Network Coding Based on Byte-wise Circular Shift and Integer Addition

Semilinear Transformations in Coding Theory: A New Technique in Code-Based Cryptography

This paper presents a new technique for disturbing the algebraic structure of linear codes in code-based cryptography. This is a new attempt to exploit Gabidulin codes in the McEliece setting and almost all the previous cryptosystems of this type have been completely or partially broken. To be specific, we introduce the so-called semilinear transformation in coding theory, which is defined through an $\mathbb{F}_q$-linear automorphism of $\mathbb{F}_{q^m}$, then apply them to construct a public key encryption scheme. Our analysis shows that this scheme can resist all the existing distinguisher…

Optimal Linear Coding Schemes for the Secure Decentralized Pliable Index Coding Problem

We study the secure decentralized Pliable Index CODing (PICOD) problem with circular side information sets at the users. The security constraint forbids every user to decode more than one message while a decentralized setting means there is no central transmitter in the system. Compared to the secure but centralized version of the problem, a converse bound from one of our previous works showed a factor of three difference in optimal code length under the constraint of linear encoding. In this paper, we first list the linearly infeasible cases, that is, problems where no linear code can simulta…

Approximate Capacity of Index Coding for Some Classes of Graphs

For a class of graphs for which the Ramsey number $R(i,j)$ is upper bounded by $ci^aj^b$, for some constants $a,b,$ and $c$, it is shown that the clique covering scheme approximates the broadcast rate of every $n$-node index coding problem in the class within a multiplicative factor of $c^{\frac{1}{a+b+1}} n^{\frac{a+b}{a+b+1}}$ for every $n$. Using this theorem and some graph theoretic arguments, it is demonstrated that the broadcast rate of planar graphs, line graphs and fuzzy circular interval graphs is approximated by the clique covering scheme within a factor of $n^{\frac{2}{3}}$.

Circular-shift Linear Network Codes with Arbitrary Odd Block Lengths

Circular-shift linear network coding (LNC) is a class of vector LNC with low encoding and decoding complexities, and with local encoding kernels chosen from cyclic permutation matrices. When $L$ is a prime with primitive root $2$, it was recently shown that a scalar linear solution over GF($2^{L-1}$) induces an $L$-dimensional circular-shift linear solution at rate $(L-1)/L$. In this work, we prove that for arbitrary odd $L$, every scalar linear solution over GF($2^{m_L}$), where $m_L$ refers to the multiplicative order of $2$ modulo $L$, can induce an $L$-dimensional circular-shift linear sol…

最大恢复性：编码理论的纽带

Maximal Recoverability: A Nexus of Coding Theory

本文综述了最大恢复性码家族，探讨了MR LRCs和GCs的恢复保证、构造方法及其与图结构刚性问题的联系。

Delay-Complexity Trade-off of Random Linear Network Coding in Wireless Broadcast

In wireless broadcast, random linear network coding (RLNC) over GF(2^L) is known to asymptotically achieve the optimal completion delay with increasing L. However, the high decoding complexity hinders the potential applicability of RLNC schemes over large GF(2^L). In this paper, a comprehensive analysis of completion delay and decoding complexity is conducted for field-based systematic RLNC schemes in wireless broadcast. In particular, we prove that the RLNC scheme over GF(2) can also asymptotically approach the optimal completion delay per packet when the packet number goes to infinity. Moreo…

The Impact of Phase Fading on the Dirty Paper Channel

The impact of phase fading on the classical Costa dirty paper coding channel is studied. We consider a variation of this channel model in which the amplitude of the interference sequence is known at the transmitter while its phase is known at the receiver. Although the capacity of this channel has already been established, it is expressed using an auxiliary random variable and as the solution of a maximization problem. To circumvent the difficulty evaluating capacity, we derive alternative inner and outer bounds and show that the two expressions are to within a finite distance. This provide an…

Invariance: a Theoretical Approach for Coding Sets of Words Modulo Literal (Anti)Morphisms

Let $A$ be a finite or countable alphabet and let $θ$ be literal (anti)morphism onto $A^*$ (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under $θ$ ($θ$-invariant for short).We establish an extension of the famous defect theorem. Moreover, we prove that for the so-called thin $θ$-invariant codes, maximality and completeness are two equivalent notions. We prove that a similar property holds in the framework of some special families of $θ$-invariant codes such as prefix (bifix) codes, codes with a finite dec…

Permute & Add Network Codes via Group Algebras

A class of network codes have been proposed in the literature where the symbols transmitted on network edges are binary vectors and the coding operation performed in network nodes consists of the application of (possibly several) permutations on each incoming vector and XOR-ing the results to obtain the outgoing vector. These network codes, which we will refer to as "permute-and-add" network codes, involve simpler operations and are known to provide lower complexity solutions than scalar linear network codes. The complexity of these codes is determined by their "degree" which is the number of …

多项式框架的结构保持纠错码

Structure-Preserving Error-Correcting Codes for Polynomial Frames

针对多项式框架传输中的静默数据损坏，提出在原始多项式环内添加系统冗余的结构保持纠错层，无需往返或格式转换，以极低开销实现高可靠纠错。

The Serializability of Network Codes

Network coding theory studies the transmission of information in networks whose vertices may perform nontrivial encoding and decoding operations on data as it passes through the network. The main approach to deciding the feasibility of network coding problems aims to reduce the problem to optimization over a polytope of entropic vectors subject to constraints imposed by the network structure. In the case of directed acyclic graphs, these constraints are completely understood, but for general graphs the problem of enumerating them remains open: it is not known how to classify the constraints im…