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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.

  • AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
  • Publishes 4 issues a year in February, May, August and November.
  • Publishes online only.
  • Indexed in Science Citation Index E, CompuMath Citation Index, Current Contents/Physics, Chemical, & Earth Sciences, INSPEC, Mathematical Reviews, MathSciNet, PASCAL/CNRS, Scopus, Web of Science, Zentralblatt MATH and dblp: computer science bibliography.
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  • Shandong University is a founding institution of AMC.
  • AMC is a publication of the American Institute of Mathematical Sciences. All rights reserved.

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Improved attacks on knapsack problem with their variants and a knapsack type ID-scheme
Konstantinos A. Draziotis and Anastasia Papadopoulou
2018, 12(3) : 429-449 doi: 10.3934/amc.2018026 +[Abstract](118) +[HTML](34) +[PDF](543.19KB)
Abstract:

In the present study we consider two variants of Schnorr-Shevchenko method (SS) for solving hard knapsack problems, which are on average faster than the SS method. Furthermore, we study the compact knapsack problem i.e. the solution space is not {0, 1} as in knapsack problem but some larger set, and we present an algorithm to attack this problem. Finally, we provide a three move sound id-scheme based on the compact knapsack problem.

The average dimension of the Hermitian hull of constayclic codes over finite fields of square order
Somphong Jitman and Ekkasit Sangwisut
2018, 12(3) : 451-463 doi: 10.3934/amc.2018027 +[Abstract](86) +[HTML](40) +[PDF](377.03KB)
Abstract:

The hulls of linear and cyclic codes have been extensively studied due to their wide applications. The dimensions and average dimension of the Euclidean hull of linear and cyclic codes have been well-studied. In this paper, the average dimension of the Hermitian hull of constacyclic codes of length \begin{document}$n$\end{document} over a finite field \begin{document}$\mathbb{F}_{q^2}$\end{document} is determined together with some upper and lower bounds. It turns out that either the average dimension of the Hermitian hull of constacyclic codes of length \begin{document}$n$\end{document} over \begin{document}$\mathbb{F}_{q^2}$\end{document} is zero or it grows the same rate as \begin{document}$n$\end{document}. Comparison to the average dimension of the Euclidean hull of cyclic codes is discussed as well.

Architecture-aware coding for distributed storage: Repairable block failure resilient codes
Gokhan Calis and O. Ozan Koyluoglu
2018, 12(3) : 465-503 doi: 10.3934/amc.2018028 +[Abstract](76) +[HTML](34) +[PDF](1343.62KB)
Abstract:

In large scale distributed storage systems (DSS) deployed in cloud computing, correlated failures resulting in simultaneous failure (or, unavailability) of blocks of nodes are common. In such scenarios, the stored data or a content of a failed node can only be reconstructed from the available live nodes belonging to the available blocks. To analyze the resilience of the system against such block failures, this work introduces the framework of Block Failure Resilient (BFR) codes, wherein the data (e.g., a file in DSS) can be decoded by reading out from a same number of codeword symbols (nodes) from a subset of available blocks of the underlying codeword. Further, repairable BFR codes are introduced, wherein any codeword symbol in a failed block can be repaired by contacting a subset of remaining blocks in the system. File size bounds for repairable BFR codes are derived, and the trade-off between per node storage and repair bandwidth is analyzed, and the corresponding minimum storage regenerating (BFR-MSR) and minimum bandwidth regenerating (BFR-MBR) points are derived. Explicit codes achieving the two operating points for a special case of parameters are constructed, wherein the underlying regenerating codewords are distributed to BFR codeword symbols according to combinatorial designs. Finally, BFR locally repairable codes (BFR-LRC) are introduced, an upper bound on the resilience is derived and optimal code construction are provided by a concatenation of Gabidulin and MDS codes. Repair efficiency of BFR-LRC is further studied via the use of BFR-MSR/MBR codes as local codes. Code constructions achieving optimal resilience for BFR-MSR/MBR-LRCs are provided for certain parameter regimes. Overall, this work introduces the framework of block failures along with optimal code constructions, and the study of architecture-aware coding for distributed storage systems.

