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Advances in Mathematics of Communications

February 2017 , Volume 11 , Issue 1

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Recursive descriptions of polar codes
Noam Presman and Simon Litsyn
2017, 11(1): 1-65 doi: 10.3934/amc.2017001 +[Abstract](6830) +[HTML](163) +[PDF](5941.51KB)

Polar codes are recursive general concatenated codes. This property motivates a recursive formalization of the known decoding algorithms: Successive Cancellation, Successive Cancellation with Lists and Belief Propagation. Using such description allows an easy development of these algorithms for arbitrary polarizing kernels. Hardware architectures for these decoding algorithms are also described in a recursive way, both for Arıkan's standard polar codes and for arbitrary polarizing kernels.

New almost perfect, odd perfect, and perfect sequences from difference balanced functions with d-form property
Yang Yang, Xiaohu Tang and Guang Gong
2017, 11(1): 67-76 doi: 10.3934/amc.2017002 +[Abstract](3876) +[HTML](79) +[PDF](331.37KB)

By using shift sequences defined by difference balanced functions with d-form property, and column sequences defined by a mutually orthogonal almost perfect sequences pair, new almost perfect, odd perfect, and perfect sequences are obtained via interleaving method. Furthermore, the proposed perfect QAM+ sequences positively answer to the problem of the existence of perfect QAM+ sequences proposed by Boztaş and Udaya.

Some designs and codes invariant under the Tits group
Jamshid Moori and Amin Saeidi
2017, 11(1): 77-82 doi: 10.3934/amc.2017003 +[Abstract](3752) +[HTML](51) +[PDF](293.5KB)

In this paper, we construct some designs and associated binary codes from a primitive permutation representation of degree 1755 of the sporadic simple Tits group $^2F_4 \left(2 \right)'$. In particular, we construct a binary code $[1755, 26,1024]_2$ on which $^2F_4 \left(2 \right)'$ acts irreducibly. This is the smallest non-trivial irreducible $GF(2)$-module for our group.

DNA cyclic codes over rings
Nabil Bennenni, Kenza Guenda and Sihem Mesnager
2017, 11(1): 83-98 doi: 10.3934/amc.2017004 +[Abstract](5801) +[HTML](80) +[PDF](376.24KB)

In this paper we construct new DNA cyclic codes over rings. Firstly, we introduce a new family of DNA cyclic codes over the ring $R=\mathbb{F}_2[u]/(u.6)$. A direct link between the elements of such a ring and the $64$ codons used in the amino acids of the living organisms is established. Using this correspondence we study the reverse-complement properties of our codes. We use the edit distance between the codewords which is an important combinatorial notion for the DNA strands. Next, we define the Lee weight, the Gray map over the ring $R$ as well as the binary image of the DNA cyclic codes allowing the transfer of studying DNA codes into studying binary codes. Secondly, we introduce another new family of DNA skew cyclic codes constructed over the ring $\tilde {R}=\mathbb{F}_2+v\mathbb{F}_2=\{0, 1, v, v+1\}, $ where $v^2=v$. The codes obtained are cyclic reverse-complement over the ring $\tilde {R}$. Further we find their binary images and construct some explicit examples of such codes.

Cyclic codes over local Frobenius rings of order 16
Steven T. Dougherty, Abidin Kaya and Esengül Saltürk
2017, 11(1): 99-114 doi: 10.3934/amc.2017005 +[Abstract](4568) +[HTML](86) +[PDF](373.91KB)

We study cyclic codes over commutative local Frobenius rings of order 16 and give their binary images under a Gray map which is a generalization of the Gray maps on the rings of order 4. We prove that the binary images of cyclic codes are quasi-cyclic codes of index 4 and give examples of cyclic codes of various lengths constructed from these techniques including new optimal quasi-cyclic codes.

Generalized Hamming weights of codes over the $\mathcal{GH}$ curve
Alonso sepúlveda Castellanos
2017, 11(1): 115-122 doi: 10.3934/amc.2017006 +[Abstract](3254) +[HTML](61) +[PDF](334.02KB)

In this work, we studied the generalized Hamming weights of algebraic geometric Goppa codes on the $\mathcal{GH}$ curve. Especially, exact results on the second generalized Hamming weight are given for almost all cases. Furthermore, we apply the results obtained to show an example where the weight hierarchy characterizes the performance of the $\mathcal{GH}$ codes on a noiseless communication channel.

