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# Special Issue "Codes, Designs, Cryptography and Optimization"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: 31 January 2021.

## Special Issue Editor

Dr. Raúl M. Falcón
Website
Guest Editor
Interests: combinatorics; Latin squares; Hadamard matrices; non-associative algebras; algebraic geometry

## Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to publish original research articles covering recent advances in any of the areas included in coding theory, cryptography, combinatorial design. and combinatorial optimization, with particular emphasis on establishing new synergies among them, and new applications to other fields and to the real world, including algebraic geometry, artificial intelligence, communication networks, computer science, hardware and software design, design of experiments, logistics, machine learning, and scheduling or transportation networks, amongst others.

Potential topics of this Special Issue include but are not limited to the following:

• Algebraic coding theory;
• Algorithm design and analysis;
• Block design theory;
• Computational complexity;
• Discrete structures: Enumeration and classification;
• Error-correcting and error-detecting codes;
• Finite geometry;
• Graph theory;
• Modeling combinatorial optimization problems;
• Network design and analysis;
• Orthogonal arrays;
• Pseudorandom sequences;
• Quantum cryptography;
• Quasigroup theory;
• Secret sharing schemes.

Dr. Raúl M. Falcón
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.dlhwdz.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

## Keywords

• Association schemes
• Block design
• Cryptosystems
• Difference sets
• Latin squares
• Lattices
• Matroids
• Networks
• Orthogonal arrays

## Published Papers (4 papers)

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# Research

Open AccessFeature PaperArticle
Pseudococyclic Partial Hadamard Matrices over Latin Rectangles
Mathematics 2021, 9(2), 113; https://doi.org/10.3390/math9020113 - 06 Jan 2021
Abstract
The classical design of cocyclic Hadamard matrices has recently been generalized by means of both the notions of the cocycle of Hadamard matrices over Latin rectangles and the pseudococycle of Hadamard matrices over quasigroups. This paper delves into this topic by introducing the [...] Read more.
The classical design of cocyclic Hadamard matrices has recently been generalized by means of both the notions of the cocycle of Hadamard matrices over Latin rectangles and the pseudococycle of Hadamard matrices over quasigroups. This paper delves into this topic by introducing the concept of the pseudococycle of a partial Hadamard matrix over a Latin rectangle, whose fundamentals are comprehensively studied and illustrated. Full article
(This article belongs to the Special Issue Codes, Designs, Cryptography and Optimization)
Open AccessArticle
Efficient Implementation of ARX-Based Block Ciphers on 8-Bit AVR Microcontrollers
Mathematics 2020, 8(10), 1837; https://doi.org/10.3390/math8101837 - 19 Oct 2020
Abstract
As the development of Internet of Things (IoT), the data exchanged through the network has significantly increased. To secure the sensitive data with user’s personal information, it is necessary to encrypt the transmitted data. Since resource-constrained wireless devices are typically used for IoT [...] Read more.
As the development of Internet of Things (IoT), the data exchanged through the network has significantly increased. To secure the sensitive data with user’s personal information, it is necessary to encrypt the transmitted data. Since resource-constrained wireless devices are typically used for IoT services, it is required to optimize the performance of cryptographic algorithms which are computation-intensive tasks. In this paper, we present efficient implementations of ARX-based Korean Block Ciphers (HIGHT and LEA) with CounTeR (CTR) mode of operation, and CTR_DRBG, one of the most widely used DRBGs (Deterministic Random Bit Generators), on 8-bit AVR Microcontrollers (MCUs). Since 8-bit AVR MCUs are widely used for various types of IoT devices, we select it as the target platform in this paper. We present an efficient implementation of HIGHT and LEA by making full use of the property of CTR mode, where the nonce value is fixed, and only the counter value changes during the encryption. On our implementation, the cost of additional function calls occurred by the generation of look-up table can be reduced. With respect to CTR_DRBG, we identified several parts that do not need to be computed. Thus, precomputing those parts in offline and using them online can result in performance improvements for CTR_DRBG. Furthermore, we applied several optimization techniques by making full use of target devices’ characteristics with AVR assembly codes on 8-bit AVR MCUs. Our proposed table generation way can reduce the cost for building a precomputation table by around 6.7% and 9.1% in the case of LEA and HIGHT, respectively. Proposed implementations of LEA and HIGHT with CTR mode on 8-bit AVR MCUs provide 6.3% and 3.8% of improved performance, compared with the previous best results, respectively. Our implementations are the fastest compared to previous LEA and HIGHT implementations on 8-bit AVR MCUs. In addition, the proposed CTR_DRBG implementations on AVR provide better performance by 37.2% and 8.7% when the underlying block cipher is LEA and HIGHT, respectively. Full article
(This article belongs to the Special Issue Codes, Designs, Cryptography and Optimization)
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Open AccessArticle
Self-Orthogonal Codes Constructed from Posets and Their Applications in Quantum Communication
Mathematics 2020, 8(9), 1495; https://doi.org/10.3390/math8091495 - 03 Sep 2020
Abstract
It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the [...] Read more.
It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length $n≥7$. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes. Full article
(This article belongs to the Special Issue Codes, Designs, Cryptography and Optimization)
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Open AccessArticle
Comparison of Entropy and Dictionary Based Text Compression in English, German, French, Italian, Czech, Hungarian, Finnish, and Croatian
Mathematics 2020, 8(7), 1059; https://doi.org/10.3390/math8071059 - 01 Jul 2020
Abstract
The rapid growth in the amount of data in the digital world leads to the need for data compression, and so forth, reducing the number of bits needed to represent a text file, an image, audio, or video content. Compressing data saves storage [...] Read more.
The rapid growth in the amount of data in the digital world leads to the need for data compression, and so forth, reducing the number of bits needed to represent a text file, an image, audio, or video content. Compressing data saves storage capacity and speeds up data transmission. In this paper, we focus on the text compression and provide a comparison of algorithms (in particular, entropy-based arithmetic and dictionary-based Lempel–Ziv–Welch (LZW) methods) for text compression in different languages (Croatian, Finnish, Hungarian, Czech, Italian, French, German, and English). The main goal is to answer a question: ”How does the language of a text affect the compression ratio?” The results indicated that the compression ratio is affected by the size of the language alphabet, and size or type of the text. For example, The European Green Deal was compressed by 75.79%, 76.17%, 77.33%, 76.84%, 73.25%, 74.63%, 75.14%, and 74.51% using the LZW algorithm, and by 72.54%, 71.47%, 72.87%, 73.43%, 69.62%, 69.94%, 72.42% and 72% using the arithmetic algorithm for the English, German, French, Italian, Czech, Hungarian, Finnish, and Croatian versions, respectively. Full article
(This article belongs to the Special Issue Codes, Designs, Cryptography and Optimization)
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