arXiv subjects

Mladen Kovačević

Publications and source records attributed to Mladen Kovačević.

At least 19 recordsLinked to original sources

arXiv2026

Bounds on Codes Correcting Adjacent Transpositions

We study the problem of correcting pairwise disjoint adjacent transpositions (or swaps) in $q$-ary strings. Equivalently, the model we assume is the radius-one instance of the so-called $\ell_\infty$-limited permutation channel. We first study the relevant combinatorial properties of the appropriately defined transposition distance, including center-specific and average ball sizes. We then derive two lower bounds and one upper bound on the asymptotic rates of optimal codes correcting $t=τn$ transpositions. The first achievability result is a generalized Gilbert--Varshamov bound, while the second follows from a construction of codes correcting all possible patterns of adjacent transpositions and therefore represents a lower bound on the zero-error capacity of this model. This construction improves the classical general-alphabet construction for $3\leqslant q\leqslant 8$ as well as the recent bounds for $q=4,5$. The upper bound is obtained by a packing argument adjusted to the run-structure of a given code. To the best of our knowledge, these are the first nonconstant, $τ$-dependent lower and upper bounds developed for the pairwise disjoint $q$-ary model throughout the linear regime. We also derive asymptotic bounds on the cardinality of optimal codes correcting $t=\textrm{const}$ pairwise disjoint adjacent transpositions.

cs.IT↗

arXiv2026

The Capacity of a Family of Sticky Channels

We determine the capacity of a family of $q$-ary sticky-insertion channels. Fix $q\geq2$ and $d\geq1$, and let $λ$ be the unique positive solution of $λ^d = (q-1) (λ^{d-1} + \cdots + λ+ 1 )$. We prove that, for every repetition law supported on $1+d\mathbb{Z}_{\geq0}$ and satisfying a coefficientwise-domination criterion with domination constant $γ\geqλ^{-d}$, the Shannon capacity equals the zero-error capacity, both being $\log_2λ$ bits per symbol. We also exhibit explicit repetition laws satisfying these conditions, one of which is given by the weighted Fuss--Catalan numbers. To the best of our knowledge, these are the first known cases of nontrivial repeat channels whose Shannon capacity has been determined exactly.

cs.IT↗

arXiv2026

An Isodiametric Theorem and Lattice Diameter-Perfect Codes in $A_3$

The root lattice $A_n$, equipped with its graph distance (equivalently, one half of the ambient $\ell_1$ metric), is isometric to $\mathbb{Z}^n$ with the asymmetric Manhattan metric. We study two extremal problems in this space -- the isodiametric problem, i.e., determining the maximum anticode cardinality, and the (non)existence of linear diameter-perfect codes, i.e., lattice tilings by optimal anticodes -- and solve them in dimension $3$. We show that, for every integer $D\ge 0$, the largest cardinality of a diameter-$D$ subset of $A_3$ is $\binom{D+3}{3}+(D+1)\lfloor D^2/4\rfloor$, and this value is attained by the balanced difference of two discrete simplices. We then prove an integrality-refined simplex-packing obstruction: a sublattice of $\mathbb{Z}^n$ of asymmetric Manhattan distance greater than $D$ induces a lattice packing by $(D+1)Δ_n$ in $\mathbb{R}^n$. Combining this observation with the exact lattice-packing density of the tetrahedron yields a complete classification in dimension $3$: lattice diameter-perfect codes in $A_3$ exist precisely for $D=1$ and $D=2$. We also give the equivalent statement for perfect $B_h$ sets of cardinality four. Finally, we formulate a conjecture regarding optimal anticodes in arbitrary dimension, and restate it as an intersection problem for uniform multisets.

