Probability of complete decoding of random codes for short messages

Abstract

A random code is a rateless erasure code with a generator matrix of randomly distributed binary values. It encodes a message of k symbols into a potentially infinite number of coded symbols. For asymptotically large k, the tail bound in Kolchin's theorem asserts that the high probability of complete decoding (PCD) is attained almost surely with k + 10 coded symbols. However, for small values of k (short messages) it is unclear if such asymptotics are useful. That the random codes achieve a high PCD with k + 10 coded symbols for small k is demonstrated. In particular, a set of lemmas is established and show that the PCD converges to five decimal digits after k = 30. A theorem extending Kolchin's work is formulated and the theorem is used to explain the complete decoding probabilities of random codes in short messages.Introduction: Rateless erasure codes for short message transmission: The current Internet architecture relies on individual flow control to manage network congestion. However, today's Internet traffic is dominated by short-lived flows with small packet sizes. Short-lived flows recovering from the loss of the last packet incur disproportionately high latency with existing flow control mechanisms. Owing to this inefficiency, the Global Environment for Network Innovations envisions future networks without congestion control [1], and erasure codes represent a significant step towards realisation of such networks.Network flows regulated by congestion control are typically modelled as a binary erasure channel whereby a transmitted packet is either received intact by the receiver, or dropped because of the network congestion or packet error. Transport protocols utilising state-of-the-art rateless erasure codes for networking have been proposed in the literature. Namely the variants of Luby transform (LT) codes [2] and Raptor codes [3] such as [4][5][6] are particularly attractive because of the low decoding complexity.Given a message of k symbols, a 'rateless' erasure code generates a potentially infinite number of coded symbols and the receiver reconstructs the original message from any k(1 + ε) coded symbols, where ε denotes the decoding inefficiency. The sender does not need a priori knowledge of the channel condition and it continues sending the coded symbols until the receiver has enough coded symbols to decode. However, the aforementioned rateless erasure codes are only efficient for long messages as reported in [7,8] whereas the network traffic is dominated by short messages instead [9,10].Erasure codes derived from LT codes such as [7,11,12] are pushing code design to address the needs of short message transmission. In particular, random code variations such as the windowed code [8] and the stepping-random code [13] have been proposed for this purpose. Decodability of these codes is well explained with Kolchin's theorem [14], i.e. the probability a random matrix of size (k + 10) × k is nonsingular is 99.9% when k → ∞. However, the theorem does not explain the non-singularity of the random matrix (i.e...

Systematic rateless erasure code for short messages transmission

“…Generally, Kolchin's theorem is an asymptotic equation and it only explains the PCD of a random matrix when k is large. On the other hand, the work in [25] complements Kolchin's theorem by showing that the random matrix (Random code) is able to achieve high PCD with k þ 10 coded symbols even for small k, i.e.…”

“…To the best of our understanding, the earliest discussion about Random code (properties of random matrix) is found in [23] and some relevant discussions about the PCD appear in [24,25]. Both Windowed code [10] and Stepping-Random (SR) code [26] are the non-systematic rateless erasure codes built on top of random matrix framework.…”

Improving the probability of complete decoding of random code by trading‐off computational complexity

“…Q RC (10) = 0.9990]. Though 1is an asymptotic equation, it has been shown in [7] that the high PCD is equally applicable for decoding short messages.…”