Maria and Giovanni Zini published their joint paper “The complete list of genera of quotients of the Fq2-maximal Hermitian curve for q≡1 (mod 4)” in Journal of Algebra. Here an open problem from 2000 is partially solved, namely that of computing all the genera of quotients of the Hermitian curve over Fq2, q prime power. Indeed such a complete list of genera is obtained here if q is congruent to 1 modulo 4. A preprint can be found here.
Freja Elbro started a few months ago as a PhD student with Lars R. Knudsen in the Cyber Security section but is from 1st May co-advised by Johan Rosenkilde of the Algebra group.
Freja’s project is quantum-safe asymmetric cryptography based on coding theory, with a focus on McEliece-like constructions.
Today’s Maria joint work with Vincenzo Pallozzi Lavorante “AG codes from the second generalization of the GK maximal curve” appeared in the journal Discrete Mathematics. Here Weierstrass semigroups and points on the second generalized GK curve are computed, as well as, some AG codes (and AG quantum codes) with new and good parameters. A preprint is available here.
Maria’s paper “MRD-codes arising from the trinomial xq+xq3+cxq5∈Fq6[x]” with co-authors Ferdinando Zullo and Giuseppe Marino has been accepted for publication in the journal Linear Algebra and its Applications. Here, using algebraic curves over finite fields, a new family of maximum rank distance codes (MRD) has been constructed. A preprint is available here.
Maria and Gabor Korchmaros (Maria’s former PhD supervisor) published their paper “Large odd prime power order automorphism groups of algebraic curves in any characteristic” in Journal of Algebra. There sharp upper bounds for the order of prime power order automorphism groups of algebraic curves are obtained. Also, a recursive way to construct extremal examples is proposed, using elliptic curves. A preprint is available here.
Group members Johan Rosenkilde and Sven Puchinger, together with several co-authors from Israel, Finland, and Germany, got four papers accepted at this year’s ISIT, the IEEE flagship conference on information theory:
Today Maria’s paper “Large automorphism groups of ordinary curves of even genus in odd characteristic” has been accepted in Communications in Algebra. This is a joint work with Pietro Speziali and gives strong upper bound for the size of the automorphism group of ordinary curves (that is, when genus and p-rank coincide) of even genus, with respect to the genus itself. A preprint is available here.
Postdoc Sven Puchinger with co-authors from TU Munich (Julian Renner and Antonia Wachter-Zeh), German Aerospace Center (Thomas Jerkovits and Hannes Bartz), and Université de Rennes 1 (Pierre Loidreau) got a paper accepted at the International Conference on Post-Quantum Cryptography (PQCrypto) 2020:
“Randomized Decoding of Gabidulin Codes Beyond the Unique Decoding Radius”, Julian Renner, Thomas Jerkovits, Hannes Bartz, Sven Puchinger, Pierre Loidreau, Antonia Wachter-Zeh [link] [arXiv]
The paper proposes a significantly faster algorithm to solve a decoding problem which the security of three code-based cryptosystems is based on.
In the period January 6-17, Peter visited Tromsø (Norway), where the Arctic University of Norway is located. His host, Prof. Trygve Johnsen has visited the algebra group in the past during his sabattical. During Peter’s visit, he finished two papers and initiated a new one, so the visit was very productive!
The Algebra group was visited by Julian Renner for one week in November 2019. Julian is a PhD student at TU Munich and has published several papers with postdoc Sven Puchinger.
During the visit, Johan, Sven, and Julian worked on generic decoding in the sum-rank metric and on low-rank parity-check codes over finite rings. Julian also gave a very nice talk on “Key Compression for Rank-Based Cryptosystems” in the algebra group colloquium.
Update: The visit resulted in two papers accepted at ISIT 2020.