Minimal codewords in Norm-Trace codes

In this paper, we consider the affine variety codes obtained evaluating the polynomials $by=a_k{x}^k+\cdots +a_{1}x+a_0$, $b,a_i\in {\mathbb{F}}_{q^r}$, at the affine ${\text{F}}_{q^r}$-rational points of the Norm-Trace curve. In particular, we investigate the weight distribution and the set of minimal codewords. Our approach, which uses tools of algebraic geometry, is based on the study of the absolutely irreducibility of certain algebraic varieties.