<p>Multi-twisted (MT) codes are an important class of linear codes. Their advantageous algebraic structures enable error-correcting coding schemes to be precisely adapted to diverse communication environments, thereby substantially enhancing communication efficiency. Moreover, MT codes contain some linear codes with better or optimal parameters and play an important role in establishing some new bounds. According to asymptotic goodness of MT codes, it facilitates the design of high-performance error-correcting codes, supports the development and optimization of practical coding schemes that effectively balance error correction capability, code rate, and complexity. In this paper, by trace representations and Gauss sums, we determine homogeneous weight distributions of MT codes over the finite chain ring <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {F}_q+u\mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo>+</mo> <mi>u</mi> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$u^{2}=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. As applications, we construct several minimal linear codes and distance-optimal linear codes over <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {F}_q+u\mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo>+</mo> <mi>u</mi> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we also give a class of strongly regular graphs from two-homogeneous weight MT codes.</p>

Homogeneous weight distributions of multi-twisted codes over a finite chain ring

期刊论文
发表日期：

2026年6月2日

查看全文

Linlin Yu,
Jian Gao,
Jiafu Mi

摘要

Multi-twisted (MT) codes are an important class of linear codes. Their advantageous algebraic structures enable error-correcting coding schemes to be precisely adapted to diverse communication environments, thereby substantially enhancing communication efficiency. Moreover, MT codes contain some linear codes with better or optimal parameters and play an important role in establishing some new bounds. According to asymptotic goodness of MT codes, it facilitates the design of high-performance error-correcting codes, supports the development and optimization of practical coding schemes that effectively balance error correction capability, code rate, and complexity. In this paper, by trace representations and Gauss sums, we determine homogeneous weight distributions of MT codes over the finite chain ring $\mathbb {F}_q+u\mathbb {F}_q$ $F_{q} + u F_{q}$ , where $u^{2}=0$ $u^{2} = 0$ . As applications, we construct several minimal linear codes and distance-optimal linear codes over $\mathbb {F}_q+u\mathbb {F}_q$ $F_{q} + u F_{q}$ . Moreover, we also give a class of strongly regular graphs from two-homogeneous weight MT codes.