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Advances in Mathematics of Communications

2009 , Volume 3 , Issue 2

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New design of quaternary LCZ and ZCZ sequence set from binary LCZ and ZCZ sequence set
Ji-Woong Jang , Young-Sik Kim and  Sang-Hyo Kim
2009, 3(2): 115-124 doi: 10.3934/amc.2009.3.115 +[Abstract](34) +[PDF](146.1KB)
Abstract:
In this paper, we suggest two construction methods of quaternary low and zero correlation zone (LCZ and ZCZ) sequence set. The new construction methods use a binary LCZ/ZCZ sequence set and the Gray mapping to produce new quaternary LCZ/ZCZ sequence sets. The parameters of the generated quaternary LCZ/ZCZ sequence set are the same as those of the employed binary LCZ/ZCZ sequence set. That means, an optimal quaternary LCZ/ZCZ sequence set can be constructed from an optimal binary LCZ/ZCZ sequence set.
Sperner capacity of small digraphs
Lasse Kiviluoto , Patric R. J. Östergård and  Vesa P. Vaskelainen
2009, 3(2): 125-133 doi: 10.3934/amc.2009.3.125 +[Abstract](37) +[PDF](162.3KB)
Abstract:
The classical concept of Shannon capacity of undirected graphs was extended by Gargano, Körner, and Vaccaro to digraphs in the early 1990s, and termed Sperner capacity. Shannon, in his seminal work, determined the capacities for all isomorphism classes of undirected graphs with up to five vertices, except for the 5-cycle, which was finally settled by Lovász in 1979. The work of Shannon is here paralleled for digraphs; the Sperner capacity is determined for all but 8 of the 9846 isomorphism classes of digraphs with at most 5 vertices.
Two binary sequence families with large merit factor
Kai-Uwe Schmidt , Jonathan Jedwab and  Matthew G. Parker
2009, 3(2): 135-156 doi: 10.3934/amc.2009.3.135 +[Abstract](44) +[PDF](265.3KB)
Abstract:
We calculate the asymptotic merit factor, under all rotations of sequence elements, of two families of binary sequences derived from Legendre sequences. The rotation is negaperiodic for the first family, and periodic for the second family. In both cases the maximum asymptotic merit factor is 6. As a consequence, we obtain the first two families of skew-symmetric sequences with known asymptotic merit factor, which is also 6 in both cases.
New linear codes with prescribed group of automorphisms found by heuristic search
Axel Kohnert and  Johannes Zwanzger
2009, 3(2): 157-166 doi: 10.3934/amc.2009.3.157 +[Abstract](59) +[PDF](167.5KB)
Abstract:
In this paper, we present a new heuristic algorithm for solving certain systems of Diophantine inequalities. A variant which involves Monte-Carlo search is also applyable to more general problems. Our goal was the construction of point sets in PG$(k-1,q)$ with fixed cardinality and small maximal intersection number with the lines. These points sets correspond to $k$-dimensional linear codes over $\mathbb F_q$ with high minimum distance. We obtained them by prescribing a certain nontrivial subgroup of GL$(k,q)$ to be contained in their automorphism group. Following a method which was first introduced by Kramer and Mesner in the 1970s, this allows a strong reduction in the size of the corresponding Diophantine systems. Doing so we found a lot of new record breaking linear codes for the cases $q = 2, 3, 4, 5, 7, 8, 9$ from which at least $6$ are optimal.
Infinite families of recursive formulas generating power moments of ternary Kloosterman sums with square arguments arising from symplectic groups
Dae San Kim
2009, 3(2): 167-178 doi: 10.3934/amc.2009.3.167 +[Abstract](39) +[PDF](171.9KB)
Abstract:
In this paper, we construct two infinite families of ternary linear codes associated with double cosets with respect to certain maximal parabolic subgroups of the symplectic group $S_p(2n, q)$. Here $q$ is a power of three. Then we obtain infinite families of recursive formulas for the power moments of Kloosterman sums with square arguments and for the even power moments of those in terms of the frequencies of weights in the codes. This is done via Pless power moment identities and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of ''Gauss sums'' for the symplectic groups $S_p(2n, q)$.
On isometries for convolutional codes
Heide Gluesing-Luerssen
2009, 3(2): 179-203 doi: 10.3934/amc.2009.3.179 +[Abstract](29) +[PDF](331.2KB)
Abstract:
In this paper we will discuss isometries and strong isometries for convolutional codes. Isometries are weight-preserving module isomorphisms whereas strong isometries are, in addition, degree-preserving. Special cases of these maps are certain types of monomial transformations. We will show a form of MacWilliams Equivalence Theorem, that is, each isometry between convolutional codes is given by a monomial transformation. Examples show that strong isometries cannot be characterized this way, but special attention paid to the weight adjacency matrices allows for further descriptions. Various distance parameters appearing in the literature on convolutional codes will be discussed as well.
Further results on implicit factoring in polynomial time
Santanu Sarkar and  Subhamoy Maitra
2009, 3(2): 205-217 doi: 10.3934/amc.2009.3.205 +[Abstract](42) +[PDF](235.3KB)
Abstract:
In PKC 2009, May and Ritzenhofen presented interesting problems related to factoring large integers with some implicit hints. One of the problems is as follows. Consider $N_1 = p_1 q_1$ and $N_2 = p_2 q_2$, where $p_1, p_2, q_1, q_2$ are large primes. The primes $p_1, p_2$ are of same bit-size with the constraint that certain amount of Least Significant Bits (LSBs) of $p_1, p_2$ are same. Further the primes $q_1, q_2$ are of same bit-size without any constraint. May and Ritzenhofen proposed a strategy to factorize both $N_1, N_2$ in poly$(\log N)$ time ($N$ is an integer with same bit-size as $N_1, N_2$) with the implicit information that $p_1, p_2$ share certain amount of LSBs. We explore the same problem with a different lattice-based strategy. In a general framework, our method works when implicit information is available related to Least Significant as well as Most Significant Bits (MSBs). Given $q_1, q_2 \approx$Nα, we show that one can factor $N_1, N_2$ simultaneously in poly$(\log N)$ time (under some assumption related to Gröbner Basis) when $p_1, p_2$ share certain amount of MSBs and/or LSBs. We also study the case when $p_1, p_2$ share some bits in the middle. Our strategy presents new and encouraging results in this direction. Moreover, some of the observations by May and Ritzenhofen get improved when we apply our ideas for the LSB case.

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