# American Institute of Mathematical Sciences

ISSN:
1930-5346

eISSN:
1930-5338

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## Advances in Mathematics of Communications

February 2010 , Volume 4 , Issue 1

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2010, 4(1): 1-22 doi: 10.3934/amc.2010.4.1 +[Abstract](1258) +[PDF](414.1KB)
Abstract:
We investigate the generalization of the Costas property in three or more dimensions, and we seek an appropriate definition; the two main complications are a) that the number of ''dots'' this multidimensional structure should have is not obvious, and b) that the notion of the multidimensional permutation needs some clarification. After proposing various alternatives for the generalization of the definition of the Costas property, based on the definitions of the Costas property in one or two dimensions, we also offer some construction methods, the main one of which is based on the idea of reshaping Costas arrays into higher-dimensional entities.
2010, 4(1): 23-47 doi: 10.3934/amc.2010.4.23 +[Abstract](1733) +[PDF](269.3KB)
Abstract:
In this article, we deal with fast arithmetic in the Picard group of hyperelliptic curves of genus 3 over binary fields. We investigate both the optimal performance curves, where $h(x)=1$, and the more general curves where the degree of $h(x)$ is 1, 2 or 3. For the optimal performance curves, we provide explicit halving and doubling formulas; not only for the most frequent case but also for all possible special cases that may occur when performing arithmetic on the proposed curves. In this situation, we show that halving offers equivalent performance to that of doubling when computing scalar multiples (by means of an halve-and-add algorithm) in the divisor class group.
For the other types of curves where halving may give performance gains (when the group order is twice an odd number), we give explicit halving formulas which outperform the corresponding doubling formulas by about 10 to 20 field multiplications per halving. These savings more than justify the use of halvings for these curves, making them significantly more efficient than previously thought. For halving on genus 3 curves there is no previous work published so far.
2010, 4(1): 49-60 doi: 10.3934/amc.2010.4.49 +[Abstract](1328) +[PDF](232.3KB)
Abstract:
We consider a concatenated code with designed distance dodi$/2$, based on an outer code with distance do and an inner code with distance di. To decode the inner code, we use a Bounded Minimum Distance decoder correcting up to (di$-1$)$/2$ errors. For decoding the outer code, we use a $\lambda$-Bounded Distance decoder correcting $\varepsilon$ errors and $\tau$ erasures if $\lambda\varepsilon+\tau \leq$do$-1$, where a real number $1<\lambda\leq 2$ is the tradeoff rate between errors and erasures for this outer decoder. A single-trial erasures-and-errors-correcting outer decoder is considered, that extends Kovalev's approach [4] for the whole given range of $\lambda$. The error-correcting radius of the proposed concatenated decoder is dido$/(\lambda +1)$ if the number $\tau$ of erasures is fixed, and (dido$/2$)∗$(1-(\frac{\lambda-1}{\lambda})^2)$ for adaptive selection of $\tau$. The error-correcting radius quickly approaches dido$/2$ with decreasing $\lambda$. These results can be applied e.g. when punctured Reed-Solomon outer codes are used.
2010, 4(1): 61-68 doi: 10.3934/amc.2010.4.61 +[Abstract](1676) +[PDF](135.3KB)
Abstract:
In this paper, two new construction methods of quaternary periodic complementary sequence (PCS) sets are proposed using a binary PCS set with even period. The proposed methods apply the Gray mapping to a binary PCS set. The only necessary condition to apply the proposed methods is that the employed binary PCS set should have an even period.
2010, 4(1): 69-81 doi: 10.3934/amc.2010.4.69 +[Abstract](1465) +[PDF](168.8KB)
Abstract:
It is shown that all non-full-rank FRH-codes, a class of perfect codes we define in this paper, are linearly equivalent to perfect codes obtainable by Phelps' construction. Moreover, it is shown by an example that the class of perfect FRH-codes also contains perfect codes that are not obtainable by Phelps construction.
2010, 4(1): 83-99 doi: 10.3934/amc.2010.4.83 +[Abstract](1425) +[PDF](246.6KB)
Abstract:
An iterative decoding algorithm for convolutional codes is presented. It successively processes $N$ consecutive blocks of the received word in order to decode the first block. A bound is presented showing which error configurations can be corrected. The algorithm can be efficiently used on a particular class of convolutional codes, known as doubly cyclic convolutional codes. Due to their highly algebraic structure those codes are well suited for the algorithm and the main step of the procedure can be carried out using Reed-Solomon decoding. Examples illustrate the decoding and a comparison with existing algorithms is made.

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