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Despite the success of Transformers on language understanding, code generation, and logical reasoning, they still fail to generalize over length on basic arithmetic tasks such as addition and multiplication. A major reason behind this…
Snake robots have showcased remarkable compliance and adaptability in their interaction with environments, mirroring the traits of their natural counterparts. While their hyper-redundant and high-dimensional characteristics add to this…
A number system coding for the permutations generated by cyclic shift is described. The system allows to find the rank of a permutation given how it has been generated, and to determine a permutation given its rank. It defines a code…
In this paper we consider a Metzner-Kapturowski-like decoding algorithm for high-order interleaved sum-rank-metric codes, offering a novel perspective on the decoding process through the concept of an error code. The error code, defined as…
A flip-swap language is a set S of binary strings of length n such that $S \cup 0^n$ is closed under two operations (when applicable): (1) Flip the leftmost 1; and (2) Swap the leftmost 1 with the bit to its right. Flip-swap languages model…
The Grassmannian space $\Gr$ is the set of all $k-$dimensional subspaces of the vector space~\smash{$\F_q^n$}. Recently, codes in the Grassmannian have found an application in network coding. The main goal of this paper is to present…
Given a binary word relation $\tau$ onto A * and a finite language X $\subseteq$ A * , a $\tau$-Gray cycle over X consists in a permutation w [i] 0$\le$i$\le$|X|--1 of X such that each word w [i] is an image under $\tau$ of the previous…
The code equivalence problem is central in coding theory and cryptography. While classical invariants are effective for Hamming and rank metrics, the sum-rank metric, which unifies both, introduces new challenges. This paper introduces new…
In this paper, we give a matrix construction method for designing DNA codes that come from group matrix rings. We show that with our construction one can obtain reversible $G^k$-codes of length $kn,$ where $k, n \in \mathbb{N},$ over the…
In this work, we provide four methods for constructing new maximum sum-rank distance (MSRD) codes. The first method, a variant of cartesian products, allows faster decoding than known MSRD codes of the same parameters. The other three…
Robust Gray codes were introduced by (Lolck and Pagh, SODA 2024). Informally, a robust Gray code is a (binary) Gray code $\mathcal{G}$ so that, given a noisy version of the encoding $\mathcal{G}(j)$ of an integer $j$, one can recover…
We introduce a new construction for the balancing of non-binary sequences that make use of Gray codes for prefix coding. Our construction provides full encoding and decoding of sequences, including the prefix. This construction is based on…
Recent advancements in State Space Models (SSMs) have attracted significant interest, particularly in models optimized for parallel training and handling long-range dependencies. Architectures like Mamba have scaled to billions of…
Rank modulation has been recently proposed as a scheme for storing information in flash memories. While rank modulation has advantages in improving write speed and endurance, the current encoding approach is based on the "push to the top"…
This paper presents an analysis of spinal codes, a class of rateless codes proposed recently. We prove that spinal codes achieve Shannon capacity for the binary symmetric channel (BSC) and the additive white Gaussian noise (AWGN) channel…
Many approaches to transform classification problems from non-linear to linear by feature transformation have been recently presented in the literature. These notably include sparse coding methods and deep neural networks. However, many of…
Maximal snake polyominoes are difficult to study numerically in large rectangles, as computing them requires the complete enumeration of all snakes for a specific grid size, which corresponds to a brute force algorithm. This technique is…
A permutation code is a nonlinear code whose codewords are permutation of a set of symbols. We consider the use of permutation code in the deletion channel, and consider the symbol-invariant error model, meaning that the values of the…
The last decade brought a significant increase in the amount of data and a variety of new inference methods for reconstructing the detailed evolutionary history of various cancers. This brings the need of designing efficient procedures for…
Insertion and deletion (insdel for short) errors are synchronization errors in communication systems caused by the loss of positional information in the message. Reed-Solomon codes have gained a lot of interest due to its encoding…