Navin Kashyap

Henry D. Pfister, Navin Kashyap, Jean-Francois Chamberland et al.

We study reliable communication over finite-state channels (FSCs) using Reed--Muller (RM) codes. Building on recent symmetry-based analyses for memoryless channels, we show that a sequence of binary RM codes (with some random scrambling) can achieve the symmetric capacity (or uniform-input information rate) of a binary-input indecomposable FSC. Our approach has three components. First, we establish a capacity-via-symmetry theorem for doubly-transitive group codes on discrete memoryless channels (DMCs) with non-binary inputs, under some symmetry and puncturing conditions. Then, we reduce a binary-input FSC to an almost memoryless non-binary channel by grouping adjacent input bits into blocks and interleaving non-binary codes onto the channel. Finally, we show that the interleaved non-binary codes can be constructed from a single binary RM code.

Prachi Mishra, Sidharth Jaggi, Navin Kashyap et al.

We investigate weakly constrained codes, in which specific patterns occur with prescribed frequencies rather than being strictly forbidden as in conventional constrained coding. We propose a capacity-achieving construction of a weakly constrained codebook based on Eulerian cycles. We then obtain, via expurgation, weakly constrained codes with linear minimum distance and positive rate, and analyze the rates achievable. Finally, we propose a practical concatenated code construction that supports polynomial-time encoding and decoding.

Geyang Wang, Alexander Barg, Navin Kashyap

In a previous paper (arXiv:2510.19746), we have studied the maximal hard-code model on the square lattice ${\mathbb Z}^2$ from the perspective of recoverable systems. Here we extend this study to the case of the triangular lattice ${\mathbb A}$. The following results are obtained: (1) We derive bounds on the capacity of the associated recoverable system on ${\mathbb A}$; (2) We show non-uniqueness of Gibbs measures in the high-activity regime; (3) We characterize extremal periodic Gibbs measures for sufficiently low values of activity.

K. Sai Mineesh Reddy, Navin Kashyap

This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal $Z$-rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group (i.e., transversal single-qubit Clifford gates are realized within a single code block, while transversal two-qubit Clifford gates are realized across two identical code blocks). Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal $T$ realizes the logical transversal $S^{\dagger}$; (b) show that the condition $C_2 \ast C_1 \subseteq C_1^{\perp}$ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal $T$ implements the logical identity and the logical transversal $T$, respectively.

Praneeth Kumar Vippathalla, Chung Chan, Navin Kashyap et al.

In this paper, we explore the connection between secret key agreement and secure omniscience within the setting of the multiterminal source model with a wiretapper who has side information. While the secret key agreement problem considers the generation of a maximum-rate secret key through public discussion, the secure omniscience problem is concerned with communication protocols for omniscience that minimize the rate of information leakage to the wiretapper. The starting point of our work is a lower bound on the minimum leakage rate for omniscience, $R_{\mathop{\mathrm{L}}}$, in terms of the wiretap secret key capacity, $C_{\mathop{\mathrm{W}}}$. Our interest is in identifying broad classes of sources for which this lower bound is met with equality, in which case we say that there is a duality between secure omniscience and secret key agreement. We show that this duality holds in the case of certain finite linear source (FLS) models, such as two-terminal FLS models and pairwise independent network models on trees with a linear wiretapper. Duality also holds for any FLS model in which $C_{\mathop{\mathrm{W}}}$ is achieved by a perfect linear secret key agreement scheme. We conjecture that the duality in fact holds unconditionally for any FLS model. On the negative side, we give an example of a (non-FLS) source model for which duality does not hold if we limit ourselves to communication-for-omniscience protocols with at most two (interactive) communications. We also address the secure function computation problem and explore the connection between the minimum leakage rate for computing a function and the wiretap secret key capacity.

Anand Jerry George, Navin Kashyap

We introduce a Markov Chain Monte Carlo (MCMC) algorithm to generate samples from probability distributions supported on a $d$-dimensional lattice $Λ= \mathbf{B}\mathbb{Z}^d$, where $\mathbf{B}$ is a full-rank matrix. Specifically, we consider lattice distributions $P_Λ$ in which the probability at a lattice point is proportional to a given probability density function, $f$, evaluated at that point. To generate samples from $P_Λ$, it suffices to draw samples from a pull-back measure $P_{\mathbb{Z}^d}$ defined on the integer lattice. The probability of an integer lattice point under $P_{\mathbb{Z}^d}$ is proportional to the density function $π= |\det(\mathbf{B})|f\circ \mathbf{B}$. The algorithm we present in this paper for sampling from $P_{\mathbb{Z}^d}$ is based on the Metropolis-Hastings framework. In particular, we use $π$ as the proposal distribution and calculate the Metropolis-Hastings acceptance ratio for a well-chosen target distribution. We can use any method, denoted by ALG, that ideally draws samples from the probability density $π$, to generate a proposed state. The target distribution is a piecewise sigmoidal distribution, chosen such that the coordinate-wise rounding of a sample drawn from the target distribution gives a sample from $P_{\mathbb{Z}^d}$. When ALG is ideal, we show that our algorithm is uniformly ergodic if $-\log(π)$ satisfies a gradient Lipschitz condition.