math.CO↗

arXiv2026

Maximum Probability of Independence in Transitive Matroids

Let $M$ be a matroid on a finite ground set $E$, and suppose that the automorphism group of $M$ acts transitively on $E$. We show the following: if $X_1,\ldots,X_K$ are sampled independently from a distribution $p$ on $E$, then the probability that the samples are distinct and that $\{X_1,\ldots,X_K\}$ is an independent set in $M$ is quasi-concave in $p$ and maximized when $p$ is uniform. As a corollary, for a random $K\times N$ matrix over a finite field whose rows are sampled independently from an arbitrary distribution on nonzero projective row classes, the uniform distribution on projective space maximizes the probability of full row rank. In this particular case we also establish the uniqueness of the maximizer and global quadratic stability, while a simple example illustrates that uniqueness and stability need not hold for arbitrary transitive matroids.

math.CO↗

arXiv2026

Maximum Entropy of Sums of Independent Ternary Random Variables

The classical problem of maximizing the Shannon entropy of a sum of independent random variables supported on a finite alphabet is considered and settled in the ternary case. Namely, the following theorem is established: if $X_1,\ldots,X_n$ are independent random variables taking values in $\{0,1,2\}$, then the entropy of $S_n=X_1+\cdots+X_n$ is maximized when $X_1,\ldots,X_{n-1}$ are uniform on $\{0,2\}$ and the probability mass function of $X_n$ is given by $\Prob(X_n=0) = \Prob(X_n=2) = w/2$, $\Prob(X_n=1) = 1-w$, where $w = \big(1 + 2^{-H(B_n)+H(B_{n-1})}\big)^{-1}$ and $B_m\sim \Bin(m,1/2)$. The statement can be seen as an extension to ternary alphabets of the Shepp--Olkin--Mateev theorem. The proof uses the Hermite--Biehler theorem, Newton's inequalities, and Yu's maximum-entropy theorem for ultra-log-concave distributions.

cs.IT↗

arXiv2025

A Zero-Knowledge Proof for the Syndrome Decoding Problem in the Lee Metric

The syndrome decoding problem is one of the NP-complete problems lying at the foundation of code-based cryptography. The variant thereof where the distance between vectors is measured with respect to the Lee metric, rather than the more commonly used Hamming metric, has been analyzed recently in several works due to its potential relevance for building more efficient code-based cryptosystems. The purpose of this article is to present a zero-knowledge proof of knowledge for this variant of the problem.

cs.CR↗

arXiv2024

DeepNcode: Encoding-Based Protection against Bit-Flip Attacks on Neural Networks

Fault injection attacks are a potent threat against embedded implementations of neural network models. Several attack vectors have been proposed, such as misclassification, model extraction, and trojan/backdoor planting. Most of these attacks work by flipping bits in the memory where quantized model parameters are stored. In this paper, we introduce an encoding-based protection method against bit-flip attacks on neural networks, titled DeepNcode. We experimentally evaluate our proposal with several publicly available models and datasets, by using state-of-the-art bit-flip attacks: BFA, T-BFA, and TA-LBF. Our results show an increase in protection margin of up to $7.6\times$ for $4-$bit and $12.4\times$ for $8-$bit quantized networks. Memory overheads start at $50\%$ of the original network size, while the time overheads are negligible. Moreover, DeepNcode does not require retraining and does not change the original accuracy of the model.

cs.CR↗

arXiv2024

Vector Multispaces and Multispace Codes

Basic algebraic and combinatorial properties of finite vector spaces in which individual vectors are allowed to have multiplicities larger than $ 1 $ are derived. An application in coding theory is illustrated by showing that multispace codes that are introduced here may be used in random linear network coding scenarios, and that they generalize standard subspace codes (defined in the set of all subspaces of $ \mathbb{F}_q^n $) and extend them to an infinitely larger set of parameters. In particular, in contrast to subspace codes, multispace codes of arbitrarily large cardinality and minimum distance exist for any fixed $ n $ and $ q $.

cs.IT↗

arXiv2024

An Information-Theoretic Analog of the Twin Paradox

We revisit the familiar scenario involving two parties in relative motion, in which Alice stays at rest while Bob goes on a journey at speed $βc$ along an arbitrary trajectory and reunites with Alice after a certain period of time. It is a well-known consequence of special relativity that the time that passes until they meet again is different for the two parties and is shorter in Bob's frame by a factor of $\sqrt{1-β^2}$. We investigate how this asymmetry manifests from an information-theoretic viewpoint. Assuming that Alice and Bob transmit signals of equal average power to each other during the whole journey, and that additive white Gaussian noise is present at both sides, we show that the maximum number of bits per second that Alice can transmit reliably to Bob is always higher than the one Bob can transmit to Alice. Equivalently, the energy per bit invested by Alice is lower than that invested by Bob, meaning that the traveler is less efficient from the communication perspective, as conjectured by Jarett and Cover.

