Subquadratic time encodable codes beating the Gilbert-Varshamov bound

We construct explicit algebraic geometry codes built from the Garcia-Stichtenoth function field tower beating the Gilbert-Varshamov bound for alphabet sizes at least 192. Messages are identied with functions in certain Riemann-Roch spaces associated with divisors supported on multiple places. Encoding amounts to evaluating these functions at degree one places. By exploiting algebraic structures particular to the Garcia-Stichtenoth tower, we devise an intricate deterministic \omega/2 < 1.19 runtime exponent encoding and 1+\omega/2 < 2.19 expected runtime exponent randomized (unique and list) decoding algorithms. Here \omega < 2.373 is the matrix multiplication exponent. If \omega = 2, as widely believed, the encoding and decoding runtimes are respectively nearly linear and nearly quadratic. Prior to this work, encoding (resp. decoding) time of code families beating the Gilbert-Varshamov bound were quadratic (resp. cubic) or worse

Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth's second tower

Asymptotically good sequences of ramp secret sharing schemes were given in [Asymptotically good ramp secret sharing schemes, arXiv:1502.05507] by using one-point algebraic geometric codes defined from asymptotically good towers of function fields. Their security is given by the relative generalized Hamming weights of the corresponding codes. In this paper we demonstrate how to obtain refined information on the RGHWs when the codimension of the codes is small. For general codimension, we give an improved estimate for the highest RGHW

Transitive and self-dual codes attaining the Tsfasman-Vladut-Zink bound

A major problem in coding theory is the question of whether the class of cyclic codes is asymptotically good. In this correspondence-as a generalization of cyclic codes-the notion of transitive codes is introduced (see Definition 1.4 in Section I), and it is shown that the class of transitive codes is asymptotically good. Even more, transitive codes attain the Tsfasman-Vladut-Zink bound over F-q, for all squares q = l(2). It is also shown that self-orthogonal and self-dual codes attain the Tsfasman-Vladut-Zink bound, thus improving previous results about self-dual codes attaining the Gilbert-Varshamov bound. The main tool is a new asymptotically optimal tower E-0 subset of E-1 subset of E-2 subset of center dot center dot center dot of function fields over F-q (with q = l(2)), where all extensions E-n/E-0 are Galois