通过扩张图构造具有良好距离剖面的树状码

Bipartite unique-neighbour expanders via Ramanujan graphs

We construct an infinite family of bounded-degree bipartite unique-neighbour expander graphs with arbitrarily unbalanced sides. Although weaker than the lossless expanders constructed by Capalbo et al., our construction is simpler and may be closer to be implementable in practice due to the smaller constants. We construct these graphs by composing bipartite Ramanujan graphs with a fixed-size gadget in a way that generalizes the construction of unique neighbour expanders by Alon and Capalbo. For the analysis of our construction we prove a strong upper bound on average degrees in small induced s…

高维展开器上的对称自双正交量子码

Symmetric Self-Dual Quantum Codes on High Dimensional Expanders

本研究在高维展开器上构建了首个具有丰富对称性的自双qLDPC码，利用非积单纯复形实现了高性能量子码及容错逻辑门。

New Cosystolic Expanders from Tensors Imply Explicit Quantum LDPC Codes with $Ω(\sqrt{n}\log^kn)$ Distance

In this work we introduce a new notion of expansion in higher dimensions that is stronger than the well studied cosystolic expansion notion, and is termed {\em Collective-cosystolic expansion}. We show that tensoring two cosystolic expanders yields a new cosystolic expander, assuming one of the complexes in the product, is not only cosystolic expander, but rather a collective cosystolic expander. We then show that the well known bounded degree cosystolic expanders, the Ramanujan complexes are, in fact, collective cosystolic expanders. This enables us to construct new bounded degree cosystolic …

Quantum Expander Codes

We present an efficient decoding algorithm for constant rate quantum hypergraph-product LDPC codes which provably corrects adversarial errors of weight $Ω(\sqrt{n})$ for codes of length $n$. The algorithm runs in time linear in the number of qubits, which makes its performance the strongest to date for linear-time decoding of quantum codes. The algorithm relies on expanding properties, not of the quantum code's factor graph directly, but of the factor graph of the original classical code it is constructed from.

在近线性时间内对基于扩展图的码进行达到容量的列表译码

List Decoding Expander-Based Codes up to Capacity in Near-Linear Time

本文提出了基于图正则性引理的新框架，利用近线性时间算法，实现了对基于谱扩展图的码进行达到容量的列表译码和列表恢复，并证明了多种显式构建的新颖组合与算法界限。

线性分组码的误差构建译码

Error-Building Decoding of Linear Block Codes

提出一种仅依赖校验矩阵的线性分组码最大似然软判决译码框架——误差构建译码，通过递归构建局部最优块实现高效搜索，在扩展汉明码上结合离线/在线排除机制大幅降低复杂度。

来自无损立方复形的好量子局部可测试码

Good Quantum Locally Testable Codes from Lossless Cubical Complexes

本文发展高维无损方法，通过局部到整体定理证明足够不平衡的双侧无损四维立方复形可构造渐近好的量子局部可测试码。

On the Complexity of Exact Maximum-Likelihood Decoding for Asymptotically Good Low Density Parity Check Codes

Since the classical work of Berlekamp, McEliece and van Tilborg, it is well known that the problem of exact maximum-likelihood (ML) decoding of general linear codes is NP-hard. In this paper, we show that exact ML decoding of a classs of asymptotically good error correcting codes--expander codes, a special case of low density parity check (LDPC) codes--over binary symmetric channels (BSCs) is possible with an expected polynomial complexity. More precisely, for any bit-flipping probability, $p$, in a nontrivial range, there exists a rate region of non-zero support and a family of asymptotically…

An Efficient Algorithm for Designing Optimal CRCs for Tail-Biting Convolutional Codes

Cyclic redundancy check (CRC) codes combined with convolutional codes yield a powerful concatenated code that can be efficiently decoded using list decoding. To help design such systems, this paper presents an efficient algorithm for identifying the distance-spectrum-optimal (DSO) CRC polynomial for a given tail-biting convolutional code (TBCC) when the target undetected error rate (UER) is small. Lou et al. found that the DSO CRC design for a given zero-terminated convolutional code under low UER is equivalent to maximizing the undetected minimum distance (the minimum distance of the concaten…

基于优多项式的量子局部可恢复码

Quantum Locally Recoverable Codes via Good Polynomials

本研究提出可使用任意优多项式的量子局部可恢复码构造方法，设计了基于仿射一般线性群子群的优多项式构建方式，并结合扩张器混合引理得到无r+1为素数限制的qLRC最小距离下界，推动量子存储编码研究。