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1930-5346
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1930-5338
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Advances in Mathematics of Communications
November 2013 , Volume 7 , Issue 4
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2013, 7(4): 379-407
doi: 10.3934/amc.2013.7.379
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Abstract:
We enumerate $H$-phase Golay sequences for $H\le 36$ and lengths up to 33. Our enumeration method is based on filtering by the power spectra. Some of the hexaphase Golay sequence pairs are new. We provide a compact way to reconstruct all these Golay sequences from specific Golay arrays. The Golay arrays are part of the three-stage construction introduced by Fiedler, Jedwab, and Parker. All such minimal Golay arrays can be constructed from a small set of Golay sequence pairs with binary, quaternary, or hexaphase alphabet adjoining 0. We also prove some non-existence results for Golay sequences when $H/2$ is odd.
We enumerate $H$-phase Golay sequences for $H\le 36$ and lengths up to 33. Our enumeration method is based on filtering by the power spectra. Some of the hexaphase Golay sequence pairs are new. We provide a compact way to reconstruct all these Golay sequences from specific Golay arrays. The Golay arrays are part of the three-stage construction introduced by Fiedler, Jedwab, and Parker. All such minimal Golay arrays can be constructed from a small set of Golay sequence pairs with binary, quaternary, or hexaphase alphabet adjoining 0. We also prove some non-existence results for Golay sequences when $H/2$ is odd.
2013, 7(4): 409-424
doi: 10.3934/amc.2013.7.409
+[Abstract](3673)
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Abstract:
Families of $m-$sequences with low correlation property have important applications in communication systems. In this paper, for a prime $p\equiv 1\ \mathrm{mod}\ 4$ and an odd integer $k$, we study the cross correlation between a $p$-ary $m$-sequence $\{s_t\}$ of period $p^n-1$ and its decimated sequence $\{s_{dt}\}$, where $d=\frac{(p^k+1)^2}{2(p^e+1)}$, $e|k$ and $n = 2k$. Using quadratic form polynomial theory, we obtain the distribution of the cross correlation which is six-valued. Specially, our results show that the magnitude of the cross correlation is upper bounded by $2\sqrt{p^n}+1$ for $p=5$ and $e=1$, which is meaningful in CDMA communication systems.
Families of $m-$sequences with low correlation property have important applications in communication systems. In this paper, for a prime $p\equiv 1\ \mathrm{mod}\ 4$ and an odd integer $k$, we study the cross correlation between a $p$-ary $m$-sequence $\{s_t\}$ of period $p^n-1$ and its decimated sequence $\{s_{dt}\}$, where $d=\frac{(p^k+1)^2}{2(p^e+1)}$, $e|k$ and $n = 2k$. Using quadratic form polynomial theory, we obtain the distribution of the cross correlation which is six-valued. Specially, our results show that the magnitude of the cross correlation is upper bounded by $2\sqrt{p^n}+1$ for $p=5$ and $e=1$, which is meaningful in CDMA communication systems.
2013, 7(4): 425-440
doi: 10.3934/amc.2013.7.425
+[Abstract](3856)
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Abstract:
For weakly regular bent functions in odd characteristic the dual function is also bent. We analyse a recently introduced construction of non-weakly regular bent functions and show conditions under which their dual is bent as well. This leads to the definition of the class of dual-bent functions containing the class of weakly regular bent functions as a proper subclass. We analyse self-duality for bent functions in odd characteristic, and characterize quadratic self-dual bent functions. We construct non-weakly regular bent functions with and without a bent dual, and bent functions with a dual bent function of a different algebraic degree.
For weakly regular bent functions in odd characteristic the dual function is also bent. We analyse a recently introduced construction of non-weakly regular bent functions and show conditions under which their dual is bent as well. This leads to the definition of the class of dual-bent functions containing the class of weakly regular bent functions as a proper subclass. We analyse self-duality for bent functions in odd characteristic, and characterize quadratic self-dual bent functions. We construct non-weakly regular bent functions with and without a bent dual, and bent functions with a dual bent function of a different algebraic degree.
2013, 7(4): 441-461
doi: 10.3934/amc.2013.7.441
+[Abstract](3255)
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Abstract:
Let $F$ be a number field with ring of integers $\boldsymbol{O}_F$ and $D$ a division $F$-algebra with a maximal cyclic subfield $K$. We study rings occurring as quotients of a natural $\boldsymbol{O}_F$-order $\Lambda$ in $D$ by two-sided ideals. We reduce the problem to studying the ideal structure of $\Lambda/q^s\Lambda$, where $q$ is a prime ideal in $\boldsymbol{O}_F$, $s\geq 1$. We study the case where $q$ remains unramified in $K$, both when $s=1$ and $s>1$. This work is motivated by its applications to space-time coded modulation.
Let $F$ be a number field with ring of integers $\boldsymbol{O}_F$ and $D$ a division $F$-algebra with a maximal cyclic subfield $K$. We study rings occurring as quotients of a natural $\boldsymbol{O}_F$-order $\Lambda$ in $D$ by two-sided ideals. We reduce the problem to studying the ideal structure of $\Lambda/q^s\Lambda$, where $q$ is a prime ideal in $\boldsymbol{O}_F$, $s\geq 1$. We study the case where $q$ remains unramified in $K$, both when $s=1$ and $s>1$. This work is motivated by its applications to space-time coded modulation.
