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12 pages, 290 KB

Open AccessArticle

Efficient Algorithms for Permutation Arrays from Permutation Polynomials

by Sergey Bereg, Brian Malouf, Linda Morales and Ivan Hal Sudborough

Entropy 2025, 27(10), 1031; https://doi.org/10.3390/e27101031 - 1 Oct 2025

We develop algorithms for computing permutation polynomials (PPs) using normalization, so-called F-maps and G-maps, and the Hermite criterion. This allows for a more efficient computation of PPs for larger degrees and for larger finite fields. We use this to improve some lower bounds [...] Read more.

We develop algorithms for computing permutation polynomials (PPs) using normalization, so-called F-maps and G-maps, and the Hermite criterion. This allows for a more efficient computation of PPs for larger degrees and for larger finite fields. We use this to improve some lower bounds for

M (n, D)

, the maximum number of permutations on n symbols with a pairwise Hamming distance of D. Full article

(This article belongs to the Special Issue Discrete Math in Coding Theory, 2nd Edition)

15 pages, 342 KB

Open AccessArticle

On the Application of a Hybrid Incomplete Exponential Sum to Aperiodic Hamming Correlation of Some Frequency-Hopping Sequences

by Peihua Li and Hongyu Han

Entropy 2025, 27(9), 988; https://doi.org/10.3390/e27090988 - 21 Sep 2025

Viewed by 227

Abstract

Frequency-hopping sequences are essential in frequency-hopping spread spectrum communication systems due to their strong anti-interference capabilities, low probability of interception, and high confidentiality. Existing research has predominantly focused on the periodic Hamming correlation properties of sequences, whereas the aperiodic Hamming correlation performance more [...] Read more.

Frequency-hopping sequences are essential in frequency-hopping spread spectrum communication systems due to their strong anti-interference capabilities, low probability of interception, and high confidentiality. Existing research has predominantly focused on the periodic Hamming correlation properties of sequences, whereas the aperiodic Hamming correlation performance more accurately reflects the actual system performance. Owing to the complexity of its application scenarios and considerable research challenges, results in this area remain scarce. In this paper, we utilize exponential sums over finite fields to derive an upper bound on a hybrid incomplete exponential sum. Then, based on this upper bound, we derive bounds on the aperiodic Hamming correlation of some frequency-hopping sequence sets constructed by trace functions. Finally, by analyzing the maximum estimation error between the average and actual frequency collision numbers of such sequence sets, the validity of the derived bound is demonstrated. Full article

(This article belongs to the Special Issue Coding Theory and Its Applications)

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20 pages, 406 KB

Open AccessArticle

Reduction and Efficient Solution of ILP Models of Mixed Hamming Packings Yielding Improved Upper Bounds

by Péter Naszvadi, Peter Adam and Mátyás Koniorczyk

Mathematics 2025, 13(16), 2633; https://doi.org/10.3390/math13162633 - 16 Aug 2025

Viewed by 471

Abstract

We consider mixed Hamming packings, addressing the maximal cardinality of codes with a minimum codeword Hamming distance. We do not rely on any algebraic structure of the alphabets. We extend known-integer linear programming models of the problem to be efficiently tractable using standard [...] Read more.

We consider mixed Hamming packings, addressing the maximal cardinality of codes with a minimum codeword Hamming distance. We do not rely on any algebraic structure of the alphabets. We extend known-integer linear programming models of the problem to be efficiently tractable using standard ILP solvers. This is achieved by adopting the concept of contact graphs from classical continuous sphere packing problems to the present discrete context, resulting in a reduction technique for the models which enables their efficient solution as well as their decomposition to smaller subproblems. Based on our calculations, we provide a systematic summary of all lower and upper bounds for packings in the smallest Hamming spaces. The known results are reproduced, with some bounds found to be sharp, and the upper bounds improved in some cases. Full article

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10 pages, 1357 KB

Open AccessArticle

Design of Balanced Wide Gap No-Hit Zone Sequences with Optimal Auto-Correlation

by Duehee Lee, Seho Lee and Jin-Ho Chung

Mathematics 2025, 13(15), 2454; https://doi.org/10.3390/math13152454 - 30 Jul 2025

Viewed by 319

Abstract

Frequency-hopping multiple access is widely adopted to blunt narrow-band jamming and limit spectral disclosure in cyber–physical systems, yet its practical resilience depends on three sequence-level properties. First, balancedness guarantees that every carrier is occupied equally often, removing spectral peaks that a jammer or [...] Read more.

