An investigation of relationships between graph theory and coding theory

The problem of th.e existence of perfect and nearly perfect codes over finite alphabets is generalised in two directions. This thesis is concerned with the existence and combinatorial properties of completely regular codes in distance-regular graphs. One of the main tools is the generalisation of Lloyd's Theorem. There are connections with designs, orthogonal latin squares and finite projective planes and various existence and non-existence results are derived for completely regular codes in three infinite families of distance-regular graphs.

[1]  Michel Deza.  Une propriete extremale des plans projectifs finis dans une classe de codes equidistants , 1973, Discret. Math..

[2]  N. Biggs.  Perfect codes in graphs , 1973 .

[3]  J. H. Lint.  On the nonexistence of perfect 5, 6 and 7-error-correcting codes over GF(q) , 1970 .

[4]  P. Hammond.  Nearly perfect codes in distance-regular graphs , 1976, Discret. Math..

[5]  A. Tietäväinen.  On the Nonexistence of Perfect Codes over Finite Fields , 1973 .

[6]  J. H. van Lint.  Recent Results on Perfect Codes and Related Topics , 1975 .

[7]  Koichiro Yamamoto.  EULER SQUARES AND INCOMPLETE EULER SQUARES OF EVEN DEGREES , 1954 .

[8]  Selmer M. Johnson.  A new upper bound for error-correcting codes , 1962, IRE Trans. Inf. Theory.

[9]  Solomon W. Golomb,et al.  Rook domains, Latin squares, affine planes, and error-distributing codes , 1964, IEEE Trans. Inf. Theory.

[10]  Philippe Delsarte,et al.  Four Fundamental Parameters of a Code and Their Combinatorial Significance , 1973, Inf. Control..

[11]  Non-existence of Steiner systems of typeS (d-1,d, 2d) , 1974 .

[12]  R. C. Bose,et al.  Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture , 1960, Canadian Journal of Mathematics.

[13]  Derek H. Smith.  Primitive and imprimitive graphs , 1971 .

[14]  S. P. Lloyd.  Binary block coding , 1957 .

[15]  E. Lloyd,et al.  The footballers of Croam , 1973 .

[16]  Derek H. Smith,et al.  Perfect codes in the graphs Ok , 1975 .

[17]  N. J. A. Sloane,et al.  A survey of constructive coding theory, and a table of binary codes of highest known rate , 1972, Discret. Math..

[18]  A. Gardiner.  Antipodal covering graphs , 1974 .

[19]  van Hca Henk Tilborg,et al.  Uniformly packed codes , 1976 .

[20]  J. Dénes,et al.  Latin squares and their applications , 1974 .

[21]  Franco P. Preparata.  A Class of Optimum Nonlinear Double-Error-Correcting Codes , 1968, Inf. Control..