from 0.…”

Time-Variant Decoding of Convolutional Network Codes

“…This has been adopted in the adaptive random construction algorithm for a CNC in [11]. It is a necessary but not sufficient condition to decode with delay L at sink r.…”

“…Efficient centralized construction of optimal convolutional network codes (CNCs) on a multicast network is discussed in [4] [6] [8] [7]. A distributed construction scheme for a CNC is proposed in [11] by adaptively and randomly assigning local encoding kernels.…”