Our main result is a reduction from worst-case lattice problems such as SVP and SIVP to a certain learning problem. This learning problem is a natural extension of the ‘learning from parity with error’ problem to higher moduli. It can also be viewed as the problem of decoding from a random linear

Abstract—The list-decodability of random linear codes is shown to be as good as that of general random codes. Specifically, for every fixed finite field Fq, p ∈ (0, 1 − 1/q) and ε>0, itis proved that with high probability a random linear code C in F n q of rate (1 − Hq(p) − ε) can be list

challenge in such systems, without the use of centralized servers. To better cope with node dynamics and failures, we propose priority random linear codes, as well as their affiliated pre-distribution protocols, to maintain measurement data in different priorities, such that critical data have a higher

, without the use of centralized servers. To cope with node failures, while providing convenient access to measured data, we propose geometric random linear codes, to encode data in a hierarchical fashion in geographic regions with different sizes, such that data are easy to access, if the original sensors

challenge in such systems, without the use of centralized servers. To better cope with node dynamics and failures, we propose priority random linear codes, as well as their affiliated pre-distribution protocols, to maintain measurement data in different priorities, such that critical data have a higher

challenge in such systems, without the use of centralized servers. To better cope with node dynamics and failures, we propose priority random linear codes, as well as their affiliated pre-distribution protocols, to maintain measurement data in different priorities, such that critical data have a higher

We show that random sparse binary linear codes are locally testable and locally decodable (under any linear encoding) with constant queries (with probability tending to one). By sparse, we mean that the code should have only polynomially many codewords. Our results are the first to show that local

random linear code.This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quantum. Hence, an efficient solution to the learning problem implies a quantum algorithm for SVP andSIVP. A main open question is whether this reduction can be made classical. Using

based storage, random linear coding based storage motivated by network coding. We demonstrate that, in principle, a traditional erasure coding based storage (eg: Reed-Solomon Codes) strategy can almost do as well as one can ask for with appropriate choice of parameters. However, the cost is a large

Abstract—Consider a lossy communication channel for unicast with zero-delay feedback. For this communication scenario, a simple retransmission scheme is optimum with respect to delay. An alternative approach is to use random linear coding in automatic repeat-request (ARQ) mode. We extend the work