Improved Decoding of Interleaved AG Codes

Abstract

Abstract. We analyze a generalization of a recent algorithm of Bleichenbacher et al. for decoding interleaved codes on the Q-ary symmetric channel for large Q. We will show that for any m and any the new algorithms can decode up to a fraction of at least βm βm+1 (1−R−2Q −1/2m )− errors, where β = ln(q m −1) ln(q m ) , and that the error probability of the decoder is upper bounded by O(1/q n ), where n is the block-length. The codes we construct do not have a-priori any bound on their length.

AG codes from vector bundles

“…interleaved AG codes considered in [3]. Thus C(X, P, E) generalizes these two previously known codes simultaneously.…”

“…Generalizing the algorithm for interleaved AG codes given in [3], Savin proposed a decoding algorithm for these vector bundle codes when E satisfies a property called weak stability and has shown that we can correct all errors with < t * − g/2, where g denotes the genus of X and t * is the designed correction capacity defined by…”

Single-Shot Security for One-Time Memories in the Isolated Qubits Model

“…Using these codes, we will get efficient implementations of one-time memories. There are several known constructions for codes that approach the capacity of the q-ary symmetric channel, and are efficiently decodable [22,23,24]. To illustrate how these techniques can be applied in our setting, we will describe one simple approach, which is based on erasure coding and error detection [23].…”

“…Our OTM's can be constructed using any code that satisfies two simple requirements: the code must be linear over GF (2), and it must approach the capacity of the q-ary symmetric channel. We show one such code in this paper; several more sophisticated constructions are known [22,23,24]. ) We will describe our OTM construction in the following section.…”

On Constructing Homomorphic Encryption Schemes from Coding Theory

“…To assess this question to some extent, we follow the usual approach by investigating a weaker variant of the scheme. 11 More precisely, we consider a variant where the used code is known to the attacker, i.e., the attacker is given (eventually) a basis for the vector space V 0 while V err remains secret. This allows for the following insights:…”