Year: 2024
Type: article
Abstract: One of the main challenges in coding theory is finding optimal binary linear codes. A binary linear code 𝐶 = [𝑛, 𝑘, 𝑑] is considered optimal if no other linear code with the same length 𝑛 and dimension 𝑘 has a greater minimum distance 𝑑. Theoretical bounds provide limits for the minimum distance 𝑑 of a linear code given 𝑛 and 𝑘. For instance, the lower bound indicates that a linear code with a specified minimum distance exists, while the upper bound can be determined using results such as the Griesmer bound. Achieving an optimal binary linear code involves constructing a code where the lower and upper bounds of 𝑑 coincide. In previously published tables, Andries Brouwer presented some optimal binary linear codes for 1 ≤ 𝑛 ≤ 256 and 𝑘 ≤ 𝑛. Markus Grassl further improved these results by detailing construction processes for codes achieving the lower bound. In this paper, we introduce new optimal binary linear codes using novel construction methods applied to existing codes. Specifically, we developed a construction algorithm that extends the list of optimal binary linear codes for 257 ≤𝑛 ≤ 300 with 𝑘 ≤ 7.
Authors: Skezeer John B. Paz, Ederlina G. Nocon
Institution: De La Salle University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W4403375195