When the extension property does not hold for vector space alphabets

In our previous work [4] we showed that the extension property holds for linear codes over a finite module alphabet with respect to the symmetrized weight composition if and only if the socle of the alphabet is cyclic. For alphabets with a noncyclic socle we proved the existence of an unextendable isometry. However, our construction was implicit. This paper deals with a particular case of an alphabet with a noncyclic socle, a vector space alphabet. Here we give an explicit construction of a code with an unextendable isometry, see Section 3.1. Moreover, the code and the isometry have nice additional properties, see Proposition 4 and Proposition 5. Unlike the general case, where we do not know the length of the constructed code, for vector space alphabet we show that the length can be relatively small. For the symmetrized weight composition built on the trivial group we give the lover bound on the length of a code with an unextendable isometry, see Proposition 6. In [4] we also introduced the notion of G-pseudo-injectivity for finite modules. It is related to the extension property for codes of the length 1. We characterize this property in the context of vector spaces, see Proposition 2.