Linear codes tied to shapes improve error correction and efficiency
Linear Codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}$ associated with Simplicial Complexes, Their Gray Images, and Subfield Codes
Information Theory
Summary
Coding theory helps send data accurately despite errors. The authors use special shapes called simplicial complexes to build new families of codes over a special type of number system. These codes have good error detection and correction properties, measured by their weight distributions, and can be translated into codes over common finite fields. They also find conditions that ensure the codes are minimal, meaning simpler and efficient. This work extends coding techniques by linking geometry-like structures to practical code designs.
What this means in practice
- •For communication engineers: Design more efficient error-correcting codes with proven minimality and optimal distance properties for data transmission systems.
- •For storage system designers: Develop storage codes that use simpler constructions from ring-based codes for better error detection and correction.
Authors
Ankit Yadav, Akanksha Tiwari, Ritumoni Sarma
Abstract
In recent years, simplicial complexes have gained considerable attention as a useful tool for constructing distance-optimal codes over finite fields. In this article, we construct four infinite families of linear codes over the ring $\mathcal{R}=\mathbb{F}_{q}+u\mathbb{F}_{q}$ with $u^2=0$ using simplicial complexes with one or two maximal elements, and completely determine their Lee weight distributions via exponential-sum techniques. By employing a Gray map on $\mathcal{R}$, we obtain infinite families of distance-optimal codes over $\mathbb{F}_{q}$, including a near-Griesmer family, and establish sufficient conditions for their minimality. Furthermore, we investigate the corresponding subfield codes and derive sufficient conditions for their distance-optimality and minimality, yielding infinite families of Griesmer and near-Griesmer codes.