Information sets of Multiplicity codes

Abstract

Abstract-We here provide a method for systematic encoding of the Multiplicity codes introduced by Kopparty, Saraf and Yekhanin in 2011. The construction is built on an idea of Kopparty. We properly define information sets for these codes and give detailed proofs of the validity of Kopparty's construction, that use generating functions. We also give a complexity estimate of the associated encoding algorithm.

Adaptive Multiplicity Codes based PIR Protocol for Multi-Cloud Plateform Services

“…Multiplicity codes were known to satisfy LCC properties at good coding rate compared to Reed Muller codes. However making them locally decodable in the strictest sense remained an open issue until came the ideas of Kopparty in [13] and the practical construction in [21] and [22] to build fast systematic Multiplicity encoding algorithm at quasi linear time. Those recent advances in complexity enable the feasibility of Multiplicity codes based PIR protocol originally introduced in [14], where the storage overhead is optimal and equals the error correcting code overhead.…”

Fast systematic encoding of multiplicity codes

“…It follows that a locally correctable code that is equipped with a systematic encoding function, i.e., one that embeds messages into their encodings, is also locally decodable. Augot, Levy-dit-Vehel and Ngô [2] provide a systematic encoding function for multiplicity codes by combining results of Kopparty [16] and Key, McDonough and Mavron [15]. By using their encoding function, multiplicity codes offer sublinear local decoding, while still allowing high rates.…”

“…However, Kopparty does not provide explicit examples of the construction. Augot, Levy-dit-Vehel and Ngô [2] subsequently provide an explicit family of information sets by supplementing Kopparty's construction with a result of Key, McDonough and Mavron [15,Theorem 1].…”

“…Consequently, if the encoding function enc s d is to be used in the private information retrieval protocol of Augot, Levy-dit-Vehel and Shikfa [3], then it is important that the function may be evaluated in time that is close to linear in σ s,n q n . Augot, Levy-dit-Vehel and Ngô [2,Appendix] show that enc s d can be evaluated in Õ(σ 3 s,n q n + k 2 ) operations in F q , where k = n+d n and the notation Õ( • ) indicates that polylogarithmic factors are omitted from the complexity. The quadratic dependency on the dimension of the code means that their algorithm is not suitable for use in the private information retrieval context, where k log 2 q must be greater than or equal to the number of bits in the database, and q is the number of (non-colluding) servers.…”