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Advances in Mathematics of Communications

2014 , Volume 8 , Issue 3

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Structural properties of Costas arrays
Jonathan Jedwab and  Jane Wodlinger
2014, 8(3): 241-256 doi: 10.3934/amc.2014.8.241 +[Abstract](32) +[PDF](1169.5KB)
We apply combinatorial arguments to establish structural constraints on Costas arrays. We prove restrictions on when a Costas array can contain a large corner region whose entries are all 0. In particular, we prove a 2010 conjecture due to Russo, Erickson and Beard. We then constrain the vectors joining pairs of 1s in a Costas array by establishing a series of results on its number of "mirror pairs," namely pairs of these vectors having the same length but opposite slopes.
Higher genus universally decodable matrices (UDMG)
Steve Limburg , David Grant and  Mahesh K. Varanasi
2014, 8(3): 257-270 doi: 10.3934/amc.2014.8.257 +[Abstract](34) +[PDF](385.8KB)
We introduce the notion of Universally Decodable Matrices of Genus $g$ (UDMG), which for $g=0$ reduces to the notion of Universally Decodable Matrices (UDM) introduced in [8]. Fix positive $K,N,L$. A UDMG is a set $\{M_i|1\leq i\leq L\}$ of matrices of size $K \times N$ over a finite field such that the rows of any matrix of $K+g$ columns formed from the initial segments of the $M_i$ are linearly independent. We show that UDMG can be used to build approximately universal codes. We then provide a dictionary between UDMG and linear codes under the $m$-metric, which quickly provides constructions of UDMG and places bounds on the size of UDMG.
On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric
Daniele Bartoli and  Leo Storme
2014, 8(3): 271-280 doi: 10.3934/amc.2014.8.271 +[Abstract](37) +[PDF](318.2KB)
We discuss the functional codes $C_h(\mathcal{Q}_N)$, for small $h\geq 3$, $q>9$, and for $N\geq 6$. This continues the study of different classes of functional codes, performed on functional codes arising from quadrics and Hermitian varieties. Here, we consider the functional codes arising from the intersections of the algebraic hypersurfaces of small degree $h$ with a given non-singular quadric $\mathcal{Q}_N$ in PG$(N,q)$.
On the dual code of points and generators on the Hermitian variety $\mathcal{H}(2n+1,q^{2})$
M. De Boeck and  P. Vandendriessche
2014, 8(3): 281-296 doi: 10.3934/amc.2014.8.281 +[Abstract](35) +[PDF](386.5KB)
We study the dual linear code of points and generators on a non-singular Hermitian variety $\mathcal{H}(2n+1,q^2)$. We improve the earlier results for $n=2$, we solve the minimum distance problem for general $n$, we classify the $n$ smallest types of code words and we characterize the small weight code words as being a linear combination of these $n$ types.
Linear complexity of cyclotomic sequences of order six and BCH codes over GF(3)
Liqin Hu , Qin Yue and  Fengmei Liu
2014, 8(3): 297-312 doi: 10.3934/amc.2014.8.297 +[Abstract](56) +[PDF](375.9KB)
In this paper, we always assume that $p=6f+1$ is a prime. First, we calculate the values of exponential sums of cyclotomic classes of orders 3 and 6 over an extension field of GF(3). Then, we give a formula to compute the linear complexity of all $p^{n+1}$-periodic generalized cyclotomic sequences of order 6 over GF(3). After that, we compute the linear complexity and the minimal polynomial of a $p^{n+1}$-periodic, balanced and generalized cyclotomic sequence of order 6 over GF(3), which is analogous to a generalized Sidelnikov's sequence. At last, we give some BCH codes with prime length $p$ from cyclotomic sequences of orders three and six.
Construction of skew cyclic codes over $\mathbb F_q+v\mathbb F_q$
Fatmanur Gursoy , Irfan Siap and  Bahattin Yildiz
2014, 8(3): 313-322 doi: 10.3934/amc.2014.8.313 +[Abstract](54) +[PDF](318.7KB)
In this paper skew cyclic codes over the the family of rings $\mathbb{F}_q+v\mathbb{F}_q$ with $v^2=v$ are studied for the first time in its generality. Structural properties of skew cyclic codes over $\mathbb{F}_q+v\mathbb{F}_q$ are investigated through a decomposition theorem. It is shown that skew cyclic codes over this ring are principally generated. The idempotent generators of skew-cyclic codes over $\mathbb{F}_q$ and $\mathbb{F}_q+v\mathbb{F}_q$ have been considered for the first time in literature. Moreover, a BCH type bound is presented for the parameters of these codes.
How to obtain division algebras used for fast-decodable space-time block codes
Susanne Pumplün
2014, 8(3): 323-342 doi: 10.3934/amc.2014.8.323 +[Abstract](41) +[PDF](388.3KB)
We present families of unital algebras obtained through a doubling process from a cyclic central simple algebra $D=(K/F,\sigma,c)$, employing a $K$-automorphism $\tau$ and an element $d\in D^\times$. These algebras appear in the construction of iterated space-time block codes. We give conditions when these iterated algebras are division which can be used to construct fully diverse iterated codes. We also briefly look at algebras (and codes) obtained from variations of this method.
A general construction for monoid-based knapsack protocols
Giacomo Micheli and  Michele Schiavina
2014, 8(3): 343-358 doi: 10.3934/amc.2014.8.343 +[Abstract](37) +[PDF](545.3KB)
We present a generalized version of the knapsack protocol proposed by D. Naccache and J. Stern at the Proceedings of Eurocrypt (1997). Our new framework will allow the construction of other knapsack protocols having similar security features. We will outline a very concrete example of a new protocol using extension fields of a finite field of small characteristic instead of the prime field $\mathbb{Z}/p\mathbb{Z}$, but more efficient in terms of computational costs for asymptotically equal information rate and similar key size.
Sets of frequency hopping sequences under aperiodic Hamming correlation: Upper bound and optimal constructions
Xing Liu and  Daiyuan Peng
2014, 8(3): 359-373 doi: 10.3934/amc.2014.8.359 +[Abstract](45) +[PDF](371.1KB)
In order to evaluate the goodness of frequency hopping (FH) sequence design, the periodic Hamming correlation function is used as an important measure. Aperiodic Hamming correlation of FH sequences matters in real applications, while it received little attraction in the literature compared with periodic Hamming correlation. In this paper, an upper bound on the family size of FH sequences, with respect to the size of the frequency slot set, the sequence length, the maximum aperiodic Hamming correlation is established. Further, a construction of optimal FH sequence sets under aperiodic Hamming correlation from Reed-Solomon codes is presented, whose parameters meet the upper bound with equality. From generalized $m$ sequences (GM sequences) and generalized Gordon-Mills-Welch sequences (GGMW sequences), two classes of optimal FH sequence sets under aperiodic Hamming correlation are also presented, whose parameters meet the upper bound with equality.

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