Bounds on Covering Radius of Some Codes Over F₂ + uF₂ + vF₂ + uvF₂

• Main Authors: Fanghui Ma, Jian Gao
• Format: Article
• Language: English
• Published: IEEE 2021-01-01
• Series: IEEE Access
• Subjects: Covering radius; repetition codes; simplex codes; MacDonald codes
• Online Access: https://ieeexplore.ieee.org/document/9385140/

Let <inline-formula> <tex-math notation="LaTeX">$R=\mathbb {F}_{2}+u\mathbb {F}_{2}+v\mathbb {F}_{2}+uv\mathbb {F}_{2}$ </tex-math></inline-formula> be a finite non-chain ring, where <inline-formula> <tex-math notation="LaTeX">$u^{2}=u$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$v^{2}=v$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$uv=vu$ </tex-math></inline-formula>. We give the lower and upper bounds on the covering radius of different types of repetition codes for Chinese Euclidean distance over <inline-formula> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula>. Furthermore, we determine the upper bound on the covering radius of block repetition codes, simplex codes of types <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula>, MacDonald codes of types <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> for Chinese Euclidean distance over <inline-formula> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula>.