An Explicit Rate-Optimal Streaming Code for Channels with Burst and Arbitrary Erasures

Abstract

In this paper, we consider transmitting a sequence of messages (a streaming source) over a packet erasure channel, where every source message must be recovered perfectly at the destination subject to a fixed decoding delay. Recently, the capacity of such a channel was established. However, the codes shown to achieve the capacity are either non-explicit constructions (proven to exist) or explicit constructions requiring large field size that scales exponentially with the delay. This work presents an explicit rate-optimal construction for all channel and delay parameters over a field size that scales only quadratically with the delay. E.. Domanovitz is with the

Explicit Rate-Optimal Streaming Codes with Smaller Field Size

“…, where c i,j = H(δ + i, a + j). This matrix is invertible since c 0,1 c 0,2 c 1,1 c 1,2 is a (2 × 2) sub-matrix of the parity check matrix of a [13,11] MDS code. Now consider H([0 : 4], [10 : 14]) which has the following structure: Then τ + 1 − a = vb + − a + 1.…”

“…A non-explicit rate-optimal streaming code, which requires a field F q 2 with prime power q ≥ τ , is presented in [6]. Subsequently, an explicit construction was presented in [13] requiring field size q 2 , for q ≥ τ + b − a, a prime power. Streaming codes for variable packet sizes are explored in [14].…”

“…Explicit rate-optimal constructions having linear field size for some {a, b, τ } parameter ranges are presented in [6], [15], [16]. However, the construction in [13] remains the smallest field size explicit rate-optimal streaming code construction that exists for all possible {a, b, τ }. Note that the field size required for the explicit code construction in [13] is larger than the field size q 2 ≥ τ 2 requirement of the code in [6].…”

“…However, the construction in [13] remains the smallest field size explicit rate-optimal streaming code construction that exists for all possible {a, b, τ }. Note that the field size required for the explicit code construction in [13] is larger than the field size q 2 ≥ τ 2 requirement of the code in [6]. In the present paper, we present an explicit rate-optimal code having the same field-size requirement q 2 , with prime power q ≥ τ , as that of the non-explicit code in [6].…”

Online Versus Offline Rate in Streaming Codes for Variable-Size Messages

“…Before the scheme for message size sequence 2 is described, we present a rate τ τ +b block code from [8] (or alternatively a block code from [5]- [7], [9]) which the scheme will build on. The block code systematically maps τ input symbols (s 0 , .…”