On the multiple threshold decoding of LDPC codes over GF(q)
Alexey Frolov and Victor Zyablov
2017, 11(1): 123-137 doi: 10.3934/amc.2017007 +[Abstract](4827) +[HTML](72) +[PDF](535.43KB)

We consider decoding of LDPC codes over GF(q) with a harddecision low-complexity majority algorithm, which is a generalization of the bit-flipping algorithm for binary LDPC codes. A modification of this algorithm with multiple thresholds is suggested. A lower estimate on the decoding radius realized by the new algorithm is derived. The estimate is shown to be better than the estimate for a single threshold majority decoder. At the same time, introducing multiple thresholds does not affect the order of decoding complexity.

AFSRs synthesis with the extended Euclidean rational approximation algorithm
Weihua Liu and Andrew Klapper
2017, 11(1): 139-150 doi: 10.3934/amc.2017008 +[Abstract](4126) +[HTML](63) +[PDF](313.8KB)

Pseudo-random sequence generators are widely used in many areas, such as stream ciphers, radar systems, Monte-Carlo simulations and multiple access systems. Generalization of linear feedback shift registers (LFSRs) and feedback with carry shift registers (FCSRs), algebraic feedback shift registers (AFSRs) [7] can generate pseudo-random sequences over an arbitrary finite field. In this paper, we present an algorithm derived from the Extended Euclidean Algorithm that can efficiently find a smallest AFSR over a quadratic field for a given sequence. It is an analog of the Extended Euclidean Rational Approximation Algorithm [1] used in solving the FCSR synthesis problem. For a given sequence $\mathbf{a}$, $2\Lambda(\alpha)+1$ terms of sequence $\mathbf{a}$ are needed to find the smallest AFSR, where $\Lambda(\alpha)$ is a complexity measure that reflects the size of the smallest AFSR that outputs $\mathbf{a}$.

Frequency hopping sequences with optimal aperiodic Hamming correlation by interleaving techniques
Xing Liu and Daiyuan Peng
2017, 11(1): 151-159 doi: 10.3934/amc.2017009 +[Abstract](3871) +[HTML](61) +[PDF](321.19KB)

Aperiodic Hamming correlation is an important criterion for evaluating the goodness of frequency hopping (FH) sequence design, while it received little attraction in the literature. In this paper, a construction of FH sequences with optimal aperiodic Hamming correlation by interleaving techniques is presented. Further, a class of one-coincidence FH sequence sets under aperiodic Hamming correlation is proposed. By employing the one-coincidence FH sequence sets, a class of FH sequence sets with optimal aperiodic Hamming correlation is also constructed by interleaving techniques.

Forward-secure identity-based encryption with direct chosen-ciphertext security in the standard model
Yang Lu and Jiguo Li
2017, 11(1): 161-177 doi: 10.3934/amc.2017010 +[Abstract](4115) +[HTML](65) +[PDF](590.78KB)

The paradigm of forward security provides a promising approach to deal with the key exposure problem as it can effectively minimize the damage caused by the key exposure. In this paper, we develop a new forward-secure identity-based encryption scheme without random oracles. We formally prove that the proposed scheme is secure against adaptive chosen-ciphertext attacks in the standard model. In the proposed scheme, the running time of the private key extraction and decryption algorithms and the sizes of the user's initial private key and the ciphertext are independent on the total number of time periods, and any other performance parameter has at most log-squared complexity in terms of the total number of time periods. Compared with the previous forward-secure identity-based encryption schemes, the proposed scheme enjoys obvious advantage in the overall performance. To the best of our knowledge, it is the first forward-secure identity-based encryption scheme that achieves direct chosen-ciphertext security in the standard model.

Codes from hall planes of odd order
J. D. Key, T. P. McDonough and V. C. Mavron
2017, 11(1): 179-185 doi: 10.3934/amc.2017011 +[Abstract](3262) +[HTML](55) +[PDF](303.42KB)

We show explicitly that the dimension of the ternary code of the Hall plane of order 9 is greater than the dimension of the ternary code of the desarguesian plane of order 9. The proof requires finding a word with some defined properties in the dual ternary code of the desarguesian plane of order 9. The idea can be generalised for other orders, provided that words in the dual code of the desarguesian projective plane that have the specified properties can be found.

More constructions of near optimal codebooks associated with binary sequences
Xiwang Cao, Wun-Seng Chou and Xiyong Zhang
2017, 11(1): 187-202 doi: 10.3934/amc.2017012 +[Abstract](3342) +[HTML](59) +[PDF](391.4KB)

An $(N, K)$ codebook $\mathcal{C}$ is a collection of unit norm vectors in a $K$-dimensional vectors space. In applications of codebooks such as CDMA, those vectors in a codebook should have a small maximum magnitude of inner products, denoted by $I_{\max}(\mathcal{C})$, between any pair of distinct code vectors. Since the famous Welch bound is a lower bound on $I_{\max}(\mathcal{C})$, it is desired to construct codebooks meeting the Welch bound strictly or asymptotically. Recently, N. Y. Yu presents a method for constructing codebooks associated with a binary sequence from a $\Phi$-transform matrix. Using this method, he discovers new classes of codebooks with nontrivial bounds on the maximum inner products from Fourier and Hadamard matrices. We construct more near optimal codebooks by Yu's idea. We first provide more choices of binary sequences. We also show more choices of the $\Phi$-transform matrices. Therefore, we can present more codebooks $\mathcal{C}$ with nontrivial bounds on their $I_{\max}(\mathcal{C})$. Our work can serve as a complement of Yu's work.