Frequency-hopping multiple access is widely adopted to blunt narrow-band jamming and limit spectral disclosure in cyber–physical systems, yet its practical resilience depends on three sequence-level properties. First, balancedness guarantees that every carrier is occupied equally often, removing spectral peaks that a jammer or energy detector could exploit. Second, a wide gap between successive hops forces any interferer to re-tune after corrupting at most one symbol, thereby containing error bursts. Third, a no-hit zone (NHZ) window with a zero pairwise Hamming correlation eliminates user collisions and self-interference when chip-level timing offsets fall inside the window. This work introduces an algebraic construction that meets the full set of requirements in a single framework. By threading a permutation over an integer ring and partitioning the period into congruent sub-blocks tied to the desired NHZ width, we generate balanced wide gap no-hit zone frequency-hopping (WG-NHZ FH) sequence sets. Analytical proofs show that (i) each sequence achieves the Lempel–Greenberger bound for auto-correlation, (ii) the family and zone sizes satisfy the Ye–Fan bound with equality, (iii) the hop-to-hop distance satisfies a provable WG condition, and (iv) balancedness holds exactly for every carrier frequency. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis for the Security and Privacy of Cyber–Physical Systems (CPSs))

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27 pages, 660 KB

Open AccessArticle

Integrating Group Setup Time Deterioration Effects and Job Processing Time Learning Effects with Group Technology in Single-Machine Green Scheduling

by Na Yin, Hongyu He, Yanzhi Zhao, Yu Chang and Ning Wang

Axioms 2025, 14(7), 480; https://doi.org/10.3390/axioms14070480 - 20 Jun 2025

Cited by 1 | Viewed by 328

Abstract

We study single-machine group green scheduling considering group setup time deterioration effects and job-processing time learning effects, where the setup time of a group is a general deterioration function on its starting setup time and the processing time of a job is a [...] Read more.

We study single-machine group green scheduling considering group setup time deterioration effects and job-processing time learning effects, where the setup time of a group is a general deterioration function on its starting setup time and the processing time of a job is a non-increasing function on its position. We focus on confirming the job schedule for each group and group schedule for minimizing the total weighted completion time. It is proved that this problem is NP-hard. According to the problem’s NP-hardness, we present some optimal properties (including lower and upper bounds) and then propose a branch-and-bound algorithm and two heuristic algorithms (including the modified Nawaz–Enscore–Ham algorithm and simulated annealing algorithm). Finally, numerical simulations are provided to indicate the effectiveness of these algorithms, which demonstrates that the branch-and-bound algorithm can solve random instances of 100 jobs and 14 groups within reasonable time and that simulated annealing is more accurate than the modified Nawaz–Enscore–Ham algorithm. Full article

(This article belongs to the Special Issue Advances in Mathematical Optimization Algorithms and Its Applications)

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17 pages, 1697 KB

Open AccessArticle

Block-Cipher No-Hit-Zone Sequence-Assisted Spectrum Control Scheme for Distributed Systems

by Wendong Gao, Lei Guan, Pei Hui, Hanwen Zhang and Zan Li

Electronics 2025, 14(9), 1802; https://doi.org/10.3390/electronics14091802 - 28 Apr 2025

Cited by 1 | Viewed by 427

Abstract

In distributed systems, the dense access of wireless devices introduces significant challenges, including severe quasi-synchronous multiple access interference (MAI) and transmission security threats, which limit the effectiveness of traditional orthogonality-based spectrum control schemes. To address these challenges, this paper proposes a new block-cipher [...] Read more.

In distributed systems, the dense access of wireless devices introduces significant challenges, including severe quasi-synchronous multiple access interference (MAI) and transmission security threats, which limit the effectiveness of traditional orthogonality-based spectrum control schemes. To address these challenges, this paper proposes a new block-cipher no-hit-zone (BC-NHZ) sequence-assisted spectrum control transmission scheme, aimed at enhancing privacy protection and improving overall communication capacity for distributed systems. The BC-NHZ scheme employs block cipher encryption and establishes control sequences to represent the spectrum usage scheme. Moreover, we mathematically prove that the parameters of the BC-NHZ scheme achieve optimal results with respect to the bound of the Hamming correlation. Numerical analysis and simulation results validate the practical feasibility of the BC-NHZ scheme, demonstrating its reliability to relax synchronous requirements while providing enhanced transmission privacy protection performance. Full article

(This article belongs to the Special Issue Security and Privacy in Distributed Machine Learning)

