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1930-5346
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1930-5338
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Advances in Mathematics of Communications
February 2012 , Volume 6 , Issue 1
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2012, 6(1): 1-26
doi: 10.3934/amc.2012.6.1
+[Abstract](3635)
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Abstract:
We introduce a new right preprocessing approach, called algebraic reduction, for the decoding of algebraic space-time block codes which are designed using maximal orders in division algebras. Unlike existing lattice-reduction aided decoding techniques, algebraic reduction exploits the multiplicative structure of the code. Its principle is to absorb part of the channel into the code, by approximating the channel matrix with a unit of the maximal order of the corresponding algebra.
In the case of $2 \times 2$ space-time codes, we propose a low-complexity algorithm to approximate a channel with a unit, and apply it to the Golden Code. Simulation results for the Golden Code evidence that algebraic reduction has the same performance of LLL reduction but significantly lower complexity.
We also show that for MIMO systems with $n$ receive and $n$ transmit antennas, algebraic reduction attains the receive diversity when followed by a simple ZF detection. However, establishing the approximation algorithm for higher dimensional codes remains an open problem.
We introduce a new right preprocessing approach, called algebraic reduction, for the decoding of algebraic space-time block codes which are designed using maximal orders in division algebras. Unlike existing lattice-reduction aided decoding techniques, algebraic reduction exploits the multiplicative structure of the code. Its principle is to absorb part of the channel into the code, by approximating the channel matrix with a unit of the maximal order of the corresponding algebra.
In the case of $2 \times 2$ space-time codes, we propose a low-complexity algorithm to approximate a channel with a unit, and apply it to the Golden Code. Simulation results for the Golden Code evidence that algebraic reduction has the same performance of LLL reduction but significantly lower complexity.
We also show that for MIMO systems with $n$ receive and $n$ transmit antennas, algebraic reduction attains the receive diversity when followed by a simple ZF detection. However, establishing the approximation algorithm for higher dimensional codes remains an open problem.
2012, 6(1): 27-38
doi: 10.3934/amc.2012.6.27
+[Abstract](2452)
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Abstract:
Assume that $G=(V,E)$ is an undirected graph with vertex set $V$ and edge set $E$. The ball $B_r(v)$ denotes the vertices within graphical distance $r$ from $v$. Let $I_r(F)=\bigcup$$v\in F$$(B_r(v) \cap C)$ be a set of codewords in the neighbourhoods of vertices $v\in F$. A subset $C\subseteq V$ is called an $(r,\leq l)$-locating-dominating code of type A if sets $I_r(F_1)$ and $I_r(F_2)$ are distinct for all subsets $F_1,F_2\subseteq V$ where $F_1\ne F_2$, $F_1\cap C= F_2 \cap C$ and $|F_1|,|F_2| \leq l$. A subset $C\subseteq V$ is an $(r,\leq l)$-locating-dominating code of type B if the sets $I_r(F)$ are distinct for all subsets $F\subseteq V\setminus C$ with at most $l$ vertices. We study $(r,\leq l)$-locating-dominating codes in the infinite king grid when $r\geq 1$ and $l=2$. The infinite king grid is the graph with vertex set $\mathbb Z^2$ and edge set $\{\{(x_1,y_1),(x_2,y_2)\}||x_1-x_2|\leq 1, |y_1-y_2|\leq 1, (x_1,y_1)\ne(x_2,y_2)\}.$
Assume that $G=(V,E)$ is an undirected graph with vertex set $V$ and edge set $E$. The ball $B_r(v)$ denotes the vertices within graphical distance $r$ from $v$. Let $I_r(F)=\bigcup$$v\in F$$(B_r(v) \cap C)$ be a set of codewords in the neighbourhoods of vertices $v\in F$. A subset $C\subseteq V$ is called an $(r,\leq l)$-locating-dominating code of type A if sets $I_r(F_1)$ and $I_r(F_2)$ are distinct for all subsets $F_1,F_2\subseteq V$ where $F_1\ne F_2$, $F_1\cap C= F_2 \cap C$ and $|F_1|,|F_2| \leq l$. A subset $C\subseteq V$ is an $(r,\leq l)$-locating-dominating code of type B if the sets $I_r(F)$ are distinct for all subsets $F\subseteq V\setminus C$ with at most $l$ vertices. We study $(r,\leq l)$-locating-dominating codes in the infinite king grid when $r\geq 1$ and $l=2$. The infinite king grid is the graph with vertex set $\mathbb Z^2$ and edge set $\{\{(x_1,y_1),(x_2,y_2)\}||x_1-x_2|\leq 1, |y_1-y_2|\leq 1, (x_1,y_1)\ne(x_2,y_2)\}.$
2012, 6(1): 39-63
doi: 10.3934/amc.2012.6.39
+[Abstract](3855)
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Abstract:
Skew polynomial rings over finite fields and over Galois rings have recently been used to study codes. In this paper, we extend this concept to finite chain rings. Properties of skew constacyclic codes generated by monic right divisors of $x^n-\lambda$, where $\lambda$ is a unit element, are exhibited. When $\lambda^2=1$, the generators of Euclidean and Hermitian dual codes of such codes are determined together with necessary and sufficient conditions for them to be Euclidean and Hermitian self-dual. Specializing to codes over the ring $\mathbb F$pm$+u\mathbb F$pm, the structure of all skew constacyclic codes is completely determined. This allows us to express the generators of Euclidean and Hermitian dual codes of skew cyclic and skew negacyclic codes in terms of the generators of the original codes. An illustration of all skew cyclic codes of length $2$ over $\mathbb F_3 + u\mathbb F_3$ and their Euclidean and Hermitian dual codes is also provided.
