Non-standard linear recurring sequence subgroups and automorphisms of irreducible cyclic codes

Let ${{\mathcal{U}}_n}$ be the multiplicative group of order n in the splitting field ${{\mathbb{F}}_{{q^m}}}$ of x
ⁿ
− 1 over a finite field ${{\mathbb{F}}_q}$. Any map of the form x → cx
^t
with $c \in {{\mathcal{U}}_n}$ and t = q
ⁱ
, 0 ≤ i < m, is ${{\mathbb{F}}_q}$-linear on ${{\mathbb{F}}_{{q^m}}}$ and fixes ${{\mathcal{U}}_n}$ set-wise; maps of this type will be called standard. Occasionally there are other, non-standard ${{\mathbb{F}}_q}$-linear maps on ${{\mathbb{F}}_{{q^m}}}$ fixing ${{\mathcal{U}}_n}$ set-wise, and in that case we say that the pair (n, q) is non-standard. We show that an irreducible cyclic code of length n over ${{\mathbb{F}}_q}$ has “extra” permutation automorphisms (others than the standard permutations generated by the cyclic shift and the Frobenius mapping that every such code has) precisely when the pair (n, q) is non-standard; we refer to such irreducible cyclic codes as non-standard or NSIC-codes. In addition, we relate these concepts to that of a non-standard linear recurring sequence subgroup as investigated in a sequence of papers by Brison and Nogueira. We present several families of NSIC-codes, and two constructions called “lifting” and “extension” to create new NSIC-codes from existing ones. We show that all NSIC-codes of dimension two can be obtained in this way, thus completing the classification for this case started by Brison and Nogueira.