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Advances in Mathematics of Communications

2013 , Volume 7 , Issue 3

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Arrays over roots of unity with perfect autocorrelation and good ZCZ cross-correlation
Samuel T. Blake, Thomas E. Hall and Andrew Z. Tirkel
2013, 7(3): 231-242 doi: 10.3934/amc.2013.7.231 +[Abstract](334) +[PDF](415.3KB)
We present a new construction for two-dimensional, perfect autocorrelation arrays over roots of unity. These perfect arrays are constructed from a block of perfect column sequences. Other blocks are constructed from the first block, to generate a block-circulant structure. The columns are then multiplied by a perfect sequence over roots of unity, which, when folded into an array commensurate with our block width has the array orthogonality property. The size of the arrays is commensurate with the length of the underlying perfect sequences. For a given size we can construct an exponential number of inequivalent perfect arrays. For each perfect array we construct a family of arrays whose pairwise cross-correlation values are almost all zero (large zero correlation zones (ZCZ)). We present experimental evidence that this construction for perfect arrays can be generalized to higher dimensions.
Improved bounds for the implicit factorization problem
Yao Lu, Rui Zhang and Dongdai Lin
2013, 7(3): 243-251 doi: 10.3934/amc.2013.7.243 +[Abstract](311) +[PDF](327.0KB)
We study the problem of integer factoring with implicit hints. This problem is described as follows: Let $N_{1}=p_{1}q_{1},\dots,N_{k}=p_{k}q_{k}$ be $k$ different RSA moduli of same bit-size, where $q_1,\dots,q_k$ are of the same bit-size too. Given the implicit information that $p_{1},\dots,p_{k}$ share some certain portions of bit pattern, under what condition is it possible to factorize $N_{1},\dots,N_{k}$ efficiently? This problem has been studied in many references recently and many interesting results have been obtained. In this paper, we modify the previous algorithm presented by Sarkar and Maitra (IEEE TIT 57(6): 4002-4013, 2011). We show that our result achieves an improved generalized bounds in the cases where $p_{1},\dots,p_{k}$ share some amount of 1) most significant bits (MSBs); 2) least significant bits (LSBs); 3) MSBs and LSBs together. As far as we are aware, our result is better than all known results.
On codes over rings invariant under affine groups
Kanat Abdukhalikov
2013, 7(3): 253-265 doi: 10.3934/amc.2013.7.253 +[Abstract](305) +[PDF](342.5KB)
We give a description of extended cyclic codes of length $p^n$ over a field and over the ring of integers modulo $p^e$ admitting the affine group $AGL_m(p^t)$, $n=mt$, as a permutation group.
The classification of complementary information set codes of lengths $14$ and $16$
Finley Freibert
2013, 7(3): 267-278 doi: 10.3934/amc.2013.7.267 +[Abstract](293) +[PDF](384.7KB)
In the paper ``A new class of codes for Boolean masking of cryptographic computations,'' Carlet, Gaborit, Kim, and Solé defined a new class of rate one-half binary codes called complementary information set (or CIS) codes. The authors then classified all CIS codes of length less than or equal to 12. CIS codes have relations to classical Coding Theory as they are a generali-zation of self-dual codes. As stated in the paper, CIS codes also have important practical applications as they may improve the cost of masking cryptographic algorithms against side channel attacks. In this paper, we give a complete classification result for length 14 CIS codes using an equivalence relation on $GL(n,\mathbb{F}_2)$. We also give a new classification for all binary $[16,8,3]$ and $[16,8,4]$ codes. We then complete the classification for length 16 CIS codes and give additional classifications for optimal CIS codes of lengths 20 and 26.
New nonexistence results for spherical designs
Peter Boyvalenkov and Maya Stoyanova
2013, 7(3): 279-292 doi: 10.3934/amc.2013.7.279 +[Abstract](314) +[PDF](391.3KB)
