Related papers: Flags of almost affine codes
There is a nice combinatorial formula of P. Beelen and M. Datta for the $r$-th generalized Hamming weight of an affine cartesian code. Using this combinatorial formula we give an easy to evaluate formula to compute the $r$-th generalized…
This paper investigates general properties of codes with the rank metric. We first investigate asymptotic packing properties of rank metric codes. Then, we study sphere covering properties of rank metric codes, derive bounds on their…
This paper considers the problem of simultaneously communicating two messages, a high-security message and a low-security message, to a legitimate receiver, referred to as the security embedding problem. An information-theoretic formulation…
Linear codes with few weights have applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper first generalizes the method of…
Linear codes with a few weights have important applications in authentication codes, secret sharing, consumer electronics, etc.. The determination of the parameters such as Hamming weight distributions and complete weight enumerators of…
In this paper, we apply two-to-one functions over $\mathbb{F}_{2^n}$ in two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) $\left(x^{2^t}+x\right)^e$…
Universal security over a network with linear network coding has been intensively studied. However, previous linear codes and code pairs used for this purpose were linear over a larger field than that used on the network, which restricts…
We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field $\mathbb K$ of characteristic $\neq 2$ from scratch. We show that the formal model…
A wiretap protocol is a pair of randomized encoding and decoding functions such that knowledge of a bounded fraction of the encoding of a message reveals essentially no information about the message, while knowledge of the entire encoding…
We propose new results on low weight codewords of affine and projective generalized Reed-Muller codes. In the affine case we prove that if the size of the working finite field is large compared to the degree of the code, the low weight…
For every non-exceptional affine Lie algebra, we explicitly construct a positive geometric crystal associated with a fundamental representation. We also show that its ultra-discretization is isomorphic to the limit of certain perfect…
This paper studies the computability of the secrecy capacity of fast-fading wiretap channels from an algorithmic perspective, examining whether it can be computed algorithmically or not. To address this question, the concept of Turing…
We introduce a class of links strictly containing quasi-alternating links for which mod 2 reduced Khovanov homology is always thin. We compute the framed instanton homology for double branched covers of such links. Aligning certain dotted…
We introduce two notions of discrepancy between binary vectors, which are not metric functions in general but nonetheless capture the mathematical structure of the binary asymmetric channel. In turn, these lead to two new fundamental…
Delsarte showed that for any projective linear code over a finite field of characteristic p with two nonzero Hamming weights w1 < w2 there exist positive integers u and s such that w1 = (p^s)u and w2 = (p^s)(u+1). Moreover, he showed that…
The fading wire-tap channel is investigated, where the source-to-destination channel and the source-to-wire-tapper channel are corrupted by multiplicative fading gain coefficients in addition to additive Gaussian noise terms. The channel…
We construct a family of two-Lee-weight codes over the ring $R_k,$ which is defined as trace codes with algebraic structure of abelian codes. The Lee weight distribution of the two-weight codes is given. Taking the Gray map, we obtain…
We show that if a graph $G$ admits a quasi-isometry $\phi$ to a graph $H$ of bounded path-width, then we can assign a non-negative integer length to each edge of $H$, such that the same function $\phi$ is a quasi-isometry to this weighted…
From an arbitrary given channel code over a discrete or Gaussian memoryless channel, we construct a wiretap code with the strong security. Our construction can achieve the wiretap capacity under mild assumptions. The key tool is the new…
We show that a large class of non-metric, non-symplectic affine holonomies can be realized, uniformly and without case by case considerations, by Weyl connections associated to the natural AHS-structures on certain generalized flag…