Skew polynomial rings over finite fields and over Galois rings have recently been used to study codes. In this paper, we extend this concept to finite chain rings. Properties of skew constacyclic codes generated by monic right divisors of $x^n-\lambda$, where $\lambda$ is a unit element, are exhibited. When $\lambda^2=1$, the generators of Euclidean and Hermitian dual codes of such codes are determined together with necessary and sufficient conditions for them to be Euclidean and Hermitian self-dual. Specializing to codes over the ring $\mathbb F$pm$+u\mathbb F$pm, the structure of all skew constacyclic codes is completely determined. This allows us to express the generators of Euclidean and Hermitian dual codes of skew cyclic and skew negacyclic codes in terms of the generators of the original codes. An illustration of all skew cyclic codes of length $2$ over $\mathbb F_3 + u\mathbb F_3$ and their Euclidean and Hermitian dual codes is also provided.
2012, 6(1): 65-68
doi: 10.3934/amc.2012.6.65
+[Abstract](3026)
+[PDF](282.0KB)
Abstract:
In a former paper we investigated the connection between $p$-ary linear codes, $p$ prime, and theta functions. Corresponding to a given code a suitable lattice and its associated theta function were defined. Using results from the theory of modular forms we got an algorithm to determine an upper bound for the minimum Lee distance of certain self-dual codes. In this note we generalize this result to $m$-ary codes, where $m$ is either a power of a prime, or $m$ is square-free. If $m$ is of a different form the generalization will not work. A class of examples to illustrate this fact is given.
In a former paper we investigated the connection between $p$-ary linear codes, $p$ prime, and theta functions. Corresponding to a given code a suitable lattice and its associated theta function were defined. Using results from the theory of modular forms we got an algorithm to determine an upper bound for the minimum Lee distance of certain self-dual codes. In this note we generalize this result to $m$-ary codes, where $m$ is either a power of a prime, or $m$ is square-free. If $m$ is of a different form the generalization will not work. A class of examples to illustrate this fact is given.
2012, 6(1): 69-94
doi: 10.3934/amc.2012.6.69
+[Abstract](3104)
+[PDF](491.4KB)
Abstract:
We determine conditions on $q$ for the nonexistence of deep holes of the standard Reed-Solomon code of dimension $k$ over $\mathbb F_q$ generated by polynomials of degree $k+d$. Our conditions rely on the existence of $q$-rational points with nonzero, pairwise-distinct coordinates of a certain family of hypersurfaces defined over $\mathbb F_q$. We show that the hypersurfaces under consideration are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of these hypersurfaces, from which the existence of $q$-rational points is established.
We determine conditions on $q$ for the nonexistence of deep holes of the standard Reed-Solomon code of dimension $k$ over $\mathbb F_q$ generated by polynomials of degree $k+d$. Our conditions rely on the existence of $q$-rational points with nonzero, pairwise-distinct coordinates of a certain family of hypersurfaces defined over $\mathbb F_q$. We show that the hypersurfaces under consideration are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of these hypersurfaces, from which the existence of $q$-rational points is established.
2012, 6(1): 95-106
doi: 10.3934/amc.2012.6.95
+[Abstract](2449)
+[PDF](382.4KB)
Abstract:
EA equivalence classes and the coarser CCZ equivalence classes of functions over $GF(p^n)$ each preserve measures of nonlinearity desirable in cryptographic functions. We identify very precisely the condition on a linear permutation defining a CCZ isomorphism between functions which ensures that the CCZ isomorphism can be rewritten as EA isomorphism. We introduce new algebraic invariants $n(f)$ of the EA isomorphism class of $f$ and $s(f)$ of the CCZ isomorphism class of $f$, with $n(f) < s(f)$, and relate them to the differential uniformity of $f$. We formulate three questions about partitioning CCZ classes into EA classes and relate these to a conjecture of Edel's about quadratic APN functions.
EA equivalence classes and the coarser CCZ equivalence classes of functions over $GF(p^n)$ each preserve measures of nonlinearity desirable in cryptographic functions. We identify very precisely the condition on a linear permutation defining a CCZ isomorphism between functions which ensures that the CCZ isomorphism can be rewritten as EA isomorphism. We introduce new algebraic invariants $n(f)$ of the EA isomorphism class of $f$ and $s(f)$ of the CCZ isomorphism class of $f$, with $n(f) < s(f)$, and relate them to the differential uniformity of $f$. We formulate three questions about partitioning CCZ classes into EA classes and relate these to a conjecture of Edel's about quadratic APN functions.
2012, 6(1): 107-120
doi: 10.3934/amc.2012.6.107
+[Abstract](3086)
+[PDF](346.3KB)
Abstract:
We show that there exists a unique maximal curve of genus $7$ over the finite field with $49$ elements, up to birational equivalence. This was the first open classification problem for maximal curves, since maximal curves over the finite fields with less than $49$ elements, as well as maximal curves over the finite field with $49$ elements with genus larger than $7$, had been previously classified. A significant role is played by some exhaustive computer searches.
We show that there exists a unique maximal curve of genus $7$ over the finite field with $49$ elements, up to birational equivalence. This was the first open classification problem for maximal curves, since maximal curves over the finite fields with less than $49$ elements, as well as maximal curves over the finite field with $49$ elements with genus larger than $7$, had been previously classified. A significant role is played by some exhaustive computer searches.
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