Hamming correlation of higher order
Ming Su and Arne Winterhof
2018, 12(3) : 505-513 doi: 10.3934/amc.2018029 +[Abstract](91) +[HTML](35) +[PDF](333.03KB)
Abstract:

We introduce a new measure of pseudorandomness, the (periodic) Hamming correlation of order \begin{document}$\ell$\end{document} which generalizes the Hamming autocorrelation (\begin{document}$\ell = 2$\end{document}). We analyze the relation between the Hamming correlation of order \begin{document}$\ell$\end{document} and the periodic analog of the correlation measure of order \begin{document}$\ell$\end{document} introduced by Mauduit and Sárközy. Roughly speaking, the correlation measure of order \begin{document}$\ell$\end{document} is a finer measure than the Hamming correlation of order \begin{document}$\ell$\end{document}. However, the latter can be much faster calculated and still detects some undesirable linear structures. We analyze examples of sequences with optimal Hamming correlation and show that they have large Hamming correlation of order \begin{document}$\ell$\end{document} for some very small \begin{document}$\ell>2$\end{document}. Thus they have some undesirable linear structures, in particular in view of cryptographic applications such as secure communications.

A note on some algebraic trapdoors for block ciphers
Marco Calderini
2018, 12(3) : 515-524 doi: 10.3934/amc.2018030 +[Abstract](61) +[HTML](32) +[PDF](325.8KB)
Abstract:

We provide sufficient conditions to guarantee that a translation based cipher is not vulnerable with respect to the partition-based trapdoor. This trapdoor has been introduced, recently, by Bannier et al. (2016) and it generalizes that introduced by Paterson in 1999. Moreover, we discuss the fact that studying the group generated by the round functions of a block cipher may not be sufficient to guarantee security against these trapdoors for the cipher.

On the linear complexities of two classes of quaternary sequences of even length with optimal autocorrelation
Pinhui Ke, Yueqin Jiang and Zhixiong Chen
2018, 12(3) : 525-539 doi: 10.3934/amc.2018031 +[Abstract](78) +[HTML](30) +[PDF](398.94KB)
Abstract:

Let \begin{document}$q$\end{document} be a prime greater than 4. In this paper, we determine the coefficients of the discrete Fourier transform over the finite field \begin{document}$\mathbb {F}_q$\end{document} of two classes of quaternary sequences of even length with optimal autocorrelation. They are quaternary sequence with period \begin{document}$2p$\end{document} derived from binary Legendre sequences and quaternary sequence with period \begin{document}$2p(p+2)$\end{document} derived from twin-prime sequences pair. As applications, the linear complexities over the finite field \begin{document}$\mathbb {F}_q$\end{document} of both of the quaternary sequences are determined.

An asymmetric ZCZ sequence set with inter-subset uncorrelated property and flexible ZCZ length
Longye Wang, Gaoyuan Zhang, Hong Wen and Xiaoli Zeng
2018, 12(3) : 541-552 doi: 10.3934/amc.2018032 +[Abstract](69) +[HTML](30) +[PDF](466.09KB)
Abstract:

In this paper, we propose a novel method for constructing new uncorrelated asymmetric zero correlation zone (UA-ZCZ) sequence sets by interleaving perfect sequences. As a type of ZCZ sequence set, an A-ZCZ sequence set consists of multiple sequence subsets. Different subsets are correlated in conventional A-ZCZ sequence set but uncorrelated in our scheme. In other words, the cross-correlation function (CCF) between two arbitrary sequences which belong to different subsets has quite a large zero cross-correlation zone (ZCCZ). Analytical results demonstrate that the UA-ZCZ sequence set proposed herein is optimal with respect to the upper bound of ZCZ sequence set. Specifically, our scheme enables the flexible selection of ZCZ length, which makes it extremely valuable for designing spreading sequences for quasi-synchronous code-division multiple-access (QS-CDMA) systems.

A first step towards the skew duadic codes
Delphine Boucher
2018, 12(3) : 553-577 doi: 10.3934/amc.2018033 +[Abstract](73) +[HTML](37) +[PDF](442.57KB)
Abstract:

This text gives a first definition of the $θ$-duadic codes where \begin{document}$θ$\end{document} is an automorphism of \begin{document}$\mathbb{F}_q$\end{document}. A link with the self-orthogonal \begin{document}$θ$\end{document}-cyclic codes is established. A construction and an enumeration are provided when \begin{document}$q$\end{document} is the square of a prime number \begin{document}$p$\end{document}. In addition, new self-dual binary codes \begin{document}$ [72, 36, 12] $\end{document} are obtained from extended $θ$-duadic codes defined on \begin{document}$\mathbb{F}_4$\end{document}.

