Computational Construction of New Projective Linear Codes of Dimension 4 over GF(11)

Abstract

This article investigates recent developments in computational research aimed at discovering new linear codes of dimension  over the Galois field . We define an  code as a -dimensional subspace of  with minimum Hamming distance . Utilizing advanced computational techniques and algorithms, we study projective linear codes of dimension  derived from caps of degree , , and  containing an elliptic quadric. Our analysis focuses on linear codes within the length range , representing the potential minimum Hamming distance for these codes across various  values. Through the computation of their weight enumerators, we introduce 969 new projective linear codes over . Furthermore, we establish the following lower bounds for the maximum size of caps of degree  in : ,  and . These results refine the known boundaries of coding theory parameters and enhance the reliability of error correction in high-order fields.