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8 pages, 252 KB

Open AccessArticle

A Construction of Optimal One-Coincidence Frequency-Hopping Sequences via Generalized Cyclotomy

by Minfeng Shao and Ying Miao

Entropy 2024, 26(11), 935; https://doi.org/10.3390/e26110935 - 31 Oct 2024

Cited by 1 | Viewed by 880

Abstract

Frequency-hopping sequences (FHSs) with low Hamming correlation are essential for synchronization and multiple-access communication systems. In this paper, we propose a novel construction of FHSs using generalized cyclotomy. Our results reveal that the constructed FHSs exhibit a one-coincidence property, meaning that the smallest [...] Read more.

Frequency-hopping sequences (FHSs) with low Hamming correlation are essential for synchronization and multiple-access communication systems. In this paper, we propose a novel construction of FHSs using generalized cyclotomy. Our results reveal that the constructed FHSs exhibit a one-coincidence property, meaning that the smallest correlation between different FHSs, aside from the trivial case, is minimized. Additionally, the new sets of FHSs achieve an optimal size in relation to a known theoretical bound. Full article

(This article belongs to the Special Issue Advances in Information and Coding Theory, the Third Edition)

11 pages, 293 KB

Open AccessArticle

Perfect Codes over Non-Prime Power Alphabets: An Approach Based on Diophantine Equations

by Pedro-José Cazorla García

Mathematics 2024, 12(11), 1642; https://doi.org/10.3390/math12111642 - 23 May 2024

Viewed by 1205

Abstract

Perfect error-correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory. Indeed, the classification of the parameters of e-error correcting perfect [...] Read more.

Perfect error-correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory. Indeed, the classification of the parameters of e-error correcting perfect codes over q-ary alphabets was a very active topic of research in the late 20th century. Consequently, all parameters of perfect e-error-correcting codes were found if

e \geq 3

, and it was conjectured that no perfect 2-error-correcting codes exist over any q-ary alphabet, where

q > 3

. In the 1970s, this was proved for q a prime power, for

q = 2^{r} 3^{s}

and for only seven other values of q. Almost 50 years later, it is surprising to note that there have been no new results in this regard and the classification of 2-error-correcting codes over non-prime power alphabets remains an open problem. In this paper, we use techniques from the resolution of the generalised Ramanujan–Nagell equation and from modern computational number theory to show that perfect 2-error-correcting codes do not exist for 172 new values of q which are not prime powers, substantially increasing the values of q which are now classified. In addition, we prove that, for any fixed value of q, there can be at most finitely many perfect 2-error-correcting codes over an alphabet of size q. Full article

(This article belongs to the Special Issue Codes, Designs, Cryptography and Optimization, 2nd Edition)

45 pages, 691 KB

Open AccessArticle

Deterministic K-Identification for Future Communication Networks: The Binary Symmetric Channel Results

by Mohammad Javad Salariseddigh, Ons Dabbabi, Christian Deppe and Holger Boche

Future Internet 2024, 16(3), 78; https://doi.org/10.3390/fi16030078 - 26 Feb 2024

Viewed by 2144

Abstract

Numerous applications of the Internet of Things (IoT) feature an event recognition behavior where the established Shannon capacity is not authorized to be the central performance measure. Instead, the identification capacity for such systems is considered to be an alternative metric, and has [...] Read more.

Numerous applications of the Internet of Things (IoT) feature an event recognition behavior where the established Shannon capacity is not authorized to be the central performance measure. Instead, the identification capacity for such systems is considered to be an alternative metric, and has been developed in the literature. In this paper, we develop deterministic K-identification (DKI) for the binary symmetric channel (BSC) with and without a Hamming weight constraint imposed on the codewords. This channel may be of use for IoT in the context of smart system technologies, where sophisticated communication models can be reduced to a BSC for the aim of studying basic information theoretical properties. We derive inner and outer bounds on the DKI capacity of the BSC when the size of the goal message set K may grow in the codeword length n. As a major observation, we find that, for deterministic encoding, assuming that K grows exponentially in n, i.e.,

K = 2^{n κ}

, where

κ

is the identification goal rate, then the number of messages that can be accurately identified grows exponentially in n, i.e.,

2^{n R}

, where R is the DKI coding rate. Furthermore, the established inner and outer bound regions reflects impact of the input constraint (Hamming weight) and the channel statistics, i.e., the cross-over probability. Full article

(This article belongs to the Special Issue Featured Papers in the Section Internet of Things)

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10 pages, 387 KB

Open AccessArticle

DNA Code Design Based on the Cosets of Codes over

Z_{4}

by Adel N. Alahmadi, Fatimah Anas Melibari and Manish K. Gupta

Mathematics 2023, 11(23), 4732; https://doi.org/10.3390/math11234732 - 22 Nov 2023

Cited by 1 | Viewed by 1583

Abstract

DNA code design is a challenging problem, and it has received great attention in the literature due to its applications in DNA data storage, DNA origami, and DNA computing. The primary focus of this paper is in constructing new DNA codes using the [...] Read more.

