Subfield Codes of Several Few-Weight Linear Codes Parameterized by Functions and Their Consequences
Li Xu, Cuiling Fan, Sihem Mesnager, Rong Luo, Haode Yan · IEEE Transactions on Information Theory · 2023
Subfield codes of linear codes over finite fields have recently received much attention since they can produce optimal codes, which may have applications in secret sharing, authentication codes and association schemes. In this paper, we first present a construction framework of 3-dimensional linear codesCf,gover Fqmparameterized by any two functionsf,gover Fqm, and then study the properties of six types ofCf,g, its punctured codeC*f,gand their corresponding subfield codes over Fq. The classification ofCf,gis based on special choices off,gas trace function, norm function, almost bent function, Boolean bent function or a combination of these functions. For the first two types ofCf,g, we explicitly determine the weight distributions and dualities ofCf,g, C*f,gand their subfield codes over Fq. The remaining four types ofCf,gare restricted toq= 2, and the weight distributions and dualities of the subfields codeC(q)f,gandC*f,g(q)are completely determined. Most of the resultant linear codes (over Fqmor over Fq) have few weights. Some of them are optimal and some have the best-known parameters according to the tables maintained at http://www.codetables.de. In fact, 16 infinite families of optimal linear codes are produced in this paper. As a byproduct, a family of [24m-2, 2m+ 1, 24m-3] quaternary Hermitian self-orthogonal codes are obtained withm≥ 2. As an application, we present several infinite families of 2-designs or 3-designs with some of the codes presented in this paper.