Abstract
A generator matrix of a linear code C over GF(q) is also a matrix of the same rank k over any extension field GF(qℓ) and generates a linear code of the same length, same dimension and same minimum distance over GF(qℓ), denoted by C(q|qℓ) and called a lifted code of C. Although C and their lifted codes C(q|qℓ) have the same parameters, they have different weight distributions and different applications. Few results about lifted linear codes are known in the literature. This paper proves some fundamental theory for lifted linear codes, and studies the 2-designs of the lifted projective Reed–Muller codes, lifted Hamming codes and lifted Simplex codes. In addition, this paper settles the weight distributions of the lifted Reed–Muller codes of certain orders, and investigates the 3-designs supported by these lifted codes. As a by-product, an infinite family of three-weight projective codes over GF(4) is obtained.
| Original language | English |
|---|---|
| Pages (from-to) | 1575-1595 |
| Number of pages | 21 |
| Journal | Designs, Codes, and Cryptography |
| Volume | 93 |
| Issue number | 6 |
| Early online date | 25 Dec 2024 |
| DOIs | |
| Publication status | Published - Jun 2025 |
Bibliographical note
Publisher Copyright:© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.
Keywords
- Hamming code
- Lifted code
- Reed–Muller code
- Simplex code
- t-Design
Fingerprint
Dive into the research topics of 'The support designs of several families of lifted linear codes'. Together they form a unique fingerprint.Cite this
- APA
- Author
- BIBTEX
- Harvard
- Standard
- RIS
- Vancouver