Codes from incidence matrices and line graphs of Paley graphs

Ghinelli, Dina; Key, J. D.

doi:10.3934/amc.2011.5.93

Cited by 21 publications

(11 citation statements)

References 14 publications

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“…It was shown in [8] that if Γ satisfies any of an extensive list of conditions, the minimum distance of C 2 (G) equals the edge-connectivity λ(Γ) of Γ, i.e., the smallest number of edges whose removal renders Γ disconnected, and there are no words whose weight is strictly between λ(Γ) and λ (Γ), where the restricted edge-connectivity λ (Γ) is defined as the smallest number of edges whose removal results in a graph that is disconnected, and in which every component has at least two vertices. The analogous results for p-ary codes, for p an odd prime, in that paper included a much smaller class of graphs, even though it had been shown, by different methods (in [13,22,20,21,15], for example) that the same results hold for all primes for all the classes studied. Thus we continue our study here to broaden our results for the p-ary case where p is odd.…”

Section: Introductionmentioning

confidence: 71%

“…Linear codes generated by |V |×|E| incidence matrices for several classes of regular graphs Γ = (V, E) were examined in, for example, [13,22,20,21,15], and shown to have certain important common properties. In order to establish this common property more generally, in [8] it was shown that several properties of these codes for regular connected graphs can be derived from properties of the graph involving edge cuts, i.e.…”

Section: Introductionmentioning

confidence: 99%

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A Characterization of Graphs by Codes from their Incidence Matrices

Dankelmann

Key

Rodrigues

2013

Electron. J. Combin.

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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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Section: Introductionmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

A Characterization of Graphs by Codes from their Incidence Matrices

Dankelmann

Key

Rodrigues

2013

Electron. J. Combin.

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“…So H is a [18,4,8] 2 code. • q = 17: here P (17) is of type (17,8,3,4), and wt(v(x)) = 32, and the minimum weight as found by Magma [3,7]: see also [14]. 2.…”

mentioning

confidence: 81%

“…the hulls of their incidence designs, can be examined. This study follows work on the codes over any field from incidence matrices of graphs: see [10,32,28,29,14] and most recently [8], where the findings of the previous papers are shown to be quite general. In particular, it is known that for many classes of connected regular graphs the code from the row span of an incidence matrix of Γ = (V, E) over F p has dimension |V | for p odd and |V | − 1 for p = 2, and the words of minimum weight are the scalar multiples of the rows of the matrix, as in the case for codes from projective planes.…”

Section: Hulls Of Codes From Incidence Matrices Of Connected Regular mentioning

confidence: 96%

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Hulls of codes from incidence matrices of connected regular graphs

Ghinelli

Key

McDonough

2012

Des. Codes Cryptogr.

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D.Ghinelli, J.D.Key, T.P.McDonough. Hulls of codes from incidence matrices of connected regular graphs. Designs Codes and Cryptography, 2014, vol 70, pg 35-54.The hulls of codes from the row span over?FpFp?, for any prime?p, of incidence matrices of connected?k-regular graphs are examined, and the dimension of the hull is given in terms of the dimension of the row span of?A?+?kI?over?FpFp?, where?A?is an adjacency matrix for the graph. Ifp?=?2, for most classes of connected regular graphs with some further form of symmetry, it was shown by Dankelmann et?al. (Des. Codes Cryptogr.?2012) that the hull is either {0} or has minimum weight at least 2k?2. Here we show that if the graph is strongly regular with parameter set (n,?k, ?,??), then, unless?k?is even and???is odd, the binary hull is non-trivial, of minimum weight generally greater than 2k?? 2, and we construct words of low weight in the hull; if?k?is even and???is odd, we show that the binary hull is zero. Further, if a graph is the line graph of a?k-regular graph,?k?? 3, that has an??-cycle for some???? 3, the binary hull is shown to be non-trivial with minimum weight at most 2?(k?2). Properties of the?p-ary hulls are also established.authorsversionPeer reviewe

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Generalised Paley Graphs with a Product Structure

Pearce¹,

Praeger

2019

Ann. Comb.

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A graph is Cartesian decomposable if it is isomorphic to a Cartesian product of (more than one) strictly smaller graphs, each of which has more than one vertex and admits no such decomposition. These smaller graphs are called the Cartesian-prime factors of the Cartesian decomposition, and were shown, by Sabidussi and Vizing independently, to be uniquely determined up to isomorphism. We characterise by their parameters those generalised Paley graphs which are Cartesian decomposable, and we prove that for such graphs, the Cartesian-prime factors are themselves smaller generalised Paley graphs. This generalises a result of Lim and the second author which deals with the case where all the Cartesianprime factors are complete graphs. These results contribute to the determination, by parameters, of generalised Paley graphs with automorphism groups larger than the 1-dimensional affine subgroups used to define them.Date: November 2016.

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Codes from incidence matrices and line graphs of Paley graphs

Cited by 21 publications

References 14 publications

A Characterization of Graphs by Codes from their Incidence Matrices

A Characterization of Graphs by Codes from their Incidence Matrices

Hulls of codes from incidence matrices of connected regular graphs

Generalised Paley Graphs with a Product Structure

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