New nonexistence results for spherical designs of odd strength and odd cardinality are proved by improvements on previously applied polynomial techniques. This implies new bounds on the designs under consideration either in small dimensions and in certain asymptotic process.
New classes of optimal frequency hopping sequences with low hit zone
Xianhua Niu, Daiyuan Peng and Zhengchun Zhou
2013, 7(3): 293-310 doi: 10.3934/amc.2013.7.293 +[Abstract](328) +[PDF](525.5KB)
In this paper, a new design of frequency hopping sequences (FHSs) sets with low hit zone (LHZ) is presented based on interleaving technique. The key idea of the new design is to use short FHSs with good Hamming correlation together with certain appropriate shift sequences to construct a set of long FHSs with LHZ. By the new design, new sets of FHSs meeting the Peng-Fan-Lee bound are obtained. It is shown that all the sequences in the proposed FHS sets are shift distinct. The proposed FHS sets are suitable for quasi-synchronous frequency hopping code division multiple access systems to eliminate multiple-access interference.
Average complexities of access structures on five participants
Motahhareh Gharahi and Massoud Hadian Dehkordi
2013, 7(3): 311-317 doi: 10.3934/amc.2013.7.311 +[Abstract](269) +[PDF](259.6KB)
In this paper, we consider the 12 access structures on five participants for which determining the exact values of the average complexities remained as open problems in Jackson and Martin's paper [6]. We establish the exact values of the average complexities of these access structures.
A 3-cycle construction of complete arcs sharing $(q+3)/2$ points with a conic
Daniele Bartoli, Alexander A. Davydov, Stefano Marcugini and Fernanda Pambianco
2013, 7(3): 319-334 doi: 10.3934/amc.2013.7.319 +[Abstract](372) +[PDF](424.0KB)
In the projective plane $PG(2,q),$ $q\equiv 2$ $(\bmod~3)$ odd prime power, $ q\geq 11,$ an explicit construction of $\frac{1}{2}(q+7)$-arcs sharing $ \frac{1}{2}(q+3)$ points with an irreducible conic is considered. The construction is based on 3-orbits of some projectivity, called 3-cycles. For every $q,$ variants of the construction give non-equivalent arcs. It allows us to obtain complete $\frac{1}{ 2}(q+7)$-arcs for $q\leq 4523.$ Moreover, for $q=17,59$ there exist variants that are incomplete arcs. Completing these variants we obtained complete $( \frac{1}{2}(q+3)+\delta)$-arcs with $ \delta =4,$ $q=17,$ and $\delta =3,$ $q=59$; a description of them as union of some symmetrical objects is given.
On the distribution of auto-correlation value of balanced Boolean functions
Yu Zhou
2013, 7(3): 335-347 doi: 10.3934/amc.2013.7.335 +[Abstract](344) +[PDF](372.0KB)
In this paper, we study the lower bound on the sum-of-square indicator of balanced Boolean functions obtained by Son, et al. in 1998, and give a sufficient and necessary condition under which balanced Boolean functions achieve this lower bound. We introduce a new general class of balanced Boolean functions in $n$ variables $(n\geq 4)$ with optimal auto-correlation distribution, and we study two sub-classes more explicitely. Finally, we study the sets of Boolean functions having a same auto-correlation distribution, and derive a lower bound on the number of elements in such set.
On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes
W. Cary Huffman
2013, 7(3): 349-378 doi: 10.3934/amc.2013.7.349 +[Abstract](338) +[PDF](566.3KB)
In [7], self-orthogonal additive codes over $\mathbb{F}_4$ under the trace inner product were connected to binary quantum codes; a similar connection was given in the nonbinary case in [33]. In this paper we consider a natural generalization of additive codes called $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. We examine a number of classical results from the theory of $\mathbb{F}_q$-linear codes, and see how they must be modified to give analogous results for $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Included in the topics examined are the MacWilliams Identities, the Gleason polynomials, the Gleason-Pierce Theorem, Mass Formulas, the Balance Principle, the Singleton Bound, and MDS codes. We also classify certain of these codes for small lengths using the theory developed.

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