A finite semifield D is a finite nonassociative ring with identity such that the set D * = D \ {0} is closed under the product. In this paper we obtain a computer-assisted description of all semifields of order 64, which completes the classification of finite semifields of order at most 125.
A finite semifield D is a finite nonassociative ring with identity such that the set D* = D \{0} is a loop under the product. Wene conjectured in [1] that any finite semifield is either right or left primitive, i.e. D* is the set of right (or left) principal powers of an element in D. In this paper we study the primitivity of finite semifields with 64 and 81 elements.
In this work we consider repeated-root multivariable codes over a finite chain ring. We show conditions for these codes to be principally generated. We consider a suitable set of generators of the code and compute its minimum distance. As an application we study the relevant example of the generalized Kerdock code in its r -dimensional cyclic version.
A quantum procedure for testing the commutativity of a finite dimensional algebra is introduced. This algorithm, based on Grover's quantum search, is shown to provide a quadratic speed-up (when the number of queries to the algebra multiplication constants is considered) over any classical algorithm (both deterministic and randomized) with equal success rate and shown to be optimal among the class of probabilistic quantum query algorithms. This algorithm can also be readily adapted to test commutativity and hermiticity of square matrices, again with quadratic speed-up. The results of the experiments carried out on a quantum computer simulator and on one of IBM's 5-qubit quantum computers are presented.
Kerdock codes (Kerdock, Inform Control 20:182-187, 1972) are a wellknown family of non-linear binary codes with good parameters admitting a linear presentation in terms of codes over the ring Z 4 (see Nechaev, Diskret Mat 1:123-139, 1989; Hammons et al., IEEE Trans Inform Theory 40:301-319, 1994). These codes have been generalized in different directions: in Calderbank et al. (Proc Lond Math Soc 75:436-480, 1997) a symplectic construction of non-linear binary codes with the same parameters of the Kerdock codes has been given. Such codes are not necessarily equivalent. On the other hand, in Kuzmin and Nechaev (Russ Math Surv 49(5), 1994) the authors give a family of non-linear codes over the finite field F of q = 2 l elements, all of them admitting a linear presentation over the Galois Ring R of cardinality q 2 and characteristic 2 2 . The aim of this article is to merge both approaches, obtaining in this way new families of non-linear codes over F that can be presented as linear codes over the Galois Ring R. The construction uses symplectic spreads.
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