In the world of coding, there are different types of codes that help us store and transmit information. Linear codes are one such type, where each codeword is a linear combination of basis vectors. Minimal codes are a particular type of linear code where all non-zero codewords are minimal, meaning their support is not properly contained in the support of another codeword. In other words, these codewords have no redundant information.
The minimum possible length of a k-dimensional linear code over Fq is denoted by m(k, q). This article sets out to determine m(7, 2), m(8, 2), and m(9, 2) as well as classify all codes attaining m(k, 2) for k ≤ 7 and those attaining m(9, 2). For m(11, 2) and m(12, 2), improved upper bounds are given.
The author reveals that in many cases, extremal codes have the property that the weights of all codewords are divisible by some constant ∆ > 1. This leads to a study of the minimum lengths of minimal codes where the weights of the codewords are divisible by ∆. By demystifying complex concepts and using everyday language, the author makes it easy for readers to understand the concepts and their significance in linear code design.
The article provides a detailed summary of the findings without oversimplifying the concepts. It captures the essence of the research without sacrificing thoroughness, making it accessible to an average adult with some background knowledge in coding theory. The use of headers for each section helps readers navigate the content more efficiently. Overall, the article is well-structured and easy to comprehend, providing valuable insights into the world of linear codes and their properties.
Generating Minimal Divisible Codes in High-Dimensional Lattices
