We provide a novel framework to study subspace codes for non-coherent communications in wireless networks. To this end, an analog operator channel is defined with inputs and outputs being subspaces of $\mathbb{C}^n$. Then a certain distance is defined to capture the performance of subspace codes in terms of their capability to recover from interference and rank-deficiency of the network. We also study the robustness of the proposed model with respect to an additive noise. Furthermore, we propose a new approach to construct subspace codes in the analog domain, also regarded as Grassmann codes, by leveraging polynomial evaluations over finite fields together with characters associated to finite fields that map their elements to the unit circle in the complex plane. The constructed codes, referred to as character-polynomial (CP) codes, are shown to perform better comparing to other existing constructions of Grassmann codes in terms of the trade-off between the rate and the normalized minimum distance, for a wide range of values for $n$.
Sparse random linear network coding (SRLNC) used as a class of erasure codes to ensure the reliability of multicast communications has been widely investigated. However, an exact expression for the decoding success probability of SRLNC is still unknown, and existing expressions are either asymptotic or approximate. In this paper, we derive an exact expression for the decoding success probability of SRLNC. The key to achieving this is to propose a criterion that a vector is contained in a subspace. To obtain this criterion, we construct a basis of a subspace, with respect to this basis, the coo…
An efficient analog network coding transmission protocol is proposed in this letter for a MIMO two way cellular network. Block signal alignment is first proposed to null the inter-user interference for multi-antenna users, which makes the dimensions of aligned space larger compared with the existing signal alignment. Two algorithms are developed to jointly design the precoding matrices at the relay and BS for outage optimization. Especially, the last algorithm is designed to maximize the effective channel gain to the effective noise gain ratio. The performance of this transmission protocol is …
We consider the problem of efficient ultra-massive multiple-input multiple-output (UM-MIMO) data detection in terahertz (THz)-band non-orthogonal multiple access (NOMA) systems. We argue that the most common THz NOMA configuration is power-domain superposition coding over quasi-optical doubly-massive MIMO channels. We propose spatial tuning techniques that modify antenna subarray arrangements to enhance channel conditions. Towards recovering the superposed data at the receiver side, we propose a family of data detectors based on low-complexity channel matrix puncturing, in which higher-order d…
Constructions and Bounds for Mixed-Dimension Subspace Codes
有限射影空间中的码被提出来用于随机线性网络编码中的差错控制。得到的所谓的\emph{Main Problem of Subspace Coding}是确定具有最小子空间距离$d$的$\operatorname{PG}(v-1,\mathbb{F}_q)$中的码的最大大小$A_q(v,d)$。在这里，我们完全解决了$d\ge v-1$的这个问题。对于$d=v-2$，我们提出了一些改进的界限，并确定了$A_q(5,3)=2q^3+2$（所有$q$），$A_2(7,5)=34$。我们还提供了一个已知的$A_q(v,2)$的测定的博览会，和一个表的确切结果和界限的数字$A_2(v,d)$，$v\leq 7$。
A Rank-Metric Approach to Error Control in Random Network Coding
随机线性网络编码中的差错控制问题是从矩阵的角度来解决的，这与Kötter和Kschischang的子空间角度密切相关。研究了一类常数维子空间码。结果表明，在这类代码可以很容易地构造秩度量码，同时保持其距离属性。此外，它表明，这种子空间码的最小距离解码可以重新制定为一个广义的解码问题的秩度量码的错误的部分信息是可用的。该部分信息可以是擦除（知道错误位置但不知道其值）和偏差（知道错误值但不知道其位置）的形式。考虑擦除和偏差（当它们发生时）严格地增加了代码的纠错能力：如果$μ$擦除和$δ$偏差发生，那么秩$t$的错误总是可以被纠正，只要$2t \leq d - 1 + μ+ δ$，其中$d$是代码的最小秩距离。针对一类重要的最大秩距离码--Gabidulin码，提出了一种有效的译码算法，该算法能够充分利用擦除和偏差。在传输长度为$M$ over $F_q$的$n$分组的网络编码应用中，解码算法的复杂度由扩展字段$F_{q^n}$中的$O(dM)$运算给出。
Subspace codes, especially cyclic constant subspace codes, are of great use in random network coding. Subspace codes can be constructed by subspaces and subspace polynomials. In particular, many researchers are keen to find special subspaces and subspace polynomials to construct subspace codes with the size and the minimum distance as large as possible. In [14], Roth, Raviv and Tamo constructed several subspace codes using Sidon spaces, and it is proved that subspace codes constructed by Sidon spaces has the largest size and minimum distance. In [12], Niu, Yue and Wu extended some results of […
Building upon the application of flags to network coding introduced by Liebhold, Nebe, and Vazquez-Castro, we develop a variant of this coding technique that uses degenerate flags. The information set is a metric affine space isometric to the space of upper triangular matrices endowed with the flag rank metric. This suggests the development of a theory for flag rank metric codes in analogy to the rank metric codes used in linear subspace coding.
One of the most fundamental topics in subspace coding is to explore the maximal possible value ${\bf A}_q(n,d,k)$ of a set of $k$-dimensional subspaces in $\mathbb{F}_q^n$ such that the subspace distance satisfies $\operatorname{d_S}(U,V) = \dim(U+V)-\dim(U\cap V) \geq d$ for any two different $k$-dimensional subspaces $U$ and $V$ in this set. In this paper, we propose a construction for constant dimension subspace codes by inserting a composite structure composing of an MRD code and its sub-codes. Its vast advantage over the previous constructions has been confirmed through extensive examples…
