# American Institute of Mathematical Sciences

ISSN:
1930-5346

eISSN:
1930-5338

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## Advances in Mathematics of Communications

2009 , Volume 3 , Issue 3

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2009, 3(3): 219-226 doi: 10.3934/amc.2009.3.219 +[Abstract](382) +[PDF](130.8KB)
Abstract:
It is well known that a quadratic function defined on a finite field of odd degree is almost bent (AB) if and only if it is almost perfect nonlinear (APN). For the even degree case there is no apparent relationship between the values in the Fourier spectrum of a function and the APN property. In this article we compute the Fourier spectrum of the quadrinomial family of APN functions from [5]. With this result, all known infinite families of APN functions now have their Fourier spectra and hence their nonlinearities computed.
2009, 3(3): 227-234 doi: 10.3934/amc.2009.3.227 +[Abstract](369) +[PDF](131.7KB)
Abstract:
In this paper we focus on two generalizations of the notion of cyclicity of codes: polycyclic codes and sequential codes. We establish a duality between these two generalizations and also show connections between them and other well-known generalizations of cyclicity such as the notions of negacyclicity and constacyclicity. In particular, it is shown that a code $C$ is sequential and polycyclic if and only if $C$ and its dual C are both sequential if and only if $C$ and its dual C are both polycyclic. Furthermore, any one of these equivalent statements characterizes the family of constacyclic codes.
2009, 3(3): 235-250 doi: 10.3934/amc.2009.3.235 +[Abstract](673) +[PDF](218.2KB)
Abstract:
We collect the main construction methods for Golomb rulers available in the literature along with their proofs. In particular, we demonstrate that the Bose-Chowla method yields Golomb rulers that appear as the main diagonal of a special subfamily of Golomb Costas arrays. We also show that Golomb rulers can be composed to yield longer Golomb rulers.
2009, 3(3): 251-263 doi: 10.3934/amc.2009.3.251 +[Abstract](362) +[PDF](204.9KB)
Abstract:
In this paper, we give optimal self-dual codes over $GF(5)$ for lengths $24$, $40$, $48$ and $56$. In particular, new inequivalent $[48, 24]$ and $[56, 28]$ self-dual codes over $GF(5)$ whose minimum weights are $14$ and $16$, are constructed using skew-Hadamard matrices of order $24$ and $28$, thus improving the only known quadratic double circulant self-dual codes of length $48$ and $56$. Moreover, $[80, 40]$ and $[88, 44]$ self-dual codes whose minimum weights are $17$ and $19$ over $GF(5)$, are constructed for the first time. These codes are derived from skew-Hadamard matrices of order $40$ and $44$, respectively. Finally, a new $[56, 28, 17]$ self-dual code is constructed over $GF(7)$ having the highest minimum weight among $[56, 28]$ self-dual codes. This new optimal code is constructed from a skew-Hadamard-matrix of order $28$, for the first time.
2009, 3(3): 265-271 doi: 10.3934/amc.2009.3.265 +[Abstract](371) +[PDF](130.9KB)
Abstract:
Recently, the minimum Hamming weights of negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$ are determined in [4]. We show that the minimum Hamming weights of such codes can also be obtained immediately using the results of [1].
2009, 3(3): 273-293 doi: 10.3934/amc.2009.3.273 +[Abstract](404) +[PDF](955.0KB)
Abstract:
Key distribution schemes play a significant role in key assignment schemes which allow participants in a network to communicate by means of symmetric cryptography in a secure way without the need of a unique key for every pair of participants. It is assumed that an adversary can eavesdrop on all communication and can corrupt up to $t$ vertices in the network. It follows that, in general, the sender needs to transmit at least $t+1$ shares of the message over different paths to the intended receiver and that each participant needs to possess at least $t+1$ encryption keys. We do assume that vertices in the network will forward messages correctly (but only the corrupted vertices will collude with the adversary to retrieve the message).
We focus on two approaches. In the first approach, the goal is to minimize the number of keys per participant. An almost complete answer is presented. The second approach is to minimize the total number of keys that are needed in the network. The number of communication paths that are needed to guarantee secure communication becomes a relevant parameter. Our security relies on the random oracle model.
2009, 3(3): 295-309 doi: 10.3934/amc.2009.3.295 +[Abstract](409) +[PDF](211.3KB)
Abstract:
The set of permutations of the coordinate set that maps a perfect code $C$ into itself is called the symmetry group of $C$ and is denoted by Sym$(C)$. It is proved that for all integers $n=2^m-1$, where $m=4,5,6,...$, and for any integer $r$, where $n-$log$(n+1)+3\leq r\leq n-1$, there are perfect codes of length $n$ and rank $r$ with a trivial symmetry group, i.e. Sym$(C)=${id}. The result is shown to be true, more generally, for the extended perfect codes of length $n+1$.

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