Mrinmoy Datta — Page 2

arXiv2015

On a conjecture of Tsfasman and an inequality of Serre for the number of points on hypersurfaces over finite fields

We give a short proof of an inequality, conjectured by Tsfasman and proved by Serre, for the maximum number of points on hypersurfaces over finite fields. Further, we consider a conjectural extension, due to Tsfasman and Boguslavsky, of this inequality to an explicit formula for the maximum number of common solutions of a system of linearly independent multivariate homogeneous polynomials of the same degree with coefficients in a finite field. This conjecture is shown to be false, in general, but is also shown to hold in the affirmative in a special case. Applications to generalized Hamming weights of projective Reed-Muller codes are outlined and a comparison with an older conjecture of Lachaud and a recent result of Couvreur is given.

math.AG↗

arXiv2015

Higher Weights of Affine Grassmann Codes and Their Duals

We consider the question of determining the higher weights or the generalized Hamming weights of affine Grassmann codes and their duals. Several initial as well as terminal higher weights of affine Grassmann codes of an arbitrary level are determined explicitly. In the case of duals of these codes, we give a formula for many initial as well as terminal higher weights. As a special case, we obtain an alternative simpler proof of the formula of Beelen et al for the minimum distance of the dual of an affine Grasmann code.

cs.IT↗