cs.IT↗

arXiv2024

Gilbert-Varshamov Bound for Codes in $L_1$ Metric using Multivariate Analytic Combinatorics

Analytic combinatorics in several variables refers to a suite of tools that provide sharp asymptotic estimates for certain combinatorial quantities. In this paper, we apply these tools to determine the Gilbert--Varshamov lower bound on the rate of optimal codes in $L_1$ metric. Several different code spaces are analyzed, including the simplex and the hypercube in $\mathbb{Z^n}$, all of which are inspired by concrete data storage and transmission models such as the sticky insertion channel, the permutation channel, the adjacent transposition (bit-shift) channel, the multilevel flash memory channel, etc.

cs.IT↗

arXiv2022

Optimal Error-Detecting Codes for General Asymmetric Channels via Sperner Theory

Several communication models that are of relevance in practice are asymmetric in the way they act on the transmitted "objects". Examples include channels in which the amplitudes of the transmitted pulses can only be decreased, channels in which the symbols can only be deleted, channels in which non-zero symbols can only be shifted to the right (e.g., timing channels), subspace channels in which the dimension of the transmitted vector space can only be reduced, unordered storage channels in which the cardinality of the stored (multi)set can only be reduced, etc. We introduce a formal definition of an asymmetric channel as a channel whose action induces a partial order on the set of all possible inputs, and show that this definition captures all the above examples. Such a general approach allows one to treat all these different models in a unified way, and to obtain a characterization of optimal error-detecting codes for many interesting asymmetric channels by using Sperner theory.

cs.IT↗

arXiv2022

Lattice Packings of Cross-polytopes from Reed-Solomon Codes and Sidon Sets

Two constructions of lattice packings of $ n $-dimensional cross-polytopes ($ \ell_1 $ balls) are described, the density of which exceeds that of any prior construction by a factor of at least $ 2^{\frac{n}{\ln n}(1 + o(1))} $ when $ n \to \infty $. The first family of lattices is explicit and is obtained by applying Construction A to a class of Reed-Solomon codes. The second family has subexponential construction complexity and is based on the notion of Sidon sets in finite Abelian groups. The construction based on Sidon sets also gives the highest known asymptotic density of packing discrete cross-polytopes of fixed radius $ r \geqslant 3 $ in $ \mathbb{Z}^n $.

math.CO↗

arXiv2022

On the Maximum Entropy of a Sum of Independent Discrete Random Variables

Let $ X_1, \ldots, X_n $ be independent random variables taking values in the alphabet $ \{0, 1, \ldots, r\} $, and $ S_n = \sum_{i = 1}^n X_i $. The Shepp--Olkin theorem states that, in the binary case ($ r = 1 $), the Shannon entropy of $ S_n $ is maximized when all the $ X_i $'s are uniformly distributed, i.e., Bernoulli(1/2). In an attempt to generalize this theorem to arbitrary finite alphabets, we obtain a lower bound on the maximum entropy of $ S_n $ and prove that it is tight in several special cases. In addition to these special cases, an argument is presented supporting the conjecture that the bound represents the optimal value for all $ n, r $, i.e., that $ H(S_n) $ is maximized when $ X_1, \ldots, X_{n-1} $ are uniformly distributed over $ \{0, r\} $, while the probability mass function of $ X_n $ is a mixture (with explicitly defined non-zero weights) of the uniform distributions over $ \{0, r\} $ and $ \{1, \ldots, r-1\} $.

cs.IT↗

arXiv2022

On the Maximum Number of Non-Confusable Strings Evolving Under Short Tandem Duplications

