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Advances in Mathematics of Communications

2007 , Volume 1 , Issue 3

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On finite fields for pairing based cryptography
Florian Luca and Igor E. Shparlinski
2007, 1(3): 281-286 doi: 10.3934/amc.2007.1.281 +[Abstract](610) +[PDF](124.9KB)
Abstract:
Here, we improve our previous bound on the number of finite fields over which elliptic curves of cryptographic interest with a given embedding degree and small complex multiplication discriminant may exist. We also give some heuristic arguments which lead to a lower bound which in some cases is close to our upper bound.
Eigenvalue bounds on the pseudocodeword weight of expander codes
Christine A. Kelley and Deepak Sridhara
2007, 1(3): 287-306 doi: 10.3934/amc.2007.1.287 +[Abstract](471) +[PDF](264.6KB)
Abstract:
Four different ways of obtaining low-density parity-check codes from expander graphs are considered. For each case, lower bounds on the minimum stopping set size and the minimum pseudocodeword weight of expander (LDPC) codes are derived. These bounds are compared with the known eigenvalue-based lower bounds on the minimum distance of expander codes. Furthermore, Tanner's parity-oriented eigenvalue lower bound on the minimum distance is generalized to yield a new lower bound on the minimum pseudocodeword weight. These bounds are useful in predicting the performance of LDPC codes under graph-based iterative decoding and linear programming decoding.
The asymptotic behavior of N-adic complexity
Andrew Klapper
2007, 1(3): 307-319 doi: 10.3934/amc.2007.1.307 +[Abstract](577) +[PDF](166.1KB)
Abstract:
We study the asymptotic behavior of stream cipher security mea- sures associated with classes of sequence generators such as linear feedback shift registers and feedback with carry shift registers. For nonperiodic sequences we consider normalized measures and study the set of accumulation points for a fixed sequence. We see that the set of accumulation points is always a closed subinterval of $[0, 1]$. For binary or ternary FCSRs we see that this interval is of the form $[B, 1-B]$, a result that is an analog of an earlier result by Dai, Jiang, Imamura, and Gong for LFSRs.
Parity properties of Costas arrays defined via finite fields
Konstantinos Drakakis, Rod Gow and Scott Rickard
2007, 1(3): 321-330 doi: 10.3934/amc.2007.1.321 +[Abstract](491) +[PDF](135.8KB)
Abstract:
A Costas array of order $n$ is an arrangement of dots and blanks into $n$ rows and $n$ columns, with exactly one dot in each row and each column, the arrangement satisfying certain specified conditions. A dot occurring in such an array is even/even if it occurs in the $i$-th row and $j$-th column, where $i$ and $j$ are both even integers, and there are similar definitions of odd/odd, even/odd and odd/even dots. Two types of Costas arrays, known as Golomb-Costas and Welch-Costas arrays, can be defined using finite fields. When $q$ is a power of an odd prime, we enumerate the number of even/even odd/odd, even/odd and odd/even dots in a Golomb-Costas array. We show that three of these numbers are equal and they differ by $\pm 1$ from the fourth. For a Welch-Costas array of order $p-1$, where $p$ is an odd prime, the four numbers above are all equal to $(p-1)/4$ when $p\equiv 1(\mod 4)$, but when $p\equiv 3(\mod 4)$, we show that the four numbers are defined in terms of the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$, and thus behave in a much less predictable manner.
A weight-based characterization of the set of correctable error patterns under list-of-2 decoding
Jonas Eriksson
2007, 1(3): 331-356 doi: 10.3934/amc.2007.1.331 +[Abstract](519) +[PDF](280.7KB)
Abstract:
List decoding of block codes is an alternative approach to the decoding problem with appealing qualities. The fairly recent development of efficient algorithms for list decoding of Reed-Solomon codes spur new fuel to the study of this decoding strategy. In this paper we give a weight-based characterization of the set of correctable error patterns under list-of-2 ecoding of $(\tau, 2$)-list-decodable linear codes with known weight distribution. We apply our characterization of the set of correctable error patterns to a few codes in a family of low-rate list-of-2 decodable Reed-Solomon codes. We study the increase in error-correction performance obtained in a symmetric AWGN channel by using list-of-2 decoding instead of traditional decoding for these codes. Some simulation results for list-of-2 decoding on QAM channels using the Guruswami-Sudan algorithm for decoding of Reed-Solomon codes are also presented.
Additive self-dual codes over $\mathbb F_4$ with an automorphism of odd prime order
W. Cary Huffman
2007, 1(3): 357-398 doi: 10.3934/amc.2007.1.357 +[Abstract](606) +[PDF](460.1KB)
Abstract:
We present a general theory for decomposing additive self-dual codes over $\mathbbF_4$ that have an automorphism of odd prime order. We apply the decomposition to codes of length $n$ with $13\leq n\leq30$ and automorphisms of prime order $r$ with $5\leq r\leq23$. Using this decomposition we classify all extremal/optimal additive self-dual codes with certain parameters in this list. In the process, we find the first $(18$, 218, $7)$, $(24$, 224, $8)$, and $(28$, 228, $10)$ Type I codes. We also improve the lower bounds on the number of known extremal/optimal additive self-dual codes for some values of $n$ with $13\leq n\leq 30$.

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