On the classification of nonsingular 2×2×2×2 hypercubes
- 153 Downloads
As a first step in the classification of nonsingular 2×2×2×2 hypercubes up to equivalence, we resolve the case where the base field is finite and the hypercubes can be written as a product of two 2×2×2 hypercubes. (Nonsingular hypercubes were introduced by D. Knuth in the context of semifields. Where semifields are related to hypercubes of dimension 3, this paper considers the next case, i.e., hypercubes of dimension 4.) We define the notion of ij-rank (with 1 ≤ i < j ≤ 4) and prove that a hypercube is the product of two 2×2×2 hypercubes if and only if its 12-rank is at most 2. We derive a ‘standard form’ for nonsingular 2×2×2×2 hypercubes of 12-rank less than 4 as a first step in the classification of such hypercubes up to equivalence. Our main result states that the equivalence class of a nonsingular 2×2×2×2 hypercube M of 12-rank 2 depends only on the value of an invariant δ 0(M) which derives in a natural way from the Cayley hyperdeterminant det0 M and another polynomial invariant det M of degree 4. As a corollary we prove that the number of equivalence classes is (q + 1)/2 or q/2 depending on whether the order q of the field is odd or even.
KeywordsNonsingular hypercube Invariant Hyperdeterminant Classification Semifield
Mathematics Subject Classification (2010)20G40 15A72 12K10
Unable to display preview. Download preview PDF.
- 2.Cayley A.: On the theory of determinants. Trans. Camb. Philos. Soc. 8, 1–16 (1843)Google Scholar
- 3.Cayley A.: On the theory of linear transformations. Camb. Math. J. 4, 193–209 (1845)Google Scholar
- 5.Glynn D., Gulliver T., Maks J., Gupta M.: The geometry of additive quantum codes. Springer, Berlin (in preparation).Google Scholar
- 6.Kantor W.M.: Finite semifields. In: Hulpke, A., Liebler, R., Penttila, T., Seress, Á. (eds.) Finite Geometries, Groups, and Computation., pp. 103–114. Walter de Gruyter GmbH & Co. KG, Berlin (2006)Google Scholar
- 8.Lavrauw M.: Finite semifields and nonsingular tensors. Des. Codes Cryptogr. (2012). doi: 10.1007/s10623-012-9710-6.
- 9.Lavrauw M., Polverino O.: Finite semifields. In: De Beule J., Storme, L. (eds.) Current Research Topics in Galois Geometry. Nova Science Publishers Inc., Hauppauge (2012)Google Scholar