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Advances in Mathematics of Communications (AMC) publishes original research papers of the highest quality in all areas of mathematics and computer science which are relevant to applications in communications technology. For this reason, submissions from many areas of mathematics are invited, provided these show a high level of originality, new techniques, an innovative approach, novel methodologies, or otherwise a high level of depth and sophistication. Any work that does not conform to these standards will be rejected.

Areas covered include coding theory, cryptology, combinatorics, finite geometry, algebra and number theory, but are not restricted to these. This journal also aims to cover the algorithmic and computational aspects of these disciplines. Hence, all mathematics and computer science contributions of appropriate depth and relevance to the above mentioned applications in communications technology are welcome.

More detailed indication of the journal's scope is given by the subject interests of the members of the board of editors.

All papers will undergo a thorough peer reviewing process unless the subject matter of the paper does not fit the journal; in this case, the author will be informed promptly. Every effort will be made to secure a decision in three months and to publish accepted papers within six months.

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Constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$
Yonglin Cao, Yuan Cao, Hai Q. Dinh, Fang-Wei Fu, Jian Gao and Songsak Sriboonchitta
2018, 12(2) : 231-262 doi: 10.3934/amc.2018016 +[Abstract](467) +[HTML](335) +[PDF](555.31KB)
Abstract:

Let \begin{document}$\mathbb{F}_{p^m}$\end{document} be a finite field of cardinality \begin{document}$p^m$\end{document} and \begin{document}$R = \mathbb{F}_{p^m}[u]/\langle u^2\rangle = \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$\end{document} \begin{document}$(u^2 = 0)$\end{document}, where \begin{document}$p$\end{document} is a prime and \begin{document}$m$\end{document} is a positive integer. For any \begin{document}$λ∈ \mathbb{F}_{p^m}^{×}$\end{document}, an explicit representation for all distinct \begin{document}$λ$\end{document}-constacyclic codes over \begin{document}$R$\end{document} of length \begin{document}$np^s$\end{document} is given by a canonical form decomposition for each code, where \begin{document}$s$\end{document} and \begin{document}$n$\end{document} are arbitrary positive integers satisfying \begin{document}${\rm gcd}(p,n) = 1$\end{document}. For any such code, using its canonical form decomposition the representation for the dual code of the code is provided. Moreover, representations for all distinct cyclic codes, negacyclic codes and their dual codes of length \begin{document}$np^s$\end{document} over \begin{document}$R$\end{document} are obtained, and self-duality for these codes are determined. Finally, all distinct self-dual negacyclic codes over \begin{document}$\mathbb{F}_5+u\mathbb{F}_5$\end{document} of length \begin{document}$2· 3^t· 5^s$\end{document} are listed for any positive integer \begin{document}$t$\end{document}.

Indiscreet logarithms in finite fields of small characteristic
Robert Granger, Thorsten Kleinjung and Jens Zumbrägel
2018, 12(2) : 263-286 doi: 10.3934/amc.2018017 +[Abstract](423) +[HTML](268) +[PDF](541.54KB)
Abstract:

Recently, several striking advances have taken place regarding the discrete logarithm problem (DLP) in finite fields of small characteristic, despite progress having remained essentially static for nearly thirty years, with the best known algorithms being of subexponential complexity. In this expository article we describe the key insights and constructions which culminated in two independent quasi-polynomial algorithms. To put these developments into both a historical and a mathematical context, as well as to provide a comparison with the cases of so-called large and medium characteristic fields, we give an overview of the state-of-the-art algorithms for computing discrete logarithms in all finite fields. Our presentation aims to guide the reader through the algorithms and their complexity analyses ab initio.

Hilbert quasi-polynomial for order domains and application to coding theory
Carla Mascia, Giancarlo Rinaldo and Massimiliano Sala
2018, 12(2) : 287-301 doi: 10.3934/amc.2018018 +[Abstract](351) +[HTML](218) +[PDF](460.17KB)
Abstract:

We present an application of Hilbert quasi-polynomials to order domains, allowing the effective check of the second order-domain condition in a direct way. We also provide an improved algorithm for the computation of the related Hilbert quasi-polynomials. This allows to identify order domain codes more easily.

