Information Theory (cs.IT)

Abstract: The MacWilliams' Extension Theorem is a classical result by Florence Jessie MacWilliams. It shows that every linear isometry between linear block-codes endowed with the Hamming distance can be extended to a linear isometry of the ambient space. Such an extension fails to exist in general for rank-metric codes, that is, one can easily find examples of linear isometries between rank-metric codes which cannot be extended to linear isometries of the ambient space. In this paper, we explore to what extent a MacWilliams' Extension Theorem may hold for rank-metric codes. We provide an extensive list of examples of obstructions to the existence of an extension, as well as a positive result.

Abstract: Coded caching schemes exploit the cumulative cache memory of the users by using simple linear encoders, outperforming uncoded schemes where cache contents are only used locally. Considering multi-AP WLANs and video-on-demand (VoD) applications where users stream videos by sequentially requesting video ``chunks", we apply existing coded caching techniques with reduced subpacketization order, and obtain a computational method to determine the theoretical throughput region of the users' content delivery rates, calculated as the number of chunks delivered per unit of time per user. We then solve the fairness scheduling problem by maximizing the desired fairness metric over the throughput region. We also provide two heuristic methods with reduced complexity, where one of them maximizes the desired fairness metric over a smaller region than the throughput region, and the other uses a greedy algorithmic approach to associate users with APs in a fair way.

Abstract: In this article, we provide the first systematic analysis of bent functions $f$ on $\mathbb{F}_2^{n}$ in the Maiorana-McFarland class $\mathcal{MM}$ regarding the origin and cardinality of their $\mathcal{M}$-subspaces, i.e., vector subspaces on which the second-order derivatives of $f$ vanish. By imposing restrictions on permutations $\pi$ of $\mathbb{F}_2^{n/2}$, we specify the conditions, such that Maiorana-McFarland bent functions $f(x,y)=x\cdot \pi(y) + h(y)$ admit a unique $\mathcal{M}$-subspace of dimension $n/2$. On the other hand, we show that permutations $\pi$ with linear structures give rise to Maiorana-McFarland bent functions that do not have this property. In this way, we contribute to the classification of Maiorana-McFarland bent functions, since the number of $\mathcal{M}$-subspaces is invariant under equivalence. Additionally, we give several generic methods of specifying permutations $\pi$ so that $f\in\mathcal{MM}$ admits a unique $\mathcal{M}$-subspace. Most notably, using the knowledge about $\mathcal{M}$-subspaces, we show that using the bent 4-concatenation of four suitably chosen Maiorana-McFarland bent functions, one can in a generic manner generate bent functions on $\mathbb{F}_2^{n}$ outside the completed Maiorana-McFarland class $\mathcal{MM}^\#$ for any even $n\geq 8$. Remarkably, with our construction methods it is possible to obtain inequivalent bent functions on $\mathbb{F}_2^8$ not stemming from two primary classes, the partial spread class $\mathcal{PS}$ and $\mathcal{MM}$. In this way, we contribute to a better understanding of the origin of bent functions in eight variables, since only a small fraction, of which size is about $2^{76}$, stems from $\mathcal{PS}$ and $\mathcal{MM}$, whereas the total number of bent functions on $\mathbb{F}_2^8$ is approximately $2^{106}$.

Abstract: Semantic communication represents a promising roadmap toward achieving end-to-end communication with reduced communication overhead and an enhanced user experience. The integration of semantic concepts with wireless communications presents novel challenges. This paper proposes a flexible simulation software that automatically transmits semantic segmentation map images over a communication channel. An additive white Gaussian noise (AWGN) channel using binary phase-shift keying (BPSK) modulation is considered as the channel setup. The well-known polar codes are chosen as the channel coding scheme. The popular COCO-Stuff dataset is used as an example to generate semantic map images corresponding to different signal-to-noise ratios (SNRs). To evaluate the proposed software, we have generated four small datasets, each containing a thousand semantic map samples, accompanied by comprehensive information corresponding to each image, including the polar code specifications, detailed image attributes, bit error rate (BER), and frame error rate (FER). The capacity to generate an unlimited number of semantic maps utilizing desired channel coding parameters and preferred SNR, in conjunction with the flexibility of using alternative datasets, renders our simulation software highly adaptable and transferable to a broad range of use cases.

Abstract: In a recent paper it has been shown that to model a diffusive molecular communication (MC) channel with multiple fully absorbing (FA) receivers, these can be interpreted as sources of negative particles from the other receivers' perspective. The barycenter point is introduced as the best position where to place the negative sources. The barycenter is obtained from the spatial mean of the molecules impinging on the surface of each FA receiver. This paper derives an expression that captures the position of the barycenter in a diffusive MC channel with multiple FA receivers. In this work, an analytical model inspired by Newton's law of gravitation is found to describe the barycenter, and the result is compared with particle-based simulation (PBS) data. Since the barycenter depends on the distance between the transmitter and receiver and the observation time, the condition that the barycenter can be assumed to be at the center of the receiver is discussed. This assumption simplifies further modeling of any diffusive MC system containing multiple FA receivers. The resulting position of the barycenter is used in channel models to calculate the cumulative number of absorbed molecules and it has been verified with PBS data in a variety of scenarios.

Abstract: Polynomial based approaches, such as the Mat-Dot and entangled polynomial (EP) codes have been used extensively within coded matrix computations to obtain schemes with good thresholds. However, these schemes are well-recognized to suffer from poor numerical stability in decoding. Moreover, the encoding process in these schemes involves linearly combining a large number of input submatrices, i.e., the encoding weight is high. For the practically relevant case of sparse input matrices, this can have the undesirable effect of significantly increasing the worker node computation time. In this work, we propose a generalization of the EP scheme by combining the idea of gradient coding along with the basic EP encoding. Our scheme allows us to reduce the weight of the encoding and arrive at schemes that exhibit much better numerical stability; this is achieved at the expense of a worse threshold. By appropriately setting parameters in our scheme, we recover several well-known schemes in the literature. Simulation results show that our scheme provides excellent numerical stability and fast computation speed (for sparse input matrices) as compared to EPC and Mat-Dot codes.