By Bouyukliev I., Fack V., Winne J.

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**Additional resources for 2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) designs with automorphisms of odd prime order, and their related Hadamard matrices and codes**

**Sample text**

The product z z¯ = x2 − α y 2 + β x y ∈ R is the square modulus of z (z z¯ = x2 + y 2 in the particular case of complex numbers). Chapter 2. N-Dimensional Commutative Hypercomplex Numbers 16 We end this section with a matrix representation (that we write in sans serif ) of the versors and of z. 12) and also α β u2 = αβ α + β2 ≡ α 0 0 α ≡ α · 1 + β · u. 10). 9) become z= 2x + 2uy ≡ x + uy; 2 z¯ = 2x − 2uy ≡ x − u y. 14) These solutions are the ones we consider for complex numbers (∆ < 0), whereas for hyperbolic numbers, because u2 = 1, it is usual to omit the versor.

In this space we introduce the metric (distance of a point from the coordinate origin), given by the N th root of the characteristic determinant. More generally, the distance between two points shall be given by substituting in the characteristic determinant the diﬀerences of the points’ coordinates. 1. Every commutative system of hypercomplex numbers generates a geometry. For such geometry, the metric is obtained from a form of degree N given by the characteristic determinant. Motions are characterized by 2N −1 parameters.

The peculiar diﬀerences between noncommutative and commutative systems are their invariants and the existence of diﬀerential calculus. As far as the invariant is concerned, for the non-commutative systems it can be an algebraic quadratic form that can be related with Euclidean geometry so that, by means of Hamilton’s quaternions, we can represent vectors in the three-dimensional Euclidean space. For commutative systems, the invariant is represented by an N -form that, for N > 2, generates new geometries.