Parameters of LCD BCH codes with two lengths
Haode Yan, Hao Liu, Chengju Li and Shudi Yang
2018, 12(3) : 579-594 doi: 10.3934/amc.2018034 +[Abstract](81) +[HTML](59) +[PDF](401.15KB)
Abstract:

In this paper, we study LCD BCH codes over the finite field GF\begin{document}$(q)$\end{document} with two types of lengths \begin{document}$n$\end{document}, where \begin{document}$n = q^l+1$\end{document} and \begin{document}$n = (q^l+1)/(q+1)$\end{document}. Several classes of LCD BCH codes are given and their parameters are determined or bounded by exploring the cyclotomic cosets modulo \begin{document}$n$\end{document}. For \begin{document}$n = q^l+1$\end{document}, we determine the dimensions of the codes with designed distance \begin{document}$δ$\end{document}, where \begin{document}$q^{\lfloor\frac{l+1}{2}\rfloor}+1 ≤ δ ≤q ^{\lfloor\frac{l+3}{2}\rfloor}+1$\end{document}. For \begin{document}$n = (q^l+1)/(q+1)$\end{document}, the dimensions of the codes with designed distance \begin{document}$δ$\end{document} are presented, where \begin{document}$2 ≤ δ ≤q ^\frac{l-1}{2}+1$\end{document}.

A connection between sumsets and covering codes of a module
Otávio J. N. T. N. dos Santos and Emerson L. Monte Carmelo
2018, 12(3) : 595-605 doi: 10.3934/amc.2018035 +[Abstract](64) +[HTML](37) +[PDF](288.05KB)
Abstract:

In this work we focus on a connection between sumsets and covering codes in an arbitrary finite module. For this purpose, bounds on a new problem on sumsets are obtained from well-known results of additive number theory, namely, the Cauchy-Davenport theorem, the Vosper theorem and a theorem due to Hamidoune-Rødseth. As an application, the approach is able to extend the Blokhuis-Lam theorems and a construction of covering codes by Honkala to an arbitrary module.

On self-orthogonal designs and codes related to Held's simple group
Crnković Dean, Vedrana Mikulić Crnković and Bernardo G. Rodrigues
2018, 12(3) : 607-628 doi: 10.3934/amc.2018036 +[Abstract](65) +[HTML](55) +[PDF](445.57KB)
Abstract:

A construction of designs acted on by simple primitive groups is used to find some 1-designs and associated self-orthogonal, decomposable and irreducible codes that admit the simple group \begin{document}${\rm He}$\end{document} of Held as an automorphism group. The properties of the codes are given and links with modular representation theory are established. Further, we introduce a method of constructing self-orthogonal binary codes from orbit matrices of weakly self-orthogonal designs. Furthermore, from the support designs of the obtained self-orthogonal codes we construct strongly regular graphs with parameters (21, 10, 3, 6), (28, 12, 6, 4), (49, 12, 5, 2), (49, 18, 7, 6), (56, 10, 0, 2), (63, 30, 13, 15), (105, 32, 4, 12), (112, 30, 2, 10) and (120, 42, 8, 18).

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](801) +[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](721) +[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](833) +[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](679) +[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](757) +[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](874) +[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](698) +[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](925) +[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](681) +[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](965) +[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](562) +[HTML](350) +[PDF](555.31KB) PDF Downloads(172)
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](499) +[HTML](287) +[PDF](541.54KB) PDF Downloads(94)
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](537) +[HTML](239) +[PDF](422.32KB) PDF Downloads(85)
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](432) +[HTML](237) +[PDF](447.58KB) PDF Downloads(83)
Duursma's reduced polynomial
Azniv Kasparian and Ivan Marinov
2017, 11(4) : 647-669 doi: 10.3934/amc.2017048 +[Abstract](613) +[HTML](333) +[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](361) +[HTML](235) +[PDF](376.9KB) PDF Downloads(79)
The weight distribution of quasi-quadratic residue codes
Nigel Boston and Jing Hao
2018, 12(2) : 363-385 doi: 10.3934/amc.2018023 +[Abstract](359) +[HTML](221) +[PDF](429.38KB) PDF Downloads(71)
\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](443) +[HTML](230) +[PDF](415.6KB) PDF Downloads(69)
New families of strictly optimal frequency hopping sequence sets
Jingjun Bao
2018, 12(2) : 387-413 doi: 10.3934/amc.2018024 +[Abstract](381) +[HTML](237) +[PDF](613.0KB) PDF Downloads(65)
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](416) +[HTML](265) +[PDF](366.85KB) PDF Downloads(62)

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