Network Error Correction for Unit-Delay, Memory-Free Networks Using Convolutional Codes

Abstract

A single source network is said to be memory-free if all of the internal nodes (those except the source and the sinks) do not employ memory but merely send linear combinations of the symbols received at their incoming edges on their outgoing edges. In this work, we introduce network-error correction for single source, acyclic, unit-delay, memory-free networks with coherent network coding for multicast. A convolutional code is designed at the source based on the network code in order to correct network-errors that correspond to any of a given set of error patterns, as long as consecutive errors are separated by a certain interval which depends on the convolutional code selected. Bounds on this interval and the field size required for constructing the convolutional code with the required free distance are also obtained. We illustrate the performance of convolutional network error correcting codes (CNECCs) designed for the unit-delay networks using simulations of CNECCs on an example network under a probabilistic error model.

Delay invariant convolutional network codes

“…Remark. Lemma 1 in [16] also proves that in an acyclic network, an -linear multicast qualifies to be a t-causal convolutional multicast with the unit delay function. Proposition 5 applies to arbitrary delay functions and networks with cycles, and is proved by a simpler method.…”

Distributed decoding of convolutional network error correction codes

“…Note that ∆(t, l) can also be regarded as a subspace determined by parameter l where l ≥ l t . For a given set of error patterns, assume G I (z) has been constructed using the scheme in [14]. We hope to find a minimum l such that Φ(t, l) ∩ ∆(t, l) = {0} where 0 is a 1 × ω(l + 1) zero vector and Φ(t, l) is the message subspace spanned by output convolutional codes generated by x l (z)G O,t (z) where x l (z) are all possible input sequences from instant 0 to l, i.e., (00 · · · 00) l , (00 · · · 01) l , · · · , (11 · · · 11) l .…”

“…Convolutional network coding is shown to have advantages in field size, small decoding delay and so on, which is more suitable for practical communications [12] [13]. Convolutional network-error correcting coding was introduced in [14] in the context of coherent network coding for acyclic instantaneous or unit-delay networks. They presented a convolutional code construction for a given acyclic instantaneous or unit-delay memory-free network that corrects a given pattern of network errors.…”

“…In this paper, we consider the decoding for a generalized convolutional network error-correction code constructed by the extended method in [14], where network errors on each edge occur with the same probability and are separated by fixed timeslots. Source encoding is always needed due to the error correction purpose in networks.…”

A Survey of Linear Network Coding and Network Error Correction Code Constructions and Algorithms

“…Their case subsumes convolutional code as a particular case. Convolutional codes for error correction were recently presented by Prasad and Rajan [37,38] LCM in acyclic networks being well known, in the next Section, we discuss construction algorithms for the acyclic case only.…”