CORRECTION & ADDITION.
On p. 44 in the middle, in the line containing the reference to
.SC Mathon
.[[
Mathon 1975 association
.]],
change `both by his and by the above' into `by his'.
On p. 44 at the end of \(sc2.1, add:
.Small
\&`Similar inequalities were given in
.SC Hobart
.[[
Hobart 1991 association scheme inequalities
.]].'
.Big
CORRECTION & ADDITION.
Proposition 2.2.2 as stated is correct only if all eigenvalues
of $L sub i$ are distinct. (And the double use of $i$ is somewhat
confusing; replace `$L sub i$' by `$L sub j$'.)
If not all eigenvalues are distinct, we can say the following:
.LP
.B "2.2.2a. Proposition"
.I
Let $lam$ be an eigenvalue of $L sub j$ and put
$H := "{" h vb P sub hj = lam "}"$.
Let $u$ be determined by $L sub j u = lam u$, $u sub 0 = 1$,
$u sup Tp DEL sub n u$ minimal.
Then the multiplicity $f$ of $lam$ as an eigenvalue of $A sub j$
equals $f = sum sub { h mo H } f sub h = n slash u sup Tp DEL sub n u$.
.LP
.B Proof.
We have $u = sum sub {h mo H} alpha sub h Q e sub h$ for certain
constants $alpha sub h$ determined by $sum alpha sub h f sub h = 1$
and $n sum alpha sub h supr(2) f sub h$ minimal. But the minimum
occurs when all $alpha sub h$ are equal (to $alpha$, say), and we
find $alpha = 1 slash sum f sub h$, and the minimum equals
$u sup Tp DEL sub n u = n slash sum f sub h$. $eop$
ADDITION.
At the end of \(sc2.2, add:
.LP
.B "C. Automorphisms"
.LP
.B "2.2.9. Proposition"
(G. Higman, cf.
.Ax "Higman, G."
.SC Cameron
.[[
Cameron Beineke Wilson 1983
.]],
Proposition 5.4).
.I
Let $sigma$ be an automorphism of the association scheme
$(X, fs R )$ and put $s sub i := # "{" x vb (x, sigma x) mo R sub i "}"$.
Then for each $j$ the number $n sup -1 sum sub i s sub i Q sub ij$
is an algebraic integer.
.LP
.B Proof.
Let $sigma$ have permutation matrix $S$. Then $S$ commutes with
all $A sub i$, and $s sub i = trace S A sub i$. It follows that
the number mentioned equals $trace S E sub j$, and hence is a sum
of eigenvalues of $S$. $eop$
ADDITION.
On p. 51, second sentence of \(sc2.4, add after [178]: `and
.SC Aschbacher
.[[
Aschbacher chromatic 1987
.]]'.
ADDITION.
On p. 54, at the end of \(sc2.4, add:
.LP
.B "Merging classes"
.LP
Let $fs X = (X, "{"R sub 0 , ... , R sub d "}")$ be an association scheme,
and let $PI$ be a partition of $"{"1, ... , d"}"$.
If $fs X sub PI := (X, "{" R sub 0 "}" cu "{" union sub { j mo J } R sub j
vb J mo PI "}" )$ again is an association scheme, then it is called
a \fIfusion scheme\fP of $fs X$.
For some examples, see \(sc4.2F.
.LP
If $fs X$ has the property that each partition
$PI$ of $"{" 1 , ... , d "}"$
yields a fusion scheme, then $fs X$ is called \fIamorphous\fP.
.SC A.V. Ivanov
.[[
Ivanov amorphic 1985
.]]
shows that in this case each $(X,R sub j )$ $(1 <= j <= d)$
is a strongly regular graph, and either all are of Latin square type,
or all are of negative Latin square type.
.SC Ikuta, Ito & Munemasa
.[[
Ikuta Ito Munemasa 1991
.]]
give a characterization of amorphous schemes in terms of
their group of `pseudo automorphisms' (i.e., automorphisms
of the Bose-Mesner algebra $fs A$).
.SC Baumert, Mills & Ward
.[[
Baumert Mills Ward 1982
.]]
determine when a cyclotomic scheme is amorphous.
.LP
.SC Goldbach & Claasen
.[[
Goldbach Claasen 1996
.]]
and
.SC Song
.[[
Song 1995
.]]
study imprimitive non-symmetric association schemes.
In [BCN] the concept of `tight design' was mentioned occasionally, but no
definition was given. Now `tight design' is dual to `perfect code', and
thinking about the right setup for tight designs in \(sc2.8 also led
to a more elegant version of Proposition 2.5.3.
.LP
.B "2.5.3A. Proposition."
.I
Let $b,c$ be two real vectors indexed by $"{"0, ... , d "}"$. Then
.EQ
sum from i=0 to d { (bQ) sub i (cQ) sub i } over { f sub i } =
n sum from j=0 to d { b sub j c sub j } over { n sub j } .
.EN
Now assume moreover that $b >= 0$, $c >= 0$, $bQ >= 0$, $cQ >= 0$ and
$b sub 0 = c sub 0 = 1$.
.IP (i)
.I
If $b sub j c sub j = 0$ for all $j != 0$, then $(bQ) sub 0 (cQ) sub 0 <= n$,
with equality if and only if $(bQ) sub i (cQ) sub i = 0$ for all $i != 0$.
.IP (ii)
.I
If $(bQ) sub i (cQ) sub i = 0$ for all $i != 0$, then $(bQ) sub 0 (cQ) sub 0 >= n$,
with equality if and only if $b sub j c sub j = 0$