Near-Optimal Streaming Codes with Linear Field Size

Streaming codes are codes designed to ensure erased packet recovery with a decoding-delay deadline. In streaming code literature, an (a, b, w) sliding-window (SW) channel model is considered, in which within any window of w time slots, either a burst of ≤ b packets, or else ≤ a random packets can be erased. An (a, b, w, τ) streaming code is a packet-level code capable of recovering messages under a decoding delay of τ time slots, from any erasure pattern produced by an (a, b, w)-SW channel. For any given (a, b, w)-SW channel, the minimum delay with which the maximum rate possible over this channel can be achieved is τ = w−1. Rate-optimal constructions of streaming codes for any parameter tuple of the form (a, b, w, τ = w−1) are known in the literature. However, these constructions require a field size that is quadratic in w in general, and linear field size constructions are known only for a subset of {a, b, w} parameters. The current paper shows that it is possible to construct linear field size streaming codes for all {a, b, w} parameters, by sacrificing a little on either delay or rate. When the allowed delay is one more than the minimum, i. e., when τ = w, we construct linear field size rate-optimal streaming codes for all possible {a, b, w} parameters. This construction also leads to an O(w) field size (a, b, w, τ = w − 1) streaming code whose rate is close to the optimal rate. Additionally, we show that our construction can be modified to get binary streaming codes without too much loss in rate.