One Class of Error-Correcting Codes

欣一朗 常盤, 和敬 木山, 高弘 山崎

Quantum error-correcting codes have been developed as one of the promising tools for protecting quantum information against quantum errors. A great deal of effort has been made mainly to construct efficient quantum random-error-correcting codes. In this article, we investigate a class of quantum codes capable of correcting quantum burst errors, and present a list of some new good quantum burst-error-correcting codes of length less than or equal to 51.

Ashleigh Meyers

Zhengyi Jiang, Wenhao Liu, Zhongyi Huang +1 more

Analog error-correcting codes (Analog ECCs), introduced by Roth [1], address errors in vector-matrix multiplication arising from analog noise and sparse outliers in in-memory computing. A fundamental open problem concerns the lower bound on the single-error correction threshold $Γ_2(\mathcal C)$ for real $[n,k]$ linear codes with fixed redundancy $r=n-k\geq 2$. Li et al. [2] recently established that for redundancy $r=2$, every real $[n,n-2]$ linear code $\mathcal{C}$ satisfies $Γ_2(\mathcal C)\geq \csc^2(\fracπ{2n})$, resolving an open problem in [1], and showed that, for every fixed $r \geq 2$, there exists a class of $[n,k]$ linear code $\mathcal{C}$ over $\mathbb{R}$ such that $Γ_2(\mathcal{C}) \leq O(n^{1+\frac{1}{r-1}})$. This paper proves the matching converse in [2]. For every $[n,k]$ linear code $\mathcal{C}\subseteq \mathbb R^n$ with fixed redundancy $2\leq r

Richard W. Hamming

Shanqi Pang, Fuyuan Yang, Rong Yan +2 more

This paper presents a method based on orthogonal arrays of constructing pure quaternary quantum error-correcting codes. As an application of the method, some infinite classes of quantum error-correcting codes with distances 2, 3, and 4 can be obtained. Moreover, the infinite class of quantum codes with distance 2 is optimal. The advantage of our method also lies in the fact that the quantum codes we obtain have less items for a basis quantum state than existing ones.