Ismael Gutierrez, Ivan Molina
In this paper we construct, using GAP System for Computational Discrete Algebra, some cyclic subspace codes, specially an optimal code over the finite field F_{2^{10}}. Further we present a definition and an example of the $q$-analogous of a $m$-quasi-cyclic subspace code over F_{2^{8}}.
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Cyclic subspace codes gained a lot of attention especially because they may be used in random network coding for correction of errors and erasures. Roth, Raviv and Tamo in 2018 established a connection between cyclic subspace codes (with certain parameters) and Sidon spaces. These latter objects were introduced by Bachoc, Serra and Zémor in 2017 in relation with the linear analogue of Vosper's Theorem. This connection allowed Roth, Raviv and Tamo to construct large classes of cyclic subspace codes with one or more orbits. In this paper we will investigate cyclic subspace codes associated to a …
Full weight spectrum one-orbit cyclic subspace codes
本文研究循环子空间码框架下的全重量谱单轨道循环子空间码，利用数论结果与组合论证，给出了该类码的完整分类。
Using multi-orbit cyclic subspace codes for constructing optical orthogonal codes
本文提出利用多轨道循环子空间码和有限域扩张的乘法结构构造大型光正交码，不同于以往基于组合与加法结构的方法，从而获得了具有不同参数的新光正交码类。
Asymptotically optimal cyclic subspace codes
本文提出一种新技术，构造具有大基数和规定最小距离的循环子空间码，在特定参数下所得码尺寸超越已知构造，并渐近达到 Johnson 型界 II。
On one-orbit cyclic subspace codes of $\mathcal{G}_q(n,3)$
本文研究三维单轨道循环子空间码，将其分为三个族，并重点分析后两个族中不等价码的性质，以推进循环子空间码的分类研究。
Cyclic codes and cyclically covering subspaces
本文研究了循环覆盖子空间的余维数，通过常循环码和不可约循环码的方法，推导了循环覆盖子空间的下界并给出了若干族n使得余维数为零。
Constructions of Constant Dimension Subspace Codes
本文研究了随机网络编码中的常维子空间码，构造了最小距离为 $2k-2$ 和 $2k$ 的大型循环常维子空间码，并推广了现有文献中的相关结果。
The distance distribution of a code is the vector whose $i^\text{th}$ entry is the number of pairs of codewords with distance $i$. We investigate the structure of the distance distribution for cyclic orbit codes, which are subspace codes generated by the action of $\mathbb{F}_{q^n}^*$ on an $\mathbb{F}_q$-subspace $U$ of $\mathbb{F}_{q^n}$. We show that for optimal full-length orbit codes the distance distribution depends only on $q,\,n$, and the dimension of $U$. For full-length orbit codes with lower minimum distance, we provide partial results towards a characterization of the distance dist…
Quasi-optimal cyclic orbit codes
本文研究循环轨道码的等价不变量与拟最优性，通过射影直线线性集获得参数新界，并证明任意特征有限域偶数维空间中拟最优码的一般存在性定理。
There is a one-to-one correspondence between $\ell$-quasi-cyclic codes over a finite field $\mathbb F_q$ and linear codes over a ring $R = \mathbb F_q[Y]/(Y^m-1)$. Using this correspondence, we prove that every $\ell$-quasi-cyclic self-dual code of length $m\ell$ over a finite field $\mathbb F_q$ can be obtained by the {\it building-up} construction, provided that char $(\mathbb F_q)=2$ or $q \equiv 1 \pmod 4$, $m$ is a prime $p$, and $q$ is a primitive element of $\mathbb F_p$. We determine possible weight enumerators of a binary $\ell$-quasi-cyclic self-dual code of length $p\ell$ (with $p$ …
This paper considers a linear quasi-cyclic product code of two given quasi-cyclic codes of relatively prime lengths over finite fields. We give the spectral analysis of a quasi-cyclic product code in terms of the spectral analysis of the row- and the column-code. Moreover, we provide a new lower bound on the minimum Hamming distance of a given quasi-cyclic code and present a new algebraic decoding algorithm.More specifically, we prove an explicit (unreduced) basis of an l\_a l\_b-quasi-cyclic product code in terms of the generator matrix in reduced Gr{ö}bner basis with respect to the position-…
We study orbit codes in the field extension ${\mathbb F}_{q^n}$. First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of ${\mathbb F}_{q^n}$. We then generalize to orbits under the normalizer of the Singer subgroup. In that situation some exceptional cases arise and some open cases remain. Finally we characterize linear isometries between such codes.
Linear quasi-cyclic product codes over finite fields are investigated. Given the generating set in the form of a reduced Gr{ö}bner basis of a quasi-cyclic component code and the generator polynomial of a second cyclic component code, an explicit expression of the basis of the generating set of the quasi-cyclic product code is given. Furthermore, the reduced Gr{ö}bner basis of a one-level quasi-cyclic product code is derived.
Practical implementation of geometric quasi-cyclic LDPC codes
本文首次给出经典有限广义四边形准循环结构的完整显式描述，据此构造了长度达400000的准循环LDPC码族，该码族覆盖广泛传输速率，易于实现且性能接近Shannon极限无错误平层。
In this thesis, we study algebraic coding theory based McEliece-type cryptosystems over quasi-cyclic codes. The main goal of this thesis is to construct a cryptosystem that resists quantum Fourier sampling making it quantum secure. We propose a new variant of Niederreiter cryptosystem over rate $\frac{m-1}{m}$ quasi-cyclic codes which is secure against quantum Fourier sampling due to indistinguishability of the hidden subgroup. The proof of indistinguishability is achieved due to two constraints over automorphism group; small size and large minimal degree. Apart from this cryptosystem, we also…
New constructions of cyclic subspace codes
本文提出两种Sidon空间构造，由此生成具有最小距离2k-2的循环子空间码，其规模优于已有结果，并在n=4k时渐近接近球填充界。
A simple connected graph G is called a p-quasi k-cyclic graph, if there exists a subset S of vertices such that |S|=p, G-S is k-cyclic and there is no a subset S` of V(G) such that |S`|<|S| and G-S` is k-cyclic. The aim of this paper is to characterize graph with maximum values of Zagreb indices among all p-quasi k-cyclic graph of order k>=3.
The ring of cyclic quasi-symmetric functions and its non-Escher subring are introduced in this paper. A natural basis consists of fundamental cyclic quasi-symmetric functions; for the non-Escher subring they arise as toric $P$-partition enumerators, for toric posets $P$ with a total cyclic order. The associated structure constants are determined by cyclic shuffles of permutations. We then prove the following positivity phenomenon: for every non-hook shape $λ$, the coefficients in the expansion of the Schur function $s_λ$ in terms of fundamental cyclic quasi-symmetric functions are nonnegative.…
In this paper, a construction of a pair of "regular" quasi-cyclic LDPC codes as ingredient codes for a quantum error-correcting code is proposed. That is, we find quantum regular LDPC codes with various weight distributions. Furthermore our proposed codes have lots of variations for length, code rate. These codes are obtained by a descrete mathematical characterization for model matrices of quasi-cyclic LDPC codes. Our proposed codes achieve a bounded distance decoding (BDD) bound, or known as VG bound, and achieve a lower bound of the code length.
Cyclic 2-Spreads in $V(6,q)$ and Flag-Transitive Affine Linear Spaces
本文利用 Pauley 和 Bamberg 的多项式方法，完全分类了特征不为 2 或 3 的有限域上 6 维向量空间中循环群传递作用的 2 维子空间铺展，解决了旗传递线性空间分类的公开情形。
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