2020/07/12 by Yang Liu, Cunsheng Ding, Liu, Yang +3 · 1 citation
Computer Science · Engineering · Mathematics · Medicine · #Algorithm #Block code #Cancer Mechanisms and Therapy #Code (set theory) #Coding theory and cryptography #Computer science #Decoding methods #Discrete mathematics #FOS: Computer and information sciences #Hamming code #Hamming distance #Information Theory (cs.IT) #Linear code #Mathematics #Puncturing #Reed–Muller code #Statistics #cs.IT #graph theory and CDMA systems #math.IT
paper · pdf · doi:10.48550/arxiv.2007.05901
arxiv created 2020/07/12 · openalex publication_date 2020/07/12 · arxiv updated 2020/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The puncturing and shortening technique are two important approaches to constructing new linear codes from old ones. In the past 70 years, a lot of progress on the puncturing technique has been made, and many works on punctured linear codes have been done. Many families of linear codes with interesting parameters have been obtained with the puncturing technique. However, little research on the shortening technique has been done and there are only a handful references on shortened linear codes. The first objective of this paper is to prove some general theory for shortened linear codes. The second objective is to study some shortened codes of the Hamming codes, Simplex codes, some Reed-Muller codes, and ovoid codes. Eleven families of optimal shortened codes with interesting parameters are presented in this paper. As a byproduct, five infinite families of 2-designs are also constructed from some of the shortened codes presented in this paper.