基于置换编码的遗传算法的线性码最小距离计算

无穷范数下的频率排列阵列译码

Decoding Frequency Permutation Arrays under Infinite norm

具有长度$n=mλ$和距离$d$的频率置换阵列（FPA）是在$m$符号上的多集合上的置换的集合，其中每个符号精确地出现$λ$次，并且阵列中的任何两个元素之间的距离至少为$d$。FPA推广了置换数组的概念。在距离度量$\ell_\infty$ -范数下，本文首先证明了FPA的大小的上下界。然后给出了一种具有高效编解码能力的FPA结构。此外，我们表明我们的设计是本地可解码的，即，我们可以通过读取至多$λ+1$符号来解码消息位，这对于私人信息检索具有有趣的应用。

Efficient Encoding Algorithm of Binary and Non-Binary LDPC Codes Using Block Triangulation

We propose an efficient encoding algorithm for the binary and non-binary low-density parity-check codes. This algorithm transforms the parity part of the parity-check matrix into a block triangular matrix with low weight diagonal submatrices by row and column permutations in the preprocessing stage. This algorithm determines the parity part of a codeword by block back-substitution to reduce the encoding complexity in the encoding stage. Numerical experiments show that this algorithm has a lower encoding complexity than existing encoding algorithms. Moreover, we show that this algorithm encodes…

喷泉码与可逆矩阵

Fountain Codes and Invertible Matrices

本文讨论喷泉码，特别是其编码矩阵，这是要求可逆的。证明了编码矩阵诱导置换的结果。而且，结果是编码矩阵形成具有乘法运算的组。编码是一种变换，其减少初始高熵输入向量的熵。构造了一种特殊的编码矩阵，与理想孤子分布所构造的矩阵相比，这种编码矩阵的熵缩减效果更好。实验结果与熵减少。

置换空间中的压缩

Compression in the Space of Permutations

我们研究了以置换形式的数据的有损压缩（信源编码）。这个问题在有序数据或排序的存储以及排序算法的分析中有直接的应用。分析了均匀分布下置换空间的率失真特性，以及恢复后允许有界失真的最小可达压缩率。我们的分析是关于不同的实际和有用的失真措施，包括肯德尔-陶距离，斯皮尔曼的foottrule，切比雪夫距离和反演- $\ell_1$距离。我们建立等价的源代码设计在一定的失真，并显示简单的显式代码设计，产生低编码/解码的复杂性和渐近最优。最后，我们表明，对于Mallow模型，一个流行的非均匀排列模型的置换空间，熵和最大失真在零速率是远远低于均匀的同行，这激励了未来的设计高效的压缩方案，这种模式。

Permutation-invariant quantum coding for quantum deletion channels

Quantum deletions, which are harder to correct than erasure errors, occur in many realistic settings. It is therefore pertinent to develop quantum coding schemes for quantum deletion channels. To date, not much is known about which explicit quantum error correction codes can combat quantum deletions. We note that {\em any} permutation-invariant quantum code that has a distance of $t+1$ can correct $t$ quantum deletions for any positive integer $t$ in both the qubit and the qudit setting. Leveraging on coding properties of permutation-invariant quantum codes under erasure errors, we derive corr…

New Theoretical Bounds and Constructions of Permutation Codes under Block Permutation Metric

Permutation codes under different metrics have been extensively studied due to their potentials in various applications. Generalized Cayley metric is introduced to correct generalized transposition errors, including previously studied metrics such as Kendall's $τ$-metric, Ulam metric and Cayley metric as special cases. Since the generalized Cayley distance between two permutations is not easily computable, Yang et al. introduced a related metric of the same order, named the block permutation metric. Given positive integers $n$ and $d$, let $\mathcal{C}_{B}(n,d)$ denote the maximum size of a pe…

基于线性约束置换矩阵的LP可译码置换码

LP Decodable Permutation Codes based on Linearly Constrained Permutation Matrices

提出了一组线性约束置换矩阵，用于构造一类置换码。利用对置换矩阵施加的线性约束，我们可以将所提出的置换码类的最小欧氏距离解码问题表示为线性规划（LP）问题。这类置换码的主要特征，称为LP可解码置换码，是这种LP可解码性。证明了这类置换码的LP译码性能是由码的码多面体的顶点来表征的。讨论了两种类型的线性约束：一种是结构化约束，另一种是随机约束。结构化约束，如纯对合导致一个有效的编码算法。另一方面，随机约束使我们能够使用概率方法来分析几个代码属性，如平均基数和平均重量分布。

Improved Rank-Modulation Codes for DNA Storage with Shotgun Sequencing

We study permutations over the set of $\ell$-grams, that are feasible in the sense that there is a sequence whose $\ell$-gram frequency has the same ranking as the permutation. Codes, which are sets of feasible permutations, protect information stored in DNA molecules using the rank-modulation scheme, and read using the shotgun sequencing technique. We construct systematic codes with an efficient encoding algorithm, and show that they are optimal in size. The length of the DNA sequences that correspond to the codewords is shown to be polynomial in the code parameters. Non-systematic with large…

基于彩色排列的最优量子车辆路径问题车辆路由

Optimal, Qubit-Efficient Quantum Vehicle Routing via Colored-Permutations

本文提出一种全局位置彩色排列编码，用于解决具有容量限制的车辆路径问题，通过K辆车辆选择不相交的排列，形成完整的n×n排列矩阵，以高效利用量子资源，减少逻辑量子比特成本。

量子匿名秘密分享与置换不变码

Quantum Anonymous Secret Sharing with Permutation Invariant Codes

本文提出一种利用置换不变量子纠错码和匿名量子传输算法实现发送者匿名的量子匿名秘密分享方案，并通过量子条件最小熵量化信息泄露。