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Advances in Mathematics of Communications

2016 , Volume 10 , Issue 3

Special issue on ALCOMA'15

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A. Kerber, M. Kiermaier, R. Laue, M. O. Pavčević and A. Wassermann
2016, 10(3): i-ii doi: 10.3934/amc.201603i +[Abstract](215) +[PDF](133.8KB)
The present issue of the Advances in Mathematics of Communications is dedicated to the conference

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A new family of linear maximum rank distance codes
John Sheekey
2016, 10(3): 475-488 doi: 10.3934/amc.2016019 +[Abstract](751) +[PDF](390.1KB)
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.
Further results on the classification of MDS codes
Janne I. Kokkala and Patric R. J.  Östergård
2016, 10(3): 489-498 doi: 10.3934/amc.2016020 +[Abstract](435) +[PDF](318.8KB)
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.
Algebraic structures of MRD codes
Javier de la Cruz, Michael Kiermaier, Alfred Wassermann and Wolfgang Willems
2016, 10(3): 499-510 doi: 10.3934/amc.2016021 +[Abstract](693) +[PDF](359.8KB)
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.
Construction of 3-designs using $(1,\sigma)$-resolution
Tran van Trung
2016, 10(3): 511-524 doi: 10.3934/amc.2016022 +[Abstract](370) +[PDF](363.3KB)
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.
Construction of subspace codes through linkage
Heide Gluesing-Luerssen and Carolyn Troha
2016, 10(3): 525-540 doi: 10.3934/amc.2016023 +[Abstract](479) +[PDF](410.8KB)
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 note on Erdős-Ko-Rado sets of generators in Hermitian polar spaces
Klaus Metsch
2016, 10(3): 541-545 doi: 10.3934/amc.2016024 +[Abstract](370) +[PDF](265.0KB)
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.
On the existence of Hadamard difference sets in groups of order 400
Joško Mandić and Tanja Vučičić
2016, 10(3): 547-554 doi: 10.3934/amc.2016025 +[Abstract](417) +[PDF](328.0KB)
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.
Self-orthogonal codes from the strongly regular graphs on up to 40 vertices
Dean Crnković, Marija Maksimović, Bernardo Gabriel Rodrigues and Sanja Rukavina
2016, 10(3): 555-582 doi: 10.3934/amc.2016026 +[Abstract](418) +[PDF](486.5KB)
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.
There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$
Sihuang Hu and Gabriele Nebe
2016, 10(3): 583-588 doi: 10.3934/amc.2016027 +[Abstract](369) +[PDF](324.6KB)
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.
Explicit constructions of some non-Gabidulin linear maximum rank distance codes
Kamil Otal and Ferruh Özbudak
2016, 10(3): 589-600 doi: 10.3934/amc.2016028 +[Abstract](683) +[PDF](319.1KB)
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.
The non-existence of $(104,22;3,5)$-arcs
Ivan Landjev and Assia Rousseva
2016, 10(3): 601-611 doi: 10.3934/amc.2016029 +[Abstract](434) +[PDF](368.2KB)
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 $q=5,k=4,d=82$.
Further results on multiple coverings of the farthest-off points
Daniele Bartoli, Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini and Fernanda Pambianco
2016, 10(3): 613-632 doi: 10.3934/amc.2016030 +[Abstract](474) +[PDF](505.0KB)
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.
On self-dual MRD codes
Gabriele Nebe and Wolfgang Willems
2016, 10(3): 633-642 doi: 10.3934/amc.2016031 +[Abstract](472) +[PDF](367.5KB)
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 $(\mathbb{F}_q)^{n\times n}$.
The weight distribution of the self-dual $[128,64]$ polarity design code
Masaaki Harada, Ethan Novak and Vladimir D. Tonchev
2016, 10(3): 643-648 doi: 10.3934/amc.2016032 +[Abstract](447) +[PDF](282.5KB)
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 $d \in \{ 20, 24 \}$.
Constructions and bounds for mixed-dimension subspace codes
Thomas Honold, Michael Kiermaier and Sascha Kurz
2016, 10(3): 649-682 doi: 10.3934/amc.2016033 +[Abstract](501) +[PDF](672.0KB)
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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