The set of all $ q $-ary strings that do not contain repeated substrings of length $ \leqslant\! 3 $ (i.e., that do not contain substrings of the form $ a a $, $ a b a b $, and $ a b c a b c $) constitutes a code correcting an arbitrary number of tandem-duplication mutations of length $ \leqslant\! 3 $. In other words, any two such strings are non-confusable in the sense that they cannot produce the same string while evolving under tandem duplications of length $ \leqslant\! 3 $. We demonstrate that this code is asymptotically optimal in terms of rate, meaning that it represents the largest set of non-confusable strings up to subexponential factors. This result settles the zero-error capacity problem for the last remaining case of tandem-duplication channels satisfying the "root-uniqueness" property.

cs.IT↗

arXiv2021

Asymptotic Behavior and Typicality Properties of Runlength-Limited Sequences

This paper studies properties of binary runlength-limited sequences with additional constraints on their Hamming weight and/or their number of runs of identical symbols. An algebraic and a probabilistic (entropic) characterization of the exponential growth rate of the number of such sequences, i.e., their information capacity, are obtained by using the methods of multivariate analytic combinatorics, and properties of the capacity as a function of its parameters are stated. The second-order term in the asymptotic expansion of the rate of these sequences is also given, and the typical values of the relevant quantities are derived. Several applications of the results are illustrated, including bounds on codes for weight-preserving and run-preserving channels (e.g., the run-preserving insertion-deletion channel), a sphere-packing bound for channels with sparse error patterns, and the asymptotics of constant-weight sub-block constrained sequences. In addition, the asymptotics of a closely related notion -- $ q $-ary sequences with fixed Manhattan weight -- is briefly discussed, and an application in coding for molecular timing channels is illustrated.

cs.IT↗

arXiv2021

Signaling to Relativistic Observers: An Einstein-Shannon-Riemann Encounter

A communication scenario is described involving a series of events triggered by a transmitter and observed by a receiver experiencing relativistic time dilation. The message selected by the transmitter is assumed to be encoded in the events' timings and is required to be perfectly recovered by the receiver, regardless of the difference in clock rates in the two frames of reference. It is shown that the largest proportion of the space of all $ k $-event signals that can be selected as a code ensuring error-free information transfer in this setting equals $ ζ(k)^{-1} $, where $ ζ$ is the Riemann zeta function.

math-ph↗

arXiv2020

Second- and Third-Order Asymptotics of the Continuous-Time Poisson Channel

The paper derives the optimal second-order coding rate for the continuous-time Poisson channel. We also obtain bounds on the third-order coding rate. This is the first instance of a second-order result for a continuous-time channel. The converse proof hinges on a novel construction of an output distribution induced by Wyner's discretized channel and the construction of an appropriate $ε$-net of the input probability simplex. While the achievability proof follows the general program to prove the third-order term for non-singular discrete memoryless channels put forth by Polyanskiy, several non-standard techniques -- such as new definitions and bounds on the probabilities of typical sets using logarithmic Sobolev inequalities -- are employed to handle the continuous nature of the channel.

cs.IT↗

arXiv2019

Sidon Sets, Difference Sets, and Codes in $ A_n $ Lattices

This chapter investigates the properties of (linear) codes in $ A_n $ lattices, the practical motivation for which is found in several communication scenarios, such as asymmetric channels, sticky-insertion channels, bit-shift channels, and permutation channels. In particular, a connection between these codes and notions of difference sets and Sidon sets in Abelian groups is demonstrated. It is shown that the $ A_n $ lattice admits a linear perfect code of radius $ 1 $ if and only if there exists an Abelian planar difference set of cardinality $ n + 1 $. Similarly, a direct link is given between linear codes of radius $ r $ in the $ A_n $ lattice and Sidon sets of order $ 2r $ and cardinality $ n + 1 $. Sidon sets of order $ 2r-1 $ are also represented geometrically in a similar way. Apart from providing geometric intuition about Sidon sets, this interpretation enables simple derivations of bounds on their parameters, which are either equivalent to, or improve upon the known bounds. In connection to the above, more general (non-planar) Abelian difference sets and perfect codes of radius $ r $ are also discussed.

math.CO↗