Linear and nonlinear codes provide infinite families of 3-designs
Infinite families of 3-designs from linear and nonlinear codes
Information Theory
Summary
Certain types of codes used for error correction can be connected to special combinatorial structures called 3-designs, which help organize data in balanced ways. The authors study both linear and nonlinear codes defined over specific mathematical fields and show that the patterns formed by their codewords correspond to these 3-designs. This work expands the known examples of such codes, particularly for nonlinear ones where examples are rare. They also find practical uses of these codes in quantum error correction and efficient data repair techniques.
What this means in practice
- •For quantum communication engineers: Build quantum error-correcting codes with specific parameters for entanglement-assisted quantum communication using the constructed dual codes.
- •For distributed storage engineers: Design locally repairable codes with locality three that meet the Singleton-type bound for efficient data recovery in distributed storage systems.
Authors
Shiyan Xiong, Xiaoqiang Wang, Dabin Zheng, Qinqin Ji
Abstract
The connection between coding theory and combinatorial $t$-designs is an important research topic at the intersection of coding theory and combinatorics. Let $q=p^m$, where $p$ is an odd prime and $m\geq 2$. In this paper, we investigate a class of linear codes $\mathcal{C}$ over $\mathbb{F}_{q^2}$ and their connection with combinatorial $3$-designs. By analyzing the relevant structural properties of $\mathcal{C}$ and $\mathcal{C}^{\perp}$, we show that the supports of the codewords of every nonzero weight in $\mathcal{C}$ and {the supports of the codewords of weight $4$ in $\mathcal{C}^{\perp}$} form $3$-designs. We further investigate a class of nonlinear codes $\mathcal{C}_2$ associated with $\mathcal{C}$, and prove that the supports of the codewords of every nonzero Hamming weight in $\mathcal{C}_2$ also form $3$-designs. These results identify further classes of linear and nonlinear codes whose codewords support combinatorial $3$-designs. In particular, the nonlinear case provides additional examples of codes supporting $3$-designs, a topic for which relatively few results are currently available. As applications, we construct from $\mathcal{C}^{\perp}$ an entanglement-assisted quantum error-correcting code with parameters $[[q+1,q-3,4;4]]_q$. We also prove that $\mathcal{C}$ is an all-symbol locally repairable code with locality $3$. Furthermore, we show that the code $\mathcal{C}$ {meets the Singleton-type bound for locally repairable codes} and hence is optimal in some cases.