Factorisation of $x^N - q$ over Q

Elliptic Curves over Q

All polynomials and rational functions in this essay are assumed to have coefficients in Q . Fix an integer n ≥ 1. An affine variety is a simultaneous irreducible system of polynomial equations in n variables. The Q -points, R -points and C points of the affine variety are all solutions of the polynomial system in Q , R n and C , respectively. Rational projective n-spaceg Q n is the set of line...

Elliptic Curves over Q

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Block Transitive Meager Squares over GF (q) for prime q

This paper shows the existence of a class of block transitive meager squares of order q on GF (q) for odd prime q, q large. Using these meager squares along with other known meager system constructions, we determine the spectrum of orders that admit meager systems: all odd n, n ≥ 9. Using this result, it is also determined that the spectrum of orders that admit 5-sparse Steiner triple systems i...

On Codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}+v^{2}\mathbb{F}_{q}$

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring $R=\F_{q}+v\F_{q}+v^{2}\F_{q}$, where $v^{3}=v$, for $q$ odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over $R$. Further, we give bounds on the minimum distance of LCD codes over $\F_q$ and extend these to codes ove...