Diameter, Covering Index, Covering Radius and Eigenvalues

Fan Chung has recently derived an upper bound on the diameter of a regular graph as a function of the second largest eigenvalue in absolute value. We generalize this bound to the case of bipartite biregular graphs, and regular directed graphs.We also observe the connection with the primitivity exponent of the adjacency matrix. This applies directly to the covering number of Finite Non Abelian Simple Groups (FINASIG). We generalize this latter problem to primitive association schemes, such as the conjugacy scheme of Paige's simple loop.By noticing that the covering radius of a linear code is the diameter of a Cayley graph on the cosets, we derive an upper bound on the covering radius of a code as a function of the scattering of the weights of the dual code. When the code has even weights, we obtain a bound on the covering radius as a function of the dual distance dl which is tighter, for d? large enough, than the recent bounds of Tietavainen.

[1]  Sung Y. Song,et al.  The Character Tables of Paige's Sample Moufang Loops and Their Relationship to the Character Tables of Psl(2, q) , 1989 .

[2]  Arthur T. White,et al.  Permutation Groups and Combinatorial Structures , 1979 .

[3]  Tor Helleseth,et al.  On the covering radius of cyclic linear codes and arithmetic codes , 1985, Discret. Appl. Math..

[4]  A. Robert Calderbank,et al.  On a Pair of Dual Subschemes of the Hamming Scheme Hn(q) , 1985, Eur. J. Comb..

[5]  Gérard D. Cohen,et al.  Covering radius - Survey and recent results , 1985, IEEE Trans. Inf. Theory.

[6]  Oscar Moreno,et al.  Exponential sums and Goppa codes: II , 1989, IEEE Trans. Inf. Theory.

[7]  Robert J. McEliece,et al.  New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities , 1977, IEEE Trans. Inf. Theory.

[8]  M.B. Pursley,et al.  Crosscorrelation properties of pseudorandom and related sequences , 1980, Proceedings of the IEEE.

[9]  Patrick Solé.  A limit law on the distance distribution of binary codes , 1990, IEEE Trans. Inf. Theory.

[10]  Norman Biggs,et al.  On Trivalent Graphs , 1971 .

[11]  Eugene Seneta,et al.  Non‐Negative Matrices , 1975 .

[12]  Noga Alon,et al.  Eigenvalues and expanders , 1986, Comb..

[13]  E. Bannai,et al.  Algebraic Combinatorics I: Association Schemes , 1984 .

[14]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[15]  Jean-Louis Dornstetter.  Some quasi-perfect cyclic codes , 1984, AAECC.

[16]  Jacques Wolfmann,et al.  New bounds on cyclic codes from algebraic curves , 1988, Coding Theory and Applications.

[17]  F. Chung.  Diameters and eigenvalues , 1989 .

[18]  A. A. Tietavainen.  An upper bound on the covering radius as a function of the dual distance , 1990 .

[19]  Lloyd R. Welch,et al.  Lower bounds on the maximum cross correlation of signals (Corresp.) , 1974, IEEE Trans. Inf. Theory.