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1930-5346
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Advances in Mathematics of Communications
August 2016 , Volume 10 , Issue 3
Special issue on ALCOMA'15
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2016, 10(3): i-ii
doi: 10.3934/amc.201603i
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Abstract:
The present issue of the Advances in Mathematics of Communications is dedicated to the conference
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The present issue of the Advances in Mathematics of Communications is dedicated to the conference
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2016, 10(3): 475-488
doi: 10.3934/amc.2016019
+[Abstract](2805)
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Abstract:
In this article we construct a new family of linear maximum rank distance (MRD) codes for all parameters. This family contains the only known family for general parameters, the Gabidulin codes, and contains codes inequivalent to the Gabidulin codes. This family also contains the well-known family of semifields known as Generalised Twisted Fields. We also calculate the automorphism group of these codes, including the automorphism group of the Gabidulin codes.
In this article we construct a new family of linear maximum rank distance (MRD) codes for all parameters. This family contains the only known family for general parameters, the Gabidulin codes, and contains codes inequivalent to the Gabidulin codes. This family also contains the well-known family of semifields known as Generalised Twisted Fields. We also calculate the automorphism group of these codes, including the automorphism group of the Gabidulin codes.
2016, 10(3): 489-498
doi: 10.3934/amc.2016020
+[Abstract](1373)
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Abstract:
An unrestricted $q$-ary maximum distance separable (MDS) code $C$ with length $n$ over an alphabet $\mathcal{A}$ (of size $q$) is a set of $q^k$ codewords that are elements of $\mathcal{A}^n$, such that the smallest Hamming distance between two distinct codewords in $C$ is $d=n-k+1$. Sets of mutually orthogonal Latin squares of orders $q\leq 9$, corresponding to $q$-ary MDS codes of size $q^2$, and $q$-ary one-error-correcting MDS codes for $q\leq 8$ have been classified in earlier studies. These results are used here to complete the classification of all $7$-ary and $8$-ary MDS codes with $d\geq 3$ using a computer search.
An unrestricted $q$-ary maximum distance separable (MDS) code $C$ with length $n$ over an alphabet $\mathcal{A}$ (of size $q$) is a set of $q^k$ codewords that are elements of $\mathcal{A}^n$, such that the smallest Hamming distance between two distinct codewords in $C$ is $d=n-k+1$. Sets of mutually orthogonal Latin squares of orders $q\leq 9$, corresponding to $q$-ary MDS codes of size $q^2$, and $q$-ary one-error-correcting MDS codes for $q\leq 8$ have been classified in earlier studies. These results are used here to complete the classification of all $7$-ary and $8$-ary MDS codes with $d\geq 3$ using a computer search.
2016, 10(3): 499-510
doi: 10.3934/amc.2016021
+[Abstract](2145)
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Abstract:
Based on results in finite geometry we prove the existence of MRD codes in $(\mathbb{F}_q)_{n,n}$ with minimum distance $n$ which are essentially different from Gabidulin codes. The construction results from algebraic structures which are closely related to those of finite fields. Some of the results may be known to experts, but to our knowledge have never been pointed out explicitly in the literature.
Based on results in finite geometry we prove the existence of MRD codes in $(\mathbb{F}_q)_{n,n}$ with minimum distance $n$ which are essentially different from Gabidulin codes. The construction results from algebraic structures which are closely related to those of finite fields. Some of the results may be known to experts, but to our knowledge have never been pointed out explicitly in the literature.
2016, 10(3): 511-524
doi: 10.3934/amc.2016022
+[Abstract](1449)
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The paper deals with recursive constructions for simple 3-designs based on other 3-designs having $(1, \sigma)$-resolution. The concept of $(1, \sigma)$-resolution may be viewed as a generalization of the parallelism for designs. We show the constructions and their applications to produce many previously unknown infinite families of simple 3-designs. We also include a discussion of $(1,\sigma)$-resolvability of the constructed designs.
The paper deals with recursive constructions for simple 3-designs based on other 3-designs having $(1, \sigma)$-resolution. The concept of $(1, \sigma)$-resolution may be viewed as a generalization of the parallelism for designs. We show the constructions and their applications to produce many previously unknown infinite families of simple 3-designs. We also include a discussion of $(1,\sigma)$-resolvability of the constructed designs.
2016, 10(3): 525-540
doi: 10.3934/amc.2016023
+[Abstract](1773)
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A construction is discussed that allows to produce subspace codes of long length using subspace codes of shorter length in combination with a rank metric code. The subspace distance of the resulting linkage code is as good as the minimum subspace distance of the constituent codes. As a special application, the construction of the best known partial spreads is reproduced. Finally, for a special case of linkage, a decoding algorithm is presented which amounts to decoding with respect to the smaller constituent codes and which can be parallelized.
A construction is discussed that allows to produce subspace codes of long length using subspace codes of shorter length in combination with a rank metric code. The subspace distance of the resulting linkage code is as good as the minimum subspace distance of the constituent codes. As a special application, the construction of the best known partial spreads is reproduced. Finally, for a special case of linkage, a decoding algorithm is presented which amounts to decoding with respect to the smaller constituent codes and which can be parallelized.
