From T(m) triangular graphs to single-error-correcting codes
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We present a way to construct an infinite family of linear codes, with d=3, from a particular type of strongly regular graph.
Using the incidence matrix and taking a tree of any nondirected and connected graph G(V,E) we can obtain the matrix of fundamental circuits and the one for fundamental cuts. These are respectively the generator and parity check matrices of a linear code.
We are interested in the construction of triangular strongly regular graphs, named T(m). From T(m) and some results on the girth and the valency of a strongly regular graph G, for any integer m≥4, we obtain a linear code C(T(m)) with parameters: n=(m(m−1)(m−2))/2, k=(m(m−1)(m−3)+2)/2, d=3 and A1=An−1, A1 being the number of codewords of weight i in the code.
Moreover we give some properties of its codewords set and its orthogonal one. Finally, an alternative method using lattice graphs Lz is proposed.
KeywordsRegular Graph Linear Code Lattice Graph Triangular Graph Parity Check Matrice
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- (1)."Graph theory, coding theory and block designs" P.J. Cameron and J.H. Van Lint Cambridge University Press, Cambridge 1975.Google Scholar
- (2)."Graph theory" W. Mayeda John Wiley and Sons, New York 1972.Google Scholar
- (3)."Error Correcting Codes" W.W. Peterson and E.J. Weldon Jr. MIT Press, Cambridge 1972.Google Scholar
- (4)."Cut-Set Matrices and Linear Codes" S.L. Hakimi and H. Frank IEEE Trans. on Inf. Theory 11 (1965) pp. 457–458.Google Scholar
- (5).Bobrow and Hakimi IEEE Trans. on Inf. Theory 17 (1971) pp. 215–221.Google Scholar
- (6)."Configuracions i grafs en la teoria de codis" J.M. Basart Dissertation for 1st. degree. Univ. Autònoma de Barcelona, Bellaterra 1985.Google Scholar