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Advances in Mathematics of Communications

2011 , Volume 5 , Issue 4

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Error bounds for repeat-accumulate codes decoded via linear programming
Idan Goldenberg and David Burshtein
2011, 5(4): 555-570 doi: 10.3934/amc.2011.5.555 +[Abstract](321) +[PDF](251.4KB)
Abstract:
We examine regular and irregular repeat-accumulate (RA) codes with repetition degrees which are all even. For these codes and with a particular choice of an interleaver, we give an upper bound on the decoding error probability of a linear-programming based decoder which is an inverse polynomial in the block length. Our bound is valid for any memoryless, binary-input, output-symmetric (MBIOS) channel. This result generalizes the bound derived by Feldman et al., which was for regular RA($2$) codes.
Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes
Steven T. Dougherty and Cristina Fernández-Córdoba
2011, 5(4): 571-588 doi: 10.3934/amc.2011.5.571 +[Abstract](417) +[PDF](409.3KB)
Abstract:
The generalized Gray map is defined for codes over $\mathbb{Z}_{2^k}$. We give bounds for the dimension of the kernel and the rank of the image of a code over $\mathbb{Z}_{2^k}$ with a given type and show that there exists such a code for each dimension in the interval for the kernel. We determine when the Gray image of a code over $\mathbb{Z}_{2^k}$ generates a linear self-dual code and give families of codes whose image generate binary self-dual codes. We investigate the Gray image of quaternary self-dual codes and examine when the Gray image of a self-dual code over $\mathbb{Z}_4$ is a binary self-dual code.
The merit factor of binary arrays derived from the quadratic character
Kai-Uwe Schmidt
2011, 5(4): 589-607 doi: 10.3934/amc.2011.5.589 +[Abstract](224) +[PDF](482.9KB)
Abstract:
We calculate the asymptotic merit factor, under all cyclic rotations of rows and columns, of two families of binary two-dimensional arrays derived from the quadratic character. The arrays in these families have size $p\times q$, where $p$ and $q$ are not necessarily distinct odd primes, and can be considered as two-dimensional generalisations of a Legendre sequence. The asymptotic values of the merit factor of the two families are generally different, although the maximum asymptotic merit factor, taken over all cyclic rotations of rows and columns, equals $36/13$ for both families. These are the first non-trivial theoretical results for the asymptotic merit factor of families of truly two-dimensional binary arrays.
On the number of bent functions from iterative constructions: lower bounds and hypotheses
Natalia Tokareva
2011, 5(4): 609-621 doi: 10.3934/amc.2011.5.609 +[Abstract](296) +[PDF](403.3KB)
Abstract:
In the paper we study lower bounds on the number of bent functions that can be obtained by iterative constructions, namely by the construction proposed by A. Canteaut and P. Charpin in 2003. The number of bent iterative functions is expressed in terms of sizes of finite sets and it is shown that evaluation of this number is closely connected to the problem of decomposing Boolean function into sum of two bent functions. A new lower bound for the number of bent iterative functions that is supposed to be asymptotically tight is given. Applying Monte-Carlo methods the number of bent iterative functions in $8$ variables is counted. Based on the performed calculations several hypotheses on the asymptotic value of the number of all bent functions are formulated.
Explicit formulas for real hyperelliptic curves of genus 2 in affine representation
Stefan Erickson, Michael J. Jacobson, Jr. and Andreas Stein
2011, 5(4): 623-666 doi: 10.3934/amc.2011.5.623 +[Abstract](365) +[PDF](562.2KB)
Abstract:
We present a complete set of efficient explicit formulas for arithmetic in the degree $0$ divisor class group of a genus two real hyperelliptic curve given in affine coordinates. In addition to formulas suitable for curves defined over an arbitrary finite field, we give simplified versions for both the odd and the even characteristic cases. Formulas for baby steps, inverse baby steps, divisor addition, doubling, and special cases such as adding a degenerate divisor are provided, with variations for divisors given in reduced and adapted basis. We describe the improvements and the correctness together with a comprehensive analysis of the number of field operations for each operation. Finally, we perform a direct comparison of cryptographic protocols using explicit formulas for real hyperelliptic curves with the corresponding protocols presented in the imaginary model.
Fast multi-sequence shift-register synthesis with the Euclidean algorithm
Alexander Zeh and Antonia Wachter
2011, 5(4): 667-680 doi: 10.3934/amc.2011.5.667 +[Abstract](332) +[PDF](524.4KB)
Abstract:
Feng and Tzeng's generalization of the Extended Euclidean Algorithm synthesizes the shortest-length linear feedback shift-register for $s \geq 1$ sequences, where each sequence has the same length $n$. In this contribution, it is shown that Feng and Tzeng's algorithm which solves this multi-sequence shift-register problem has time complexity $\mathcal{O}(sn^2)$. An acceleration based on the Divide and Conquer strategy is proposed and it is proven that subquadratic time complexity is achieved.
All binary linear codes of lengths up to 18 or redundancy up to 10 are normal
Tsonka Baicheva
2011, 5(4): 681-686 doi: 10.3934/amc.2011.5.681 +[Abstract](277) +[PDF](279.6KB)
Abstract:
We show that all binary codes of lengths 16, 17 and 18, or redundancy 10, are normal. These results have applications in the construction of codes that attain $t[n,k]$, the smallest covering radius of any binary linear code.

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