2016, 10(3): 541-545
doi: 10.3934/amc.2016024
+[Abstract](1431)
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The size of the largest Erdős-Ko-Rado set of generators in a finite classical polar space is known for all polar spaces except for $H(2d-1,q^2)$ when $d\ge 5$ is odd. We improve the known upper bound in this remaining case by using a variant of the famous Hoffman's bound.
The size of the largest Erdős-Ko-Rado set of generators in a finite classical polar space is known for all polar spaces except for $H(2d-1,q^2)$ when $d\ge 5$ is odd. We improve the known upper bound in this remaining case by using a variant of the famous Hoffman's bound.
2016, 10(3): 547-554
doi: 10.3934/amc.2016025
+[Abstract](1455)
+[PDF](328.0KB)
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This paper concerns the problem of the existence of Hadamard difference sets in nonabelian groups of order 400. By introducing a new construction method, we construct new difference sets in 20 groups. We survey non-existence results, verifying non-existence in 45 groups.
This paper concerns the problem of the existence of Hadamard difference sets in nonabelian groups of order 400. By introducing a new construction method, we construct new difference sets in 20 groups. We survey non-existence results, verifying non-existence in 45 groups.
2016, 10(3): 555-582
doi: 10.3934/amc.2016026
+[Abstract](1595)
+[PDF](486.5KB)
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This paper outlines a method for constructing self-orthogonal codes from orbit matrices of strongly regular graphs admitting an automorphism group $G$ which acts with orbits of length $w$, where $w$ divides $|G|$. We apply this method to construct self-orthogonal codes from orbit matrices of the strongly regular graphs with at most 40 vertices. In particular, we construct codes from adjacency or orbit matrices of graphs with parameters $(36, 15, 6, 6)$, $(36, 14, 4, 6)$, $(35, 16, 6, 8)$ and their complements, and from the graphs with parameters $(40, 12, 2, 4)$ and their complements. That completes the classification of self-orthogonal codes spanned by the adjacency matrices or orbit matrices of the strongly regular graphs with at most 40 vertices. Furthermore, we construct ternary codes of $2$-$(27,9,4)$ designs obtained as residual designs of the symmetric $(40, 13, 4)$ designs (complementary designs of the symmetric $(40, 27, 18)$ designs), and their ternary hulls. Some of the obtained codes are optimal, and some are best known for the given length and dimension.
This paper outlines a method for constructing self-orthogonal codes from orbit matrices of strongly regular graphs admitting an automorphism group $G$ which acts with orbits of length $w$, where $w$ divides $|G|$. We apply this method to construct self-orthogonal codes from orbit matrices of the strongly regular graphs with at most 40 vertices. In particular, we construct codes from adjacency or orbit matrices of graphs with parameters $(36, 15, 6, 6)$, $(36, 14, 4, 6)$, $(35, 16, 6, 8)$ and their complements, and from the graphs with parameters $(40, 12, 2, 4)$ and their complements. That completes the classification of self-orthogonal codes spanned by the adjacency matrices or orbit matrices of the strongly regular graphs with at most 40 vertices. Furthermore, we construct ternary codes of $2$-$(27,9,4)$ designs obtained as residual designs of the symmetric $(40, 13, 4)$ designs (complementary designs of the symmetric $(40, 27, 18)$ designs), and their ternary hulls. Some of the obtained codes are optimal, and some are best known for the given length and dimension.
2016, 10(3): 583-588
doi: 10.3934/amc.2016027
+[Abstract](1612)
+[PDF](324.6KB)
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We show that there is no $[24,12,9]$ doubly-even self-dual code over $\mathbb{F}_4$ by attempting to construct the generator matrix of this code directly.
We show that there is no $[24,12,9]$ doubly-even self-dual code over $\mathbb{F}_4$ by attempting to construct the generator matrix of this code directly.
2016, 10(3): 589-600
doi: 10.3934/amc.2016028
+[Abstract](2141)
+[PDF](319.1KB)
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We investigate rank metric codes using univariate linearized polynomials and multivariate linearized polynomials together. We examine the construction of maximum rank distance (MRD) codes and the test of equivalence between two codes in the polynomial representation. Using this approach, we present new classes of some non-Gabidulin linear MRD codes.
We investigate rank metric codes using univariate linearized polynomials and multivariate linearized polynomials together. We examine the construction of maximum rank distance (MRD) codes and the test of equivalence between two codes in the polynomial representation. Using this approach, we present new classes of some non-Gabidulin linear MRD codes.
2016, 10(3): 601-611
doi: 10.3934/amc.2016029
+[Abstract](1341)
+[PDF](368.2KB)
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Using some recent results about multiple extendability of arcs and codes, we prove the nonexistence of $(104,22)$-arcs in $PG(3,5)$. This implies the non-existence of Griesmer $[104,4,82]_5$-codes and settles one of the four remaining open cases for the main problem of coding theory for $q=5,k=4,d=82$.