Several infinite families of p-ary weakly regular bent functions
Yanfeng Qi, Chunming Tang, Zhengchun Zhou and Cuiling Fan
2018, 12(2) : 303-315 doi: 10.3934/amc.2018019 +[Abstract](347) +[HTML](247) +[PDF](366.85KB)
Abstract:

As an optimal combinatorial object, bent functions have been an interesting research object due to their important applications in cryptography, coding theory, and sequence design. The characterization and construction of bent functions are challenging problems in general. The objective of this paper is to present a construction of p-ary weakly regular bent functions from known weakly regular bent functions. This generalizes some earlier constructions of Boolean bent functions and p-ary bent functions, and produces several infinite families of p-ary weakly regular bent functions from known ones. Some infinite families of p-ary rotation symmetric bent functions are obtained as well.

Locally recoverable codes with availability t≥2 from fiber products of curves
Kathryn Haymaker, Beth Malmskog and Gretchen L. Matthews
2018, 12(2) : 317-336 doi: 10.3934/amc.2018020 +[Abstract](318) +[HTML](220) +[PDF](572.2KB)
Abstract:

We generalize the construction of locally recoverable codes on algebraic curves given by Barg, Tamo and Vlăduţ [4] to those with arbitrarily many recovery sets by exploiting the structure of fiber products of curves. Employing maximal curves, we create several new families of locally recoverable codes with multiple recovery sets, including codes with two recovery sets from the generalized Giulietti and Korchmáros (GK) curves and the Suzuki curves, and new locally recoverable codes with many recovery sets based on the Hermitian curve, using a fiber product construction of van der Geer and van der Vlugt. In addition, we consider the relationship between local error recovery and global error correction as well as the availability required to locally recover any pattern of a fixed number of erasures.

Completely regular codes by concatenating Hamming codes
Joaquim Borges, Josep Rifà and Victor Zinoviev
2018, 12(2) : 337-349 doi: 10.3934/amc.2018021 +[Abstract](314) +[HTML](212) +[PDF](327.08KB)
Abstract:

We construct new families of completely regular codes by concatenation methods. By combining parity check matrices of cyclic Hamming codes, we obtain families of completely regular codes. In all cases, we compute the intersection array of these codes. As a result, we find some non-equivalent completely regular codes, over the same finite field, with the same parameters and intersection array. We also study when the extension of these codes gives completely regular codes. Some of these new codes are completely transitive.

Further results on the existence of super-simple pairwise balanced designs with block sizes 3 and 4
Guangzhou Chen, Yue Guo and Yong Zhang
2018, 12(2) : 351-362 doi: 10.3934/amc.2018022 +[Abstract](294) +[HTML](236) +[PDF](388.62KB)
Abstract:

In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Super-simple pairwise balanced designs are useful in constructing other types of super-simple designs which can be applied to codes and designs. In this paper, the super-simple pairwise balanced designs with block sizes 3 and 4 are investigated and it is proved that the necessary conditions for the existence of a super-simple \begin{document}$(v, \{3,4\}, λ)$\end{document}-PBD for \begin{document}$λ = 7,9$\end{document} and \begin{document}$λ = 2k$\end{document}, \begin{document}$k≥1$\end{document}, are sufficient with seven possible exceptions. In the end, several optical orthogonal codes and superimposed codes are given.

The weight distribution of quasi-quadratic residue codes
Nigel Boston and Jing Hao
2018, 12(2) : 363-385 doi: 10.3934/amc.2018023 +[Abstract](309) +[HTML](209) +[PDF](429.38KB)
Abstract:

We investigate a family of codes called quasi-quadratic residue (QQR) codes. We are interested in these codes mainly for two reasons: Firstly, they have close relations with hyperelliptic curves and Goppa's Conjecture, and serve as a strong tool in studying those objects. Secondly, they are very good codes. Computational results show they have large minimum distances when \begin{document}$p\equiv 3 \pmod 8$\end{document}.

