The number of invariant subspaces under a linear operator on finite vector spaces

Abstract

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb F$<SUB><I>q</I></SUB> and $T$ a linear operator on $V$. For each $k\in\{1,\ldots,n\}$ we determine the number of $k$-dimensional $T$-invariant subspaces of $V$. Finally, this method is applied for the enumeration of all monomially nonisometric linear $(n,k)$-codes over $\mathbb F$<SUB><I>q</I></SUB>.