Using some recent results about multiple extendability of arcs and codes, we prove the nonexistence of $(104,22)$-arcs in $PG(3,5)$. This implies the non-existence of Griesmer $[104,4,82]_5$-codes and settles one of the four remaining open cases for the main problem of coding theory for $q=5,k=4,d=82$.
2016, 10(3): 613-632
doi: 10.3934/amc.2016030
+[Abstract](1688)
+[PDF](505.0KB)
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Multiple coverings of the farthest-off points ($(R,\mu)$-MCF codes) and the corresponding $(\rho,\mu)$-saturating sets in projective spa\-ces $\mathrm{PG}(N,q)$ are considered. We propose some methods which allow us to obtain new small $(1,\mu)$-saturating sets and short $(2,\mu)$-MCF codes with $\mu$-density either equal to 1 (optimal saturating sets and almost perfect MCF-codes) or close to 1 (roughly $1+1/cq$, $c\ge1$). In particular, we provide some algebraic constructions and bounds. Also, we classify minimal and optimal $(1,\mu)$-saturating sets in $\mathrm{PG}(2,q)$, $q$ small.
Multiple coverings of the farthest-off points ($(R,\mu)$-MCF codes) and the corresponding $(\rho,\mu)$-saturating sets in projective spa\-ces $\mathrm{PG}(N,q)$ are considered. We propose some methods which allow us to obtain new small $(1,\mu)$-saturating sets and short $(2,\mu)$-MCF codes with $\mu$-density either equal to 1 (optimal saturating sets and almost perfect MCF-codes) or close to 1 (roughly $1+1/cq$, $c\ge1$). In particular, we provide some algebraic constructions and bounds. Also, we classify minimal and optimal $(1,\mu)$-saturating sets in $\mathrm{PG}(2,q)$, $q$ small.
2016, 10(3): 633-642
doi: 10.3934/amc.2016031
+[Abstract](1756)
+[PDF](367.5KB)
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We investigate self-dual MRD codes. In particular we prove that a Gabidulin code in $(\mathbb{F}_q)^{n\times n}$ is equivalent to a self-dual code if and only if its dimension is $n^2/2$, $n \equiv 2 \pmod 4$, and $q \equiv 3 \pmod 4$. On the way we determine the full automorphism group of Gabidulin codes in $(\mathbb{F}_q)^{n\times n}$.
We investigate self-dual MRD codes. In particular we prove that a Gabidulin code in $(\mathbb{F}_q)^{n\times n}$ is equivalent to a self-dual code if and only if its dimension is $n^2/2$, $n \equiv 2 \pmod 4$, and $q \equiv 3 \pmod 4$. On the way we determine the full automorphism group of Gabidulin codes in $(\mathbb{F}_q)^{n\times n}$.
2016, 10(3): 643-648
doi: 10.3934/amc.2016032
+[Abstract](1661)
+[PDF](282.5KB)
Abstract:
The weight distribution of the binary self-dual $[128,64]$ code being the extended code $C^{*}$ of the code $C$ spanned by the incidence vectors of the blocks of the polarity design in $PG(6,2)$ [11] is computed. It is shown also that $R(3,7)$ and $C^{*}$ have no self-dual $[128,64,d]$ neighbor with $d \in \{ 20, 24 \}$.
The weight distribution of the binary self-dual $[128,64]$ code being the extended code $C^{*}$ of the code $C$ spanned by the incidence vectors of the blocks of the polarity design in $PG(6,2)$ [11] is computed. It is shown also that $R(3,7)$ and $C^{*}$ have no self-dual $[128,64,d]$ neighbor with $d \in \{ 20, 24 \}$.
2016, 10(3): 649-682
doi: 10.3934/amc.2016033
+[Abstract](1704)
+[PDF](672.0KB)
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Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. The resulting so-called Main Problem of Subspace Coding is to determine the maximum size $A_q(v,d)$ of a code in $PG(v-1,\mathbb{F}_q)$ with minimum subspace distance $d$. Here we completely resolve this problem for $d\ge v-1$. For $d=v-2$ we present some improved bounds and determine $A_q(5,3)=2q^3+2$ (all $q$), $A_2(7,5)=34$. We also provide an exposition of the known determination of $A_q(v,2)$, and a table with exact results and bounds for the numbers $A_2(v,d)$, $v\leq 7$.
Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. The resulting so-called Main Problem of Subspace Coding is to determine the maximum size $A_q(v,d)$ of a code in $PG(v-1,\mathbb{F}_q)$ with minimum subspace distance $d$. Here we completely resolve this problem for $d\ge v-1$. For $d=v-2$ we present some improved bounds and determine $A_q(5,3)=2q^3+2$ (all $q$), $A_2(7,5)=34$. We also provide an exposition of the known determination of $A_q(v,2)$, and a table with exact results and bounds for the numbers $A_2(v,d)$, $v\leq 7$.
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