International workshop, Nogaro, France (August 25 to 30 2019)

In some applications, it is desirable to recover a single (or small number of) codeword symbol(s) by accessing only a few, say r, particular symbols of a received word. This leads to the notion of local recovery. Codes designed for this purpose are called locally recoverable codes, or LRCs for short. In principle, the locality r  should be small so as to limit network traffic though this can adversely impact other code parameters. In this talk, we discuss local recovery using codes from curves.

Extending Gabidulin codes, Martinez-Penas recently defined linearized Reed-Solomon codes and proved that these codes exhibit an optimal minimal distance for the sum-rank metric. Martinez-Penas’ construction is based on Ore’s skew polynomials. The aim of this talk is to introduce a linearized version of geometric Goppa codes in the spirit of Martinez-Penas’ construction. Precisely, I will first give a meaning to the notion skew rational functions and outline a theory of residues for them. This will eventually be the key for adapting the definition of classical Goppa codes in the skew setting (at least over the projective line) and showing that linearized Goppa codes are duals of Martinez-Penas’ linearized Reed-Solomod codes.

Itzhak Tamo: Optimal repair of polynomials

Anand Kumar Narayanan: Computing Riemann-Roch spaces on Garcia-Stichtenoth function field tower

Error-correcting codes enable reliable transmission of information over an erroneous channel. One typically desires codes to transmit information at a high rate while still being able to correct a large fraction of errors. However, rate and relative distance (which quantifies the fraction of errors corrected) are competing quantities with a trade off. The Gilbert-Varshamov bound assures for every rate R, relative distance D and alphabet size Q, there exists an infinite family of codes with R + H_Q(D) > = 1-\epsilon. Here H_Q is the Q-ary entropy. Explicitly constructing codes meeting or beating the Gilbert-Varshamov bound remained a long-standing open problem, until the advent of algebraic geometry codes by Goppa. For Q ≥ 7^2, algebraic geometry codes constructed from function fields attaining the Drinfeld-Vladut bound (first constructed by Tsfasman-Vladut-Zink) beat the Gilbert-Varshamov bound. For codes to find use in practice, one often requires fast encoding and decoding algorithms in addition to a good rate-minimum distance trade off. A natural question, which remains unresolved, is if there exist linear time encodable and decodable codes meeting or beating the Gilbert-Varshamov bound.

Aurel Page: Codes from unit groups of division algebras over number fields

Lenstra and Guruswami described number field analogues of the algebraic geometry codes of Goppa. Recently, Maire and Oggier generalised these constructions to other arithmetic groups: unit groups in number fields and orders in division algebras; they suggested to use unit groups in quaternion algebras but could not completely analyse the resulting codes. In this talk, I will present joint work with Christian Maire, were we prove that the noncommutative unit group construction yields asymptotically good families of codes for division algebras of any degree. In particular, I will highlight the differences between the commutative and the noncommutative cases, and introduce the appropriate tools to study noncommutative unit groups.

Cédric Lecouvey: Algebraic extensions of additive number theory

The aim of the talk is to prove how results and techniques coming from additive number theory extend naturally to linear structures (fields, skew fields, algebras). Obtained analogs entail frequently the original theorems and have interesting applications to other mathematical areas (among others : coding theory and representation theory).