Information retrieval and the average number of input clues
Tero Laihonen
2017, 11(1): 203-223 doi: 10.3934/amc.2017013 +[Abstract](3464) +[HTML](58) +[PDF](464.02KB)

Information retrieval in an associative memory was introduced in a recent paper by Yaakobi and Bruck. The associative memory is represented by a graph where the vertices correspond to the stored information units and the edges to associations between them. The goal is to find a stored information unit in the memory using input clues. In this paper, we study the minimum average number of input clues needed to find the sought information unit in the infinite king grid. We provide a geometric approach to determine the minimum number of input clues. Using this approach we are able to find optimal results and bounds on the number of input clues. The model by Yaakobi and Bruck has also applications to sensor networks monitoring and Levenshtein's sequence reconstruction problem.

On defining generalized rank weights
Relinde Jurrius and Ruud Pellikaan
2017, 11(1): 225-235 doi: 10.3934/amc.2017014 +[Abstract](3415) +[HTML](60) +[PDF](341.0KB)

This paper investigates the generalized rank weights, with a definition implied by the study of the generalized rank weight enumerator. We study rank metric codes over $L$, where $L$ is a finite extension of a field $K$. This is a generalization of the case where $K=\mathbb{F}_q $ and $L={\mathbb{F}_{{q^m}}}$ of Gabidulin codes to arbitrary characteristic. We show equivalence to previous definitions, in particular the ones by Kurihara-Matsumoto-Uyematsu [12,13], Oggier-Sboui [16] and Ducoat [6]. As an application of the notion of generalized rank weights, we discuss codes that are degenerate with respect to the rank metric.

On the arithmetic autocorrelation of the Legendre sequence
Richard Hofer and Arne Winterhof
2017, 11(1): 237-244 doi: 10.3934/amc.2017015 +[Abstract](4321) +[HTML](76) +[PDF](273.67KB)

The Legendre sequence possesses several desirable features of pseudorandomness in view of different applications such as a high linear complexity (profile) for cryptography and a small (aperiodic) autocorrelation for radar, gps, or sonar. Here we prove the first nontrivial bound on its arithmetic autocorrelation, another figure of merit introduced by Mandelbaum for errorcorrecting codes.

Some results on the structure of constacyclic codes and new linear codes over GF(7) from quasi-twisted codes
Nuh Aydin, Nicholas Connolly and Markus Grassl
2017, 11(1): 245-258 doi: 10.3934/amc.2017016 +[Abstract](5503) +[HTML](69) +[PDF](361.66KB)

One of the most important and challenging problems in coding theory is to construct codes with good parameters. There are various methods to construct codes with the best possible parameters. A promising and fruitful approach has been to focus on the class of quasi-twisted (QT) codes which includes constacyclic codes as a special case. This class of codes is known to contain many codes with good parameters. Although constacyclic codes and QT codes have been the subject of numerous studies and computer searches over the last few decades, we have been able to discover a new fundamental result about the structure of constacyclic codes which was instrumental in our comprehensive search method for new QT codes over GF(7). We have been able to find 41 QT codes with better parameters than the previously best known linear codes. Furthermore, we derived a number of additional new codes via Construction X as well as standard constructions, such as shortening and puncturing.

5-SEEDs from the lifted Golay code of length 24 over Z4
Jianying Fang
2017, 11(1): 259-266 doi: 10.3934/amc.2017017 +[Abstract](3852) +[HTML](83) +[PDF](316.52KB)

Spontaneous emission error designs (SEEDs) are combinatorial objects that can be used to construct quantum jump codes. The lifted Golay code $G_{24}$ of length $24$ over $\mathbb{Z}_4$ is cyclic self-dual code. It is known that $G_{24}$ yields $5$-designs. In this paper, by using the generator matrices of bordered double circulant codes, we obtain $22$ mutually disjoint $5$-$(24, k, \lambda)$ designs with $(k, \lambda)=(8, 1), $ $(10, 36), $ $(12,1584)$ and $5$-$(24, k;22)$-SEEDs for $k=8, $ $10, $ $12, $ $13$ from $G_{24}$.

2020 Impact Factor: 0.935
5 Year Impact Factor: 0.976
2020 CiteScore: 1.5




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