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Abstract-A table of binary constant weight codes of length n <; 28 is presented. Explicit constructions are given for most of the 600 codes in the table; the majority of these codes are new. The known techniques for constructing constant weight codes are surveyed, and also a table is given of (unrestricted) binary codes of length J1 <; 28. I. I'ITRODUCTIONT HE MAIN GOAL of this paper is to givc an exten sive table of lower bounds on A(n, d, w), the maximal possible number of binary vectors of length n, Hamming distance at least d apart, and constant weight w. We also give a table of lower bounds on A(n, d), the maximal possible number of binary vectors of length n and Ham ming distance at least d apart (with no restriction on weight).These functions have been studied by many authors, and were tabulated for n:<:: 24 in [13], [45], [72], [132]. In the present paper we extend the tables to length n:<:: 28.Our main concern is with Table I, the table of constant weight codes. The majority of the 600 codes in this table are new, either because we have discovered nicer versions of existing codes, or (more frequcntly) because we have found better codes than were known before.Our goal has been to give either an explicit construc tion or a reference for every code in the table. With some exceptions a rcadcr should be able to reconstruct any of these codes from the information given here. (This is in contrast to [13], where several codes are simply describcd as being found by an unstatcd "miscellaneous construc tion".) However, because of space limitations, we have not included explicit listings for the codes constructed in Section XII (indicated by "y" in Table 1) when they contain more than 1500 codewords.Although [13] gives both upper and lower bounds on A(n, d, w) and A(n, d), in the present paper we give only lower bounds, i.e. tables of actual codes. We have not To save space we have sometimes written vectors in hexa decimal, using 0 = 0000,' . " 9 = 1001, A = 1010, . . " F = 1111, usually omitting leading zeros (so the vectors are right-justified). Superscripts (for example in Table XV) indicate the number of vectors in an orbit. Parenthe scs inside a vector (for example in Tables XII-XIV) indicate that all simultaneous cyclic shifts of the paren thesized sections are to be used. For example (110)(10) is an abbreviation for the six vectors 11010,01101,10110, 11001,01110,10101.A design (X, . 9]) is a set X (of "points") together with a collection [f8 of subscts of X (called "blocks"). A t -(V,k,A) design is a design in which IXI = v, all blocks contain exactly k points, and any t distinct points of X More generally an (r, A)-design is a design in which cach point belongs to exactly r blocks and each pair of points belong to exactly A blocks (but the blocks need not all contain the same number of points). A symmetric design

Remark. It may be verified that K4 is minimal in the sense that for every v E K4, v~B(K4'-{v}, 1) holds. 6.3. Group divisible designs with block-size 4 Lemma 6.8. Let h ~ t, t E T(5, A) and {3h + m, 3t + m} C GD(4, A, m); then 3(4t + h) + m E GD(4, A, m). Proof. By deleting t-h points from one group of a transversal design T[5, A; t] we obtain a pairwise group divisible design GD[{4, 5}, A, {h, t}; 4t + h ]. Since by Lemma 4.10, {12, 15} C GD(4, 1, 3) it follows by Lemma 2.27 that 3(4t + h) E GD(4, A, {3h, 3t}) and by Lemma 2.29 our lemma follows. Lemma 6.9. IfvEGD(4,A,m) and vET(4,A), then 4vEGD(4,A,m). Proof. v E T(4, A) means 4v E GD(4, A, v). Further apply Lemma 2.28. Lemma 6.10. If q = 1(mod6) is a prime-power, then 2q E GD(4, 1, 2). Proof. X = Z(2) x GF(q, f(x) = 0). Let d = (q-1)/6. P = {(0; 0), (0, a), (0; a+ 2d), (O; a+ 4d)/ mod (2; q), a = 0, 1, ... , d-1. Lemma 6.11. If n = 1(mod3) and n I 4, then v = 2n E GD(4, 1, 2) holds. Proof. Let v = 2n = 6s + 2, s ~ 1. For s = 0 the lemma is trivial. Considering Lemma 6.8 with t = 0 (mod 4), h = 0 (mod 2), A = 1 and m = 2 and applying Lemma 3.24 it suffices to prove the lemma for s E S =

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ABsTR.ACT. In this paper we exploit the relations between near polygons with lines of size 3 and Fischer spaces to classify near hexagons with quads and with lines of size three. We also construct some infinite families of near polygons. NEAR POLYGONSA near polygon is a connected partial linear space (X, L) such that given a point x and a line L there is a unique point on L closest to x (where distances are measured in the collinearity graph: two distinct points are adjacent when they are collinear). A near polygon of diameter n is called a near 2n-gon, and for n = 3 a near hexagon. , any quadrangle (in the collinearity graph of a near polygon) of which at least one side lies on a line with at least three points is contained in a unique geodetically closed subspace of diameter 2, necessarily a nondegenerate generalized quadrangle. Such a subspace is called a quad, and a near polygon is said to 'have quads' when any two points at distance 2 determine a quad containing them. When all lines have at least three points this is equivalent to asking that any two points at distance 2 have at least two common neighbours.In this paper we construct some infinite families of near polygons, and classify near hexagons with lines of length 3 and with quads. This paper is a compilation of the three reports Brouwer et al. [6], Brouwer and Wilbrink [7], and Brouwer [ 4] together with the contributions of the third author, who was referee of [6].Our main goal is the following theorem.1.1. THEOREM. Let (X. L) be a near hexagon with lines of size 3 and such that any two points at distance 2 have at least two common neighbours. Then Geometriae Dedlcata 49: 349-368, 1994.

Strongly regular graphs lie at the intersection of statistical design, group theory, finite geometry, information and coding theory, and extremal combinatorics. This monograph collects all the major known results together for the first time in book form, creating an invaluable text that researchers in algebraic combinatorics and related areas will refer to for years to come. The book covers the theory of strongly regular graphs, polar graphs, rank 3 graphs associated to buildings and Fischer groups, cyclotomic graphs, two-weight codes and graphs related to combinatorial configurations such as Latin squares, quasi-symmetric designs and spherical designs. It gives the complete classification of rank 3 graphs, including some new constructions. More than 100 graphs are treated individually. Some unified and streamlined proofs are featured, along with original material including a new approach to the (affine) half spin graphs of rank 5 hyperbolic polar spaces.