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Given $\Sigma\subset\mathbb K[x_1,\ldots,x_k]$, any finite collection oflinear forms, some possibly proportional, and any $1\leq a\leq |\Sigma|$, ithas been conjectured that $I_a(\Sigma)$, the ideal generated by all $a$-foldproducts of $\Sigma$, has linear graded free resolution. In this article weshow the validity of this conjecture for two cases: the first one is when$a=d+1$ and $\Sigma$ is dual to the columns of a generating matrix of a linearcode of minimum distance $d$; and the second one is when $k=3$ and $\Sigma$defines a line arrangement in $\mathbb P^2$ (i.e., there are no proportionallinear forms). For the second case we investigate what are the graded bettinumbers of $I_a(\Sigma)$.

On some ideals with linear free resolutions