Our studies focus on the weight distributions of these codes. We will prove a new discovery about their weight polynomials, i.e. they are divisible by \begin{document}$(x^2 + y^2)^{d-1}$\end{document}, where \begin{document}$d$\end{document} is the corresponding minimum distance. We also show that the weight distributions of these codes are asymptotically normal. Based on the relation between QQR codes and hyperelliptic curves, we will also prove a result on the point distribution on hyperelliptic curves.

New families of strictly optimal frequency hopping sequence sets
Jingjun Bao
2018, 12(2) : 387-413 doi: 10.3934/amc.2018024 +[Abstract](324) +[HTML](221) +[PDF](613.0KB)
Abstract:

Frequency hopping sequences (FHSs) with favorable partial Hamming correlation properties have important applications in many synchronization and multiple-access systems. In this paper, we investigate constructions of FHS sets with optimal partial Hamming correlation. We present several direct constructions for balanced nested cyclic difference packings (BNCDPs) and balanced nested cyclic relative difference packings (BNCRDPs) by using trace functions and discrete logarithm. We also show three recursive constructions for FHS sets with partial Hamming correlation, which are based on cyclic difference matrices and discrete logarithm. Combing these BNCDPs, BNCRDPs and three recursive constructions, we obtain infinitely many new strictly optimal FHS sets with respect to the Peng-Fan bounds.

On some classes of codes with a few weights
Yiwei Liu and Zihui Liu
2018, 12(2) : 415-428 doi: 10.3934/amc.2018025 +[Abstract](323) +[HTML](224) +[PDF](376.9KB)
Abstract:

We generalize the code constructed recently by Wang et al, and obtain many classes of codes with a few weights. The weight distribution of these codes is completely determined, and the minimum distance of the duals of these codes is determined. We also show that some subclasses of the duals of these codes are optimal. Furthermore, some parameters of the generalized Hamming weight of these codes are calculated in certain cases.