2013, 7(4): 463-474
doi: 10.3934/amc.2013.7.463
+[Abstract](2621)
+[PDF](346.8KB)
Abstract:
In this paper we introduce the notion of cyclic ($f(t),\sigma,\delta$)-codes for $f(t)\in A[t;\sigma,\delta]$. These codes generalize the $\theta$-codes as introduced by D. Boucher, F. Ulmer, W. Geiselmann [2]. We construct generic and control matrices for these codes. As a particular case the ($\sigma,\delta$)-$W$-code associated to a Wedderburn polynomial are defined and we show that their control matrices are given by generalized Vandermonde matrices. All the Wedderburn polynomials of $\mathbb F_q[t;\theta]$ are described and their control matrices are presented. A key role will be played by the pseudo-linear transformations.
In this paper we introduce the notion of cyclic ($f(t),\sigma,\delta$)-codes for $f(t)\in A[t;\sigma,\delta]$. These codes generalize the $\theta$-codes as introduced by D. Boucher, F. Ulmer, W. Geiselmann [2]. We construct generic and control matrices for these codes. As a particular case the ($\sigma,\delta$)-$W$-code associated to a Wedderburn polynomial are defined and we show that their control matrices are given by generalized Vandermonde matrices. All the Wedderburn polynomials of $\mathbb F_q[t;\theta]$ are described and their control matrices are presented. A key role will be played by the pseudo-linear transformations.
2013, 7(4): 475-484
doi: 10.3934/amc.2013.7.475
+[Abstract](3328)
+[PDF](375.2KB)
Abstract:
In this paper, new constructions of the binary sequence families of period $q-1$ with large family size and low correlation, derived from multiplicative characters of finite fields for odd prime powers, are proposed. For $m ≥ 2$, the maximum correlation magnitudes of new sequence families $\mathcal{S}_m$ are bounded by $(2m-2)\sqrt{q}+2m+2$, and the family sizes of $\mathcal{S}_m$ are given by $q-1$ for $m=2$, $2(q-1)-1$ for $m=3$, $(q^2-1)q^{\frac{m-4}{2}}$ for $m$ even, $m>2$, and $2(q-1)q^{\frac{m-3}{2}}$ for $m$ odd, $m>3$. It is shown that the known binary Sidel'nikov-based sequence families are equivalent to the new constructions for the case $m=2$.
In this paper, new constructions of the binary sequence families of period $q-1$ with large family size and low correlation, derived from multiplicative characters of finite fields for odd prime powers, are proposed. For $m ≥ 2$, the maximum correlation magnitudes of new sequence families $\mathcal{S}_m$ are bounded by $(2m-2)\sqrt{q}+2m+2$, and the family sizes of $\mathcal{S}_m$ are given by $q-1$ for $m=2$, $2(q-1)-1$ for $m=3$, $(q^2-1)q^{\frac{m-4}{2}}$ for $m$ even, $m>2$, and $2(q-1)q^{\frac{m-3}{2}}$ for $m$ odd, $m>3$. It is shown that the known binary Sidel'nikov-based sequence families are equivalent to the new constructions for the case $m=2$.
2013, 7(4): 485-502
doi: 10.3934/amc.2013.7.485
+[Abstract](3751)
+[PDF](485.8KB)
Abstract:
A significant amount of effort has been devoted to improving divisor arithmetic on low-genus hyperelliptic curves via explicit versions of generic algorithms. Moderate and high genus curves also arise in cryptographic applications, for example, via the Weil descent attack on the elliptic curve discrete logarithm problem, but for these curves, the generic algorithms are to date the most efficient available. Nagao [22] described how some of the techniques used in deriving efficient explicit formulas can be used to speed up divisor arithmetic using Cantor's algorithm on curves of arbitrary genus. In this paper, we describe how Nagao's methods, together with a sub-quadratic complexity partial extended Euclidean algorithm using the half-gcd algorithm can be applied to improve arithmetic in the degree zero divisor class group. We present numerical results showing which combination of techniques is more efficient for hyperelliptic curves over $\mathbb{F}_{2^n}$ of various genera.
A significant amount of effort has been devoted to improving divisor arithmetic on low-genus hyperelliptic curves via explicit versions of generic algorithms. Moderate and high genus curves also arise in cryptographic applications, for example, via the Weil descent attack on the elliptic curve discrete logarithm problem, but for these curves, the generic algorithms are to date the most efficient available. Nagao [22] described how some of the techniques used in deriving efficient explicit formulas can be used to speed up divisor arithmetic using Cantor's algorithm on curves of arbitrary genus. In this paper, we describe how Nagao's methods, together with a sub-quadratic complexity partial extended Euclidean algorithm using the half-gcd algorithm can be applied to improve arithmetic in the degree zero divisor class group. We present numerical results showing which combination of techniques is more efficient for hyperelliptic curves over $\mathbb{F}_{2^n}$ of various genera.
2013, 7(4): 503-510
doi: 10.3934/amc.2013.7.503
+[Abstract](3883)
+[PDF](332.4KB)
Abstract:
A computer calculation with $M$AGMA shows that there is no extremal self-dual binary code $\mathcal{C}$ of length $72$ whose automorphism group contains the symmetric group of degree $3$, the alternating group of degree $4$ or the dihedral group of order $8$. Combining this with the known results in the literature one obtains that $Aut(\mathcal{C})$ has order at most $5$ or is isomorphic to the elementary abelian group of order $8$.
A computer calculation with $M$AGMA shows that there is no extremal self-dual binary code $\mathcal{C}$ of length $72$ whose automorphism group contains the symmetric group of degree $3$, the alternating group of degree $4$ or the dihedral group of order $8$. Combining this with the known results in the literature one obtains that $Aut(\mathcal{C})$ has order at most $5$ or is isomorphic to the elementary abelian group of order $8$.
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