Lee weight distributions of trace codes over $\mathbb{F}_q+u\mathbb{F}_q$ from irreducible cyclic codes

Abstract

The construction of two or few-weight codes from trace codes over the ring $\mathbb{F}_q+u\mathbb{F}_q$, where $u^2=0$, was recently presented in [4]. For such construction, the defining sets for the trace codes are given in terms of cyclotomic classes, and for some of these classes, it is shown that it is possible to obtain the Lee weight distributions of the corresponding trace codes. Motivated by this construction, and by the $p$-ary semiprimitive irreducible cyclic codes over a prime field $\mathbb{F}_p$, the Lee weight distributions of an infinite family of $p$-ary three-weight codes from trace codes over the ring $\mathbb{F}_p+u\mathbb{F}_p$, was recentely found in [11]. In this work, we prove that the Lee weight distribution problem for the trace codes constructed in accordance with either [4] or [11], is equivalent to the weight distribution problem for the irreducible cyclic codes. With this equivalence in mind, and by using the already known weight distributions of an infinite family irreducible cyclic codes (semiprimitive and not semiprimitive), we follow the open problem suggested in the Conclusion of [11] to determine the Lee weight distribution of an infinite family of trace codes over the ring $\mathbb{F}_q+u\mathbb{F}_q$, that includes the infinite family found in [11].

Received: 16 March 2021 Accepted: 31 October 2022