Rank metric convolutional codes for Random Linear Network Coding

Abstract

So far, in the area of Random Linear Network Coding, attention has been given to the so-called one-shot network coding, meaning that the network is used just once to propagate the information. In contrast, one can use the network more than once to spread redundancy over different shots. In this paper, we propose rank metric convolutional codes for this purpose. The framework we present is slightly more general than the one which can be found in the literature. We introduce a rank distance, which is suitable for convolutional codes, and derive a new Singleton-like upper bound. Codes achieving this bound are called Maximum Rank Distance (MRD) convolutional codes. Finally, we prove that this bound is optimal by showing a concrete construction of a family of MRD convolutional codes.

Convolutional Codes in Rank Metric With Application to Random Network Coding

“…For k (1) = 1 ≤ k, the upper bound on the slope is attained. If we compare this to the construction from [21], we see that both constructions attain the upper bound on the free rank distance for k (1) < k. It depends on the explicit values of n, k and k (1) , which construction has a higher slope.…”

“…, µ, are k × n-matrices and µ denotes the memory of G, see [28] and Definition 3. Each codeword of C is a sequence of N + µ blocks of length n over F, i.e., c = (c (0) c (1) . .…”

“…. c (N +µ−1) ), represented equivalently as a sequence of m × n matrices over F q , i.e., C = Φ β (c) = (C (0) C (1) . .…”

“…However, at the same time, if k + k (1) > n, there have to be linearly dependent rows in G (0) G (1) . Therefore, we define these matrices such that ϕ denotes the number of rows which are contained (amongst others) in both, G (0) and G (1) . We define a full-rank (k+k (1)…”

“…Parts of this work were presented at the IEEE International Symposium on Network Coding 2012 (NETCOD), Cambridge, MA, USA, 2012 [1]. A.…”