Partial spreads and vector space partitions

Constant-dimension codes with the maximum possible minimum distance have been studied under the name of partial spreads in Finite Geometry for several decades. Not surprisingly, for this subclass typically the sharpest bounds on the maximal code size are known. The seminal works of Beutelspacher and Drake \& Freeman on partial spreads date back to 1975, and 1979, respectively. From then until recently, there was almost no progress besides some computer-based constructions and classifications. It turns out that vector space partitions provide the appropriate theoretical framework and can be used to improve the long-standing bounds in quite a few cases. Here, we provide a historic account on partial spreads and an interpretation of the classical results from a modern perspective. To this end, we introduce all required methods from the theory of vector space partitions and Finite Geometry in a tutorial style. We guide the reader to the current frontiers of research in that field, including a detailed description of the recent improvements.Comment: 30 pages, 1 tabl

The Hardness of Code Equivalence over Fq\mathbf{F}_q and its Application to Code-based Cryptography

International audienceThe code equivalence problem is to decide whether two linear codes over F_q are equivalent, that is identical up to a linear isometry of the Hamming space. In this paper, we review the hardness of code equivalence over F_q due to some recent negative results and argue on the possible implications in code-based cryptography. In particular, we present an improved version of the three-pass identification scheme of Girault and discuss on a connection between code equivalence and the hidden subgroup problem

Parallel Fast Möbius (Reed-Muller) Transform and its Implementation with CUDA on GPUs

One of the most important cryptographic characteristics of the Boolean and vector Boolean functions is the algebraic degree which is connected with the Algebraic Normal Form. In this paper, we present an algorithm for computing the Algebraic Normal Form of a Boolean function using binary Fast Möbius (Reed-Muller) Transform implemented in CUDA for parallel execution on GPU. In the end, we give some experimental results

Projective two-weight codes with small parameters and their corresponding graphs

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