We present a family of constant–dimension codes for random linear network coding
called spread codes. This is a family of optimal codes with maximum minimum distance.
A spread code is constructed starting from the algebra defined by the companion matrix
of an irreducible polynomial. We give a minimum distance decoding algorithm that is
particularly efficient when the dimension of the codewords is small. The decoding algo-
rithm takes advantage of the structure of the algebra and it uses an original result on
minors of a matrix and the factorization of polynomials over finite fields.