Suppose there exists a Hadamard 2-(m,m−12,m−34) design having skew incidence matrix. If there exists a conference graph on 2m−1 vertices, then there exists a regular Hadamard matrix of order 4m2. A ...conference graph on 2m+3 vertices yields a regular Hadamard matrix of order 4(m+1)2.
LCD subspace codes Crnković, Dean; Švob, Andrea
Designs, codes, and cryptography,
10/2023, Letnik:
91, Številka:
10
Journal Article
Recenzirano
A subspace code is a nonempty set of subspaces of a vector space
F
q
n
. Linear codes with complementary duals, or LCD codes, are linear codes whose intersection with their duals is trivial. In this ...paper, we introduce a notion of LCD subspace codes. We show that the minimum distance decoding problem for an LCD subspace code reduces to a problem that is simpler than for a general subspace code. Further, we show that under some conditions equitable partitions of association schemes yield such LCD subspace codes and as an illustration of the method give some examples from distance-regular graphs. We also give constructions from mutually unbiased weighing matrices, that include constructions from mutually unbiased Hadamard matrices.
In this paper, we give constructions of self-orthogonal codes from orbit matrices of Deza graphs, normally regular digraphs and Deza digraphs in terms of a definition given by Wang and Feng. These ...constructions can also be applied to adjacency matrices of the mentioned graphs. Since a lot of constructions of Deza graphs, normally regular digraphs and Deza digraphs in the sense of Wang and Feng have been known, the methods presented in this paper give us a rich source of matrices that span self-orthogonal codes.
In this paper we construct distance-regular graphs admitting a vertex transitive action of the five sporadic simple groups discovered by E. Mathieu, the Mathieu groups
M
11
,
M
12
,
M
22
,
M
23
and
M
...24
. From the binary code spanned by an adjacency matrix of the strongly regular graph with parameters (176,70,18,34) we obtain block designs having the full automorphism groups isomorphic to the Higman-Sims finite simple group. Moreover, from that code we obtain eight 2-designs having the full automorphism group isomorphic to
M
22
, whose existence cannot be explained neither by the Assmus-Mattson theorem nor by a transitivity argument. Further, we discuss a possibility of permutation decoding of the codes spanned by adjacency matrices of the graphs constructed and find small PD-sets for some of the codes.
Switching for 2-designs Crnković, Dean; Švob, Andrea
Designs, codes, and cryptography,
07/2022, Letnik:
90, Številka:
7
Journal Article
Recenzirano
In this paper, we introduce a switching for 2-designs, which defines a type of trade. We illustrate this method by applying it to some symmetric (64, 28, 12) designs, showing that the switching ...introduced in this paper in some cases can be applied directly to orbit matrices. In that way we obtain six new symmetric (64, 28, 12) designs. Further, we show that this type of switching (of trades) can be applied to any symmetric design related to a Bush-type Hadamard matrix and construct symmetric designs with parameters (36, 15, 6) leading to new Bush-type Hadamard matrices of order 36, and symmetric (100, 45, 20) designs yielding Bush-type Hadamard matrices of order 100.
Linear codes with complementary duals, shortly named LCD codes, are linear codes whose intersection with their duals is trivial. In this paper, we outline a construction for LCD codes over finite ...fields from the adjacency matrices of two-class association schemes. These schemes consist of either strongly regular graphs (SRGs) or doubly regular tournaments (DRTs). Under certain conditions, the method yields formally self-dual codes. Further, we propose a decoding algorithm that can be feasible for the LCD codes obtained using one of the given methods.
q-analogs of strongly regular graphs Braun, Michael; Crnković, Dean; De Boeck, Maarten ...
Linear algebra and its applications,
07/2024, Letnik:
693
Journal Article
Recenzirano
Odprti dostop
We introduce the notion of q-analogs of strongly regular graphs and give several examples of such structures. We prove a necessary condition on the parameters, show the connection to designs over ...finite fields, and present a classification.
We propose a method of constructing block designs which combine genetic algorithm and a method for constructing designs with a prescribed automorphism group using tactical decompositions (i.e., orbit ...matrices). We apply this method to construct new Steiner systems with parameters S(2,5,45) $S(2,5,45)$ and new symmetric designs with parameters (71, 15, 3).
LDPC codes constructed from cubic symmetric graphs Crnković, Dean; Rukavina, Sanja; Šimac, Marina
Applicable algebra in engineering, communication and computing,
11/2022, Letnik:
33, Številka:
5
Journal Article
Recenzirano
Odprti dostop
Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction of LDPC codes from ...cubic symmetric graphs. The codes constructed are (3, 3)-regular and the vast majority of the corresponding Tanner graphs have girth greater than four. We analyse properties of the codes obtained and present bounds for the code parameters, the dimension and the minimum distance. Furthermore, we give an expression for the variance of the syndrome weight of the codes constructed. Information on the LDPC codes constructed from bipartite cubic symmetric graphs with less than 200 vertices is presented as well. Some of the codes constructed are optimal, and some have an additional property of being self-orthogonal or linear codes with complementary dual (LCD codes).
We give a method of constructing self-orthogonal codes from equitable partitions of association schemes. By applying this method, we construct self-orthogonal codes from some distance-regular graphs. ...Some of the obtained codes are optimal. Further, we introduce a notion of self-orthogonal subspace codes. We show that under some conditions equitable partitions of association schemes yield such self-orthogonal subspace codes and we give some examples from distance-regular graphs.
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