Bounding the sum of µ-invariants on pair symbol weights over some irreducible codes

Official URL: https://dx.doi.org/10.1016/j.ifacol.2024.10.185

Let Fq be a finite field with q = 3 mod 4. Let w be a primitive element of F*q4 and let C(w) be the irreducible cyclic code of length (q4 - 1)/(q - 1) of dimension 4 defined by w. The symbol-pair weight enumerator of C(w) is determined exactly by the invariant µ(w) introduced in Zhu et al. (2022). The determination of good upper and lower bounds on this invariant remains an open problem. Let W be the set of all primitive elements of F*q4. In this paper, by using algebraic and arithmetical methods, particularly those from the theory of algebraic curves over finite fields, we derive effective upper and lower bounds on ?w?W µ(w).