A new almost perfect nonlinear function which is not quadratic
Yves Edel and Alexander Pott
2009, 3(1) : 59-81 doi: 10.3934/amc.2009.3.59 +[Abstract](718) +[PDF](288.3KB) Cited By(42)
Skew constacyclic codes over Galois rings
Delphine Boucher, Patrick Solé and Felix Ulmer
2008, 2(3) : 273-292 doi: 10.3934/amc.2008.2.273 +[Abstract](658) +[PDF](247.3KB) Cited By(28)
A survey of perfect codes
Olof Heden
2008, 2(2) : 223-247 doi: 10.3934/amc.2008.2.223 +[Abstract](757) +[PDF](323.1KB) Cited By(24)
A review of the available construction methods for Golomb rulers
Konstantinos Drakakis
2009, 3(3) : 235-250 doi: 10.3934/amc.2009.3.235 +[Abstract](602) +[PDF](218.2KB) Cited By(22)
Public key cryptography based on semigroup actions
Gérard Maze, Chris Monico and Joachim Rosenthal
2007, 1(4) : 489-507 doi: 10.3934/amc.2007.1.489 +[Abstract](691) +[PDF](248.6KB) Cited By(22)
A new family of linear maximum rank distance codes
John Sheekey
2016, 10(3) : 475-488 doi: 10.3934/amc.2016019 +[Abstract](763) +[PDF](390.1KB) Cited By(19)
On the order bounds for one-point AG codes
Olav Geil, Carlos Munuera, Diego Ruano and Fernando Torres
2011, 5(3) : 489-504 doi: 10.3934/amc.2011.5.489 +[Abstract](637) +[PDF](378.6KB) Cited By(18)
Efficient implementation of elliptic curve cryptography in wireless sensors
Diego F. Aranha, Ricardo Dahab, Julio López and Leonardo B. Oliveira
2010, 4(2) : 169-187 doi: 10.3934/amc.2010.4.169 +[Abstract](827) +[PDF](304.4KB) Cited By(17)
Geometric constructions of optimal optical orthogonal codes
T. L. Alderson and K. E. Mellinger
2008, 2(4) : 451-467 doi: 10.3934/amc.2008.2.451 +[Abstract](606) +[PDF](239.2KB) Cited By(16)
Linear nonbinary covering codes and saturating sets in projective spaces
Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini and Fernanda Pambianco
2011, 5(1) : 119-147 doi: 10.3934/amc.2011.5.119 +[Abstract](872) +[PDF](566.6KB) Cited By(15)
\begin{document}$np^s$\end{document} over \begin{document}$\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$\end{document}" >Constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$
Yonglin Cao, Yuan Cao, Hai Q. Dinh, Fang-Wei Fu, Jian Gao and Songsak Sriboonchitta
2018, 12(2) : 231-262 doi: 10.3934/amc.2018016 +[Abstract](467) +[HTML](335) +[PDF](555.31KB) PDF Downloads(150)
New constructions of systematic authentication codes from three classes of cyclic codes
Yunwen Liu, Longjiang Qu and Chao Li
2018, 12(1) : 1-16 doi: 10.3934/amc.2018001 +[Abstract](442) +[HTML](238) +[PDF](422.32KB) PDF Downloads(84)
Reduced access structures with four minimal qualified subsets on six participants
Motahhareh Gharahi and Shahram Khazaei
2018, 12(1) : 199-214 doi: 10.3934/amc.2018014 +[Abstract](381) +[HTML](236) +[PDF](447.58KB) PDF Downloads(82)
Indiscreet logarithms in finite fields of small characteristic
Robert Granger, Thorsten Kleinjung and Jens Zumbrägel
2018, 12(2) : 263-286 doi: 10.3934/amc.2018017 +[Abstract](423) +[HTML](268) +[PDF](541.54KB) PDF Downloads(82)
Duursma's reduced polynomial
Azniv Kasparian and Ivan Marinov
2017, 11(4) : 647-669 doi: 10.3934/amc.2017048 +[Abstract](569) +[HTML](309) +[PDF](439.95KB) PDF Downloads(81)
On some classes of codes with a few weights
Yiwei Liu and Zihui Liu
2018, 12(2) : 415-428 doi: 10.3934/amc.2018025 +[Abstract](323) +[HTML](224) +[PDF](376.9KB) PDF Downloads(69)
\begin{document}${{\mathbb{Z}}}_{p^r}{{\mathbb{Z}}}_{p^s}$\end{document}-additive cyclic codes" >On ${{\mathbb{Z}}}_{p^r}{{\mathbb{Z}}}_{p^s}$-additive cyclic codes
Joaquim Borges, Cristina Fernández-Córdoba and Roger Ten-Valls
2018, 12(1) : 169-179 doi: 10.3934/amc.2018011 +[Abstract](391) +[HTML](229) +[PDF](415.6KB) PDF Downloads(68)
The weight distribution of quasi-quadratic residue codes
Nigel Boston and Jing Hao
2018, 12(2) : 363-385 doi: 10.3934/amc.2018023 +[Abstract](309) +[HTML](209) +[PDF](429.38KB) PDF Downloads(64)
New families of strictly optimal frequency hopping sequence sets
Jingjun Bao
2018, 12(2) : 387-413 doi: 10.3934/amc.2018024 +[Abstract](324) +[HTML](221) +[PDF](613.0KB) PDF Downloads(60)
Several infinite families of p-ary weakly regular bent functions
Yanfeng Qi, Chunming Tang, Zhengchun Zhou and Cuiling Fan
2018, 12(2) : 303-315 doi: 10.3934/amc.2018019 +[Abstract](347) +[HTML](247) +[PDF](366.85KB) PDF Downloads(57)

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