Combinatorial Classification of Linear Codes Using (s,3)-Arcs of the Galois Plane PG(2,11)

Abstract

Linear codes are the cornerstone of many contemporary technologies such as communication and data storage. But, a full description of the structural features of codes with lengths between the extremes are still not available. In this paper, we provide the total number of linear codes of lengths  which are not isomorphic to each other for the case when . Through the interplay of coding theory and finite geometry, we identify the codes by their relationships to -arcs in projective planes. The paper describes the technique of an iterative expansion in code generation where each step relies on the code created in the previous one using the smallest geometry foundation. Using the described method, a large number of code classes were obtained in lengths 8 to 21. Generator matrices and weight enumerators of these codes were explicitly provided. Interestingly, quantitative evaluation revealed that code with the same [s,k,d] parameter may still have different weight distributions which we explained as the consequence of the different geometric multiplet structures (unisecants bisecants trisecants) involved in the underlying construction process. These types of classification will not only give us very basic data to be used for optimizing codes, but also give code constructors a geometrical platform for the creation of error-correcting codes that maximize performance