Quasi-optimal cyclic orbit codes

Abstract We focus on two aspects of cyclic orbit codes: invariants under equivalence and quasi-optimality. Regarding the first aspect, we establish a connection between the cyclic orbit code generated by a subspace U of $${{\mathbb {F}}}_{q^n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:msup> <mml:mi>q</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:msub> </mml:math> and the associated linear set $$L_{U\times U}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>×</mml:mo> <mml:mi>U</mml:mi> </mml:mrow> </mml:msub> </mml:math> . Relating the size of the linear set to the number of fractions formed by the elements of U allows us to derive new bounds on the parameters of the cyclic orbit code. In the second part, we study a particular family of (quasi-)optimal cyclic orbit codes. With the aid of these codes we establish the existence of quasi-optimal codes in even-dimensional vector spaces over finite fields of any characteristic. Finally, for the particular code family we determine the automorphism groups in various general linear group, depending on the assumed ground field, and their orbits under the Galois group over the prime field.