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In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, which were recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking…
The problem of error-control in random linear network coding is considered. A ``noncoherent'' or ``channel oblivious'' model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer…
Recently, code language models have achieved notable advancements in addressing a diverse array of essential code comprehension and generation tasks. Yet, the field lacks a comprehensive deep dive and understanding of the code embeddings of…
We determine the minimal equivariant embedding dimension of orthgonal groups acting on real flag manifolds and unitary groups acting on complex flag manifolds. The minimal embedding dimension is achieved at isospectral model.
An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We show a dichotomy for the size of the smallest identifying code in classes…
Fractional repetition (FR) codes are a class of repair efficient erasure codes that can recover a failed storage node with both optimal repair bandwidth and complexity. In this paper, we study the minimum distance of FR codes, which is the…
A basic problem for constant dimension codes is to determine the maximum possible size $A_q(n,d;k)$ of a set of $k$-dimensional subspaces in $\mathbb{F}_q^n$, called codewords, such that the subspace distance satisfies…
After the optimal parameters of additive quaternary codes of dimension $k\le 3$ have been determined there is some recent activity to settle the next case of dimension $k=3.5$. Here we complete dimension $k=3.5$ and $k=4$. We also solve the…
Receptive field profiles registered by cell recordings have shown that mammalian vision has developed receptive fields tuned to different sizes and orientations in the image domain as well as to different image velocities in space-time.…
Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. In this paper we prove an upper bound on the minimal…
A basic problem in constant dimension subspace coding is to determine the maximal possible size ${\bf A}_q(n,d,k)$ of a set of $k$-dimensional subspaces in ${\bf F}_q^n$ such that the subspace distance satisfies…
We employ analytic number theoretic techniques, specifically character sums and Weil type estimates, to study the covering radius of the generalized Zetterberg codes over all finite fields. Although the even and odd field cases require…
Embedding representations power machine intelligence in many applications, including recommendation systems, but they are space intensive -- potentially occupying hundreds of gigabytes in large-scale settings. To help manage this outsized…
Geometric relational embeddings map relational data as geometric objects that combine vector information suitable for machine learning and structured/relational information for structured/relational reasoning, typically in low dimensions.…
Two new constructions are presented for coils and snakes in the hypercube. Improvements are made on the best known results for snake-in-the-box coils of dimensions 9, 10 and 11, and for some other circuit codes of dimensions between 8 and…
In many applications of neural network, it is common to introduce huge amounts of input categorical features, as well as output labels. However, since the required network size should have rapid growth with respect to the dimensions of…
Neural embeddings are a popular set of methods for representing words, phrases or text as a low dimensional vector (typically 50-500 dimensions). However, it is difficult to interpret these dimensions in a meaningful manner, and creating…
For a finite group $G$, the representation dimension is the smallest integer realizable as the degree of a complex faithful representation of $G$. In this article, we compute representation dimension for some $p$-groups, their direct…
Embedding-based representations in Euclidean space $\mathbb{R}^d$ are a cornerstone of modern machine learning, where a major goal is to use the \emph{smallest dimension} that faithfully captures data relations. In this work, we prove sharp…
Maximally recoverable codes are a class of codes which recover from all potentially recoverable erasure patterns given the locality constraints of the code. In earlier works, these codes have been studied in the context of codes with…