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Browsing by Author "Cotardo, Giuseppe"

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    Rank-Metric Lattices
    Cotardo, Giuseppe; Ravagnani, Alberto (Electronic Journal Of Combinatorics, 2023-01-13)
    We introduce the class of rank-metric geometric lattices and initiate the study of their structural properties. Rank-metric lattices can be seen as the q-analogues of higher-weight Dowling lattices, defined by Dowling himself in 1971. We fully characterize the supersolvable rank-metric lattices and compute their characteristic polynomials. We then concentrate on small rank-metric lattices whose characteristic polynomial we cannot compute, and provide a formula for them under a polyno-miality assumption on their Whitney numbers of the first kind. The proof relies on computational results and on the theory of vector rank-metric codes, which we review in this paper from the perspective of rank-metric lattices. More precisely, we introduce the notion of lattice-rank weights of a rank-metric code and inves-tigate their properties as combinatorial invariants and as code distinguishers for inequivalent codes.Mathematics Subject Classifications: 05A15, 06C10, 94B99
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    Squares of bivariate Goppa codes
    Basener, Wesley; Cotardo, Giuseppe; Krebs, Jenna; Liu, Yihan; Matthews, Gretchen L.; Ufferman, Eric (2023-10-13)
    In this paper, we study squares of bivariate Goppa codes, as they relate to the Goppa code distinguishing problem for bivariate Goppa codes. Introduced in 2021, multivariate Goppa codes are subfield subcodes of certain evaluation codes defined by evaluating polynomials in m variables. The evaluation codes are augmented Cartesian codes, a generalization of Reed-Muller codes. Classical Goppa codes are obtained by taking m=1. The multivariate Goppa code distinguishing problem is to distinguish efficiently a generator matrix of a multivariate Goppa code from a randomly drawn one. Because a randomly drawn code has a large square, codes with small squares may be considered distinguishable, revealing structure which facilitates private key recovery in a code-based cryptosystem.