Linear codes from incidence matrices of unit graphs

In this paper we consider binary linear codes spanned by incidence matrices of Steiner 2-designs associated with maximal arcs in projective planes of even order, and their dual codes. Upper and lower bounds on the 2-rank of the incidence matrices are derived. A lower bound on the minimum distance of the dual codes is proved, and it is shown that the bound is achieved if and only if the related maximal arc contains a hyperoval of the plane. The binary linear codes of length 52 spanned by the incidence matrices of 2-$(52,4,1)$ designs associated with previously known and some newly found maximal arcs of degree 4 in projective planes of order 16 are analyzed and classified up to equivalence. The classification shows that some designs associated with maximal arcs in nonisomorphic planes generate equivalent codes. This phenomenon establishes new links between several of the known planes. A conjecture concerning the codes of maximal arcs in $PG(2,2^m)$ is formulated.

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Codes from incidence matrices of some regular graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s179383092150035x ◽

2020 ◽

pp. 2150035

Author(s):

R. Saranya ◽

C. Durairajan

Keyword(s):

Linear Codes ◽

Automorphism Groups ◽

Regular Graphs ◽

Permutation Decoding ◽

Incidence Matrices ◽

Hamming Metric ◽

Vertex Set ◽

Lee Distance ◽

Lee Metric ◽

Transitive Subgroup

We examine the [Formula: see text]-ary linear codes with respect to Lee metric from incidence matrix of the Lee graph with vertex set [Formula: see text] and two vertices being adjacent if their Lee distance is one. All the main parameters of the codes are obtained as [Formula: see text] if [Formula: see text] is odd and [Formula: see text] if [Formula: see text] is even. We examine also the [Formula: see text]-ary linear codes with respect to Hamming metric from incidence matrices of Desargues graph, Pappus graph, Folkman graph and the main parameters of the codes are [Formula: see text], respectively. Any transitive subgroup of automorphism groups of these graphs can be used for full permutation decoding using the corresponding codes. All the above codes can be used for full error correction by permutation decoding.

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A Characterization of Graphs by Codes from their Incidence Matrices

The Electronic Journal of Combinatorics ◽

10.37236/2770 ◽

2013 ◽

Vol 20 (3) ◽

Cited By ~ 4

Author(s):

Peter Dankelmann ◽

Jennifer D. Key ◽

Bernardo G. Rodrigues

Keyword(s):

Incidence Matrix ◽

Linear Codes ◽

Minimum Weight ◽

Weight Enumerator ◽

Edge Connectivity ◽

Connected Graphs ◽

Incidence Matrices ◽

Wide Range ◽

The Matrix

We continue our earlier investigation of properties of linear codes generated by the rows of incidence matrices of $k$-regular connected graphs on $n$ vertices. The notion of edge connectivity is used to show that, for a wide range of such graphs, the $p$-ary code, for all primes $p$, from an $n \times \frac{1}{2}nk$ incidence matrix has dimension $n$ or $n-1$, minimum weight $k$, the minimum words are the scalar multiples of the rows, there is a gap in the weight enumerator between $k$ and $2k-2$, and the words of weight $2k-2$ are the scalar multiples of the differences of intersecting rows of the matrix. For such graphs, the graph can thus be retrieved from the code.

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Codes from incidence matrices of some Bouwer graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500270 ◽

2020 ◽

pp. 2150027

Author(s):

R. Saranya ◽

C. Durairajan

Keyword(s):

Error Correction ◽

Incidence Matrix ◽

Linear Codes ◽

Permutation Decoding ◽

Incidence Matrices ◽

Vertex Set

We examine the [Formula: see text]-ary linear codes from incidence matrix of the Bouwer graph [Formula: see text] with vertex set [Formula: see text] and two vertices are adjacent if they can be written as [Formula: see text] and [Formula: see text] where either [Formula: see text] or [Formula: see text] differs from [Formula: see text] in exactly one position, say the [Formula: see text]th position, where [Formula: see text] All the main parameters of the codes are obtained as [Formula: see text] Also, we determine linear codes from incidence matrices of Bouwer graphs [Formula: see text] and all the main parameters of the codes are obtained as [Formula: see text] All the above codes can be used for full error correction by permutation decoding.

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