# American Institute of Mathematical Sciences

ISSN:
1930-5346

eISSN:
1930-5338

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## Advances in Mathematics of Communications

2013 , Volume 7 , Issue 2

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2013, 7(2): 113-125 doi: 10.3934/amc.2013.7.113 +[Abstract](640) +[PDF](371.4KB)
Abstract:
A pair of two sequences is called the even periodic (odd periodic) complementary sequence pair if the sum of their even periodic (odd periodic) correlation function is a delta function. The well-known Golay aperiodic complementary sequence pair (Golay pair) is a special case of even periodic (odd periodic) complementary sequence pair. In this paper, we presented several classes of even periodic and odd periodic complementary pairs based on the generalized Boolean functions, but which do not form Gloay pairs. The proposed sequences could be used to design signal sets, which have been applied in direct sequence code division multiple (DS-CDMA) cellular communication systems.
2013, 7(2): 127-145 doi: 10.3934/amc.2013.7.127 +[Abstract](696) +[PDF](427.1KB)
Abstract:
We apply the semidefinite programming method to derive bounds for projective codes over a finite field.
2013, 7(2): 147-160 doi: 10.3934/amc.2013.7.147 +[Abstract](716) +[PDF](339.0KB)
Abstract:
We define linear and semilinear isometry for general subspace codes, used for random network coding. Furthermore, some results on isometry classes and automorphism groups of known constant dimension code constructions are derived.
2013, 7(2): 161-174 doi: 10.3934/amc.2013.7.161 +[Abstract](676) +[PDF](365.4KB)
Abstract:
In this paper we present a method for constructing self-orthogonal codes from orbit matrices of $2$-designs that admit an automorphism group $G$ which acts with orbit lengths $1$ and $w$, where $w$ divides $|G|$. This is a generalization of an earlier method proposed by Tonchev for constructing self-orthogonal codes from orbit matrices of $2$-designs with a fixed-point-free automorphism of prime order. As an illustration of our method we provide a classification of self-orthogonal codes obtained from the non-fixed parts of the orbit matrices of the symmetric $2$-$(56,11,2)$ designs, some symmetric designs $2$-$(71,15,3)$ (and their residual designs), and some non-symmetric $2$-designs, namely those with parameters $2$-$(15,3,1)$, $2$-$(25,4,1)$, $2$-$(37,4,1)$, and $2$-$(45,5,1)$, respectively with automorphisms of order $p$, where $p$ is an odd prime. We establish that the codes with parameters $[10,4,6]_3$ and $[11,4,6]_3$ are optimal two-weight codes. Further, we construct an optimal binary self-orthogonal $[16,5,8]$ code from the non-fixed part of the orbit matrix of the $2$-$(64,8,1)$ design with respect to an automorphism group of order four.
2013, 7(2): 175-186 doi: 10.3934/amc.2013.7.175 +[Abstract](512) +[PDF](336.9KB)
Abstract:
The polarity designs, introduced in [9], are combinatorial 2-designs having the same parameters as a projective geometry design $PG_s(2s,q)$ formed by the $s$-subspaces of $PG(2s,q)$, $s\ge 2$, $q=p^t$, $p$ prime. If $q=p$ is a prime, a polarity design has also the same $p$-rank as $PG_s(2s,p)$. If $q=2$, any polarity 2-design is extendable to a 3-design having the same parameters and 2-rank as an affine geometry design $AG_{s+1}(2s+1,2)$ formed by the $(s+1)$-subspaces of $AG(2s+1,2)$. It is shown in this paper that a linear code being the null space of the incidence matrix of a polarity design can correct by majority-logic decoding the same number of errors as the projective geometry code based on $PG_s(2s,q)$. In the binary case, any polarity 3-design yields a binary self-dual code with the same parameters, minimum distance, and correcting the same number of errors by majority-logic decoding as the Reed-Muller code of length $2^{2s+1}$ and order $s$.
2013, 7(2): 187-195 doi: 10.3934/amc.2013.7.187 +[Abstract](623) +[PDF](325.5KB)
Abstract:
In this paper we study the hardness of some discrete logarithm like problems defined in linear recurring sequences over finite fields from a point of view as general as possible. The intractability of these problems plays a key role in the security of the class of public key cryptographic constructions based on linear recurring sequences. We define new discrete logarithm, Diffie-Hellman and decisional Diffie-Hellman problems for any nontrivial linear recurring sequence in any finite field whose minimal polynomial is irreducible. Then, we prove that these problems are polynomially equivalent to the discrete logarithm, Diffie-Hellman and decisional Diffie-Hellman problems in the subgroup generated by any root of the minimal polynomial of the sequence.
2013, 7(2): 197-217 doi: 10.3934/amc.2013.7.197 +[Abstract](779) +[PDF](401.1KB)
Abstract:
This paper extends the work of F. Didier (IEEE Transactions on Information Theory, Vol. 52(10): 4496-4503, October 2006) on the algebraic immunity of random balanced Boolean functions, into an asymptotic lower bound on the algebraic immunity of random balanced multi-output Boolean functions.
2013, 7(2): 219-229 doi: 10.3934/amc.2013.7.219 +[Abstract](594) +[PDF](341.7KB)
Abstract:
A lift of binary self-dual codes to the ring $R_2$ is described. By using this lift, a family of self-dual codes over $R_2$ of length $17$ are constructed. Taking the binary images of these codes, extremal binary self-dual codes of length $68$ are obtained. For the first time in the literature, extremal binary codes of length $68$ with $\gamma=4$ and $\gamma = 6$ in $W_{68,2}$ have been obtained. In addition to these, six new codes with $\gamma = 0$ and fourteen new codes with $\gamma = 2$ in $W_{68,2}$ have also been found.

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