DNA code design is a challenging problem, and it has received great attention in the literature due to its applications in DNA data storage, DNA origami, and DNA computing. The primary focus of this paper is in constructing new DNA codes using the cosets of linear codes over the ring

Z_{4}

. The Hamming distance constraint, GC-content constraint, and homopolymers constraint are all considered. In this study, we consider the cosets of Simplex alpha code, Kerdock code, Preparata code, and Hadamard code. New DNA codes of lengths four, eight, sixteen, and thirty-two are constructed using a combination of an algebraic coding approach and a variable neighborhood search approach. In addition, good lower bounds for DNA codes that satisfy important constraints have been successfully established using Magma software V2.24-4 and Python 3.10 programming in our comprehensive methodology. Full article

(This article belongs to the Section E: Applied Mathematics)

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17 pages, 376 KB

Open AccessArticle

Bounds on the Probability of Undetected Error for q-Ary Codes

by Xuan Wang, Huizhou Liu and Patrick Solé

Entropy 2023, 25(9), 1349; https://doi.org/10.3390/e25091349 - 17 Sep 2023

Viewed by 1671

Abstract

We study the probability of an undetected error for general q-ary codes. We give upper and lower bounds on this quantity, by the Linear Programming and the Polynomial methods, as a function of the length, size, and minimum distance. Sharper bounds are [...] Read more.

We study the probability of an undetected error for general q-ary codes. We give upper and lower bounds on this quantity, by the Linear Programming and the Polynomial methods, as a function of the length, size, and minimum distance. Sharper bounds are obtained in the important special case of binary Hamming codes. Finally, several examples are given to illustrate the results of this paper. Full article

(This article belongs to the Special Issue Discrete Math in Coding Theory)

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13 pages, 315 KB

Open AccessArticle

Lower Bound on the Minimum Distance of Single-Generator Quasi-Twisted Codes

by Adel Alahmadi, Patrick Solé and Ramy Taki Eldin

Mathematics 2023, 11(11), 2539; https://doi.org/10.3390/math11112539 - 31 May 2023

Viewed by 1521

Abstract

We recall a classic lower bound on the minimum Hamming distance of constacyclic codes over finite fields, analogous to the well-known BCH bound for cyclic codes. This BCH-like bound serves as a foundation for proposing some minimum-distance lower bounds for single-generator quasi-twisted (QT) [...] Read more.

We recall a classic lower bound on the minimum Hamming distance of constacyclic codes over finite fields, analogous to the well-known BCH bound for cyclic codes. This BCH-like bound serves as a foundation for proposing some minimum-distance lower bounds for single-generator quasi-twisted (QT) codes. Associating each QT code with a constacyclic code over an extension field, we obtain the first bound. This is the QT analogue to a result in the literature for quasi-cyclic codes. We point out some weaknesses in this bound and propose a novel bound that takes into account the Chinese remainder theorem approach to QT codes as well as the BCH bound of constacyclic codes. This proposed bound, in contrast to previous bounds in the literature, does not presuppose a specific form of code generator and does not require calculations in any extension field. We illustrate that our bound meets the one in the literature when the code generator adheres to the specific form assumed in that study. Various numerical examples enable us to compare and discuss these bounds. Full article

(This article belongs to the Special Issue Advances in Algebraic Coding Theory and Cryptography)

18 pages, 2273 KB

Open AccessArticle

Approximate Solutions for Time-Fractional Fornberg–Whitham Equation with Variable Coefficients

by Fahad Alsidrani, Adem Kılıçman and Norazak Senu

Fractal Fract. 2023, 7(3), 260; https://doi.org/10.3390/fractalfract7030260 - 14 Mar 2023

Cited by 1 | Viewed by 2199

Abstract

In this research, three numerical methods, namely the variational iteration method, the Adomian decomposition method, and the homotopy analysis method are considered to achieve an approximate solution for a third-order time-fractional partial differential Equation (TFPDE). The equation is obtained from the classical (FW) [...] Read more.

In this research, three numerical methods, namely the variational iteration method, the Adomian decomposition method, and the homotopy analysis method are considered to achieve an approximate solution for a third-order time-fractional partial differential Equation (TFPDE). The equation is obtained from the classical (FW) equation by replacing the integer-order time derivative with the Caputo fractional derivative of order

η = (0, 1]

with variable coefficients. We consider homogeneous boundary conditions to find the approximate solutions for the bounded space variable

l < χ < L

and

l, L \in R

. To confirm the effectiveness of the proposed methods of non-integer order

η

, the computation of two test problems was presented. A comparison is made between the obtained results of the (VIM), (ADM), and (HAM) through tables and graphs. The numerical results demonstrate the effectiveness of the three numerical methods. Full article

(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)

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16 pages, 305 KB

Open AccessArticle

Bounds for Coding Theory over Rings

by Niklas Gassner, Marcus Greferath, Joachim Rosenthal and Violetta Weger

Entropy 2022, 24(10), 1473; https://doi.org/10.3390/e24101473 - 16 Oct 2022

Cited by 5 | Viewed by 2632

Abstract

Coding theory where the alphabet is identified with the elements of a ring or a module has become an important research topic over the last 30 years. It has been well established that, with the generalization of the algebraic structure to rings, there [...] Read more.

Coding theory where the alphabet is identified with the elements of a ring or a module has become an important research topic over the last 30 years. It has been well established that, with the generalization of the algebraic structure to rings, there is a need to also generalize the underlying metric beyond the usual Hamming weight used in traditional coding theory over finite fields. This paper introduces a generalization of the weight introduced by Shi, Wu and Krotov, called overweight. Additionally, this weight can be seen as a generalization of the Lee weight on the integers modulo 4 and as a generalization of Krotov’s weight over the integers modulo 2^s for any positive integer s. For this weight, we provide a number of well-known bounds, including a Singleton bound, a Plotkin bound, a sphere-packing bound and a Gilbert–Varshamov bound. In addition to the overweight, we also study a well-known metric on finite rings, namely the homogeneous metric, which also extends the Lee metric over the integers modulo 4 and is thus heavily connected to the overweight. We provide a new bound that has been missing in the literature for homogeneous metric, namely the Johnson bound. To prove this bound, we use an upper estimate on the sum of the distances of all distinct codewords that depends only on the length, the average weight and the maximum weight of a codeword. An effective such bound is not known for the overweight. Full article

(This article belongs to the Special Issue Information Theoretic Methods for Future Communication Systems)

15 pages, 739 KB

Open AccessArticle

Cooperative DF Protocol for MIMO Systems Using One-Bit ADCs

by Tae-Kyoung Kim

Sensors 2022, 22(20), 7843; https://doi.org/10.3390/s22207843 - 15 Oct 2022

Cited by 2 | Viewed by 1717

Abstract

This study considers a detection scheme for cooperative multi-input–multi-output (MIMO) systems using one-bit analog-to-digital converters (ADCs) in a decode-and-forward (DF) relay protocol. The use of one-bit ADCs is a promising technique for reducing the power consumption, which is necessary for supporting future wireless [...] Read more.

This study considers a detection scheme for cooperative multi-input–multi-output (MIMO) systems using one-bit analog-to-digital converters (ADCs) in a decode-and-forward (DF) relay protocol. The use of one-bit ADCs is a promising technique for reducing the power consumption, which is necessary for supporting future wireless systems comprising a large number of antennas. However, the use of a large number of antennas remains still limited to mobile devices owing to their size. Cooperative communication using a DF relay can resolve this limitation; however, detection errors at the relay make it difficult to employ cooperative communication directly. This difficulty is more severe in a MIMO system using one-bit ADCs due to its nonlinear nature. To efficiently address the difficulty, this paper proposes a detection scheme that mitigates the error propagation effect. The upper bound of the pairwise error probability (PEP) of one-bit ADCs is first derived in a weighted Hamming distance form. Then, using the derived PEP, the proposed detection for the DF relay protocol is derived as a single weighted Hamming distance. Finally, the complexity of the proposed detection is analyzed in terms of real multiplications. The simulation results show that the proposed detection method efficiently mitigates the error propagation effect but has a relatively low level of complexity when compared to conventional detection methods. Full article

(This article belongs to the Special Issue Cooperative Communication in 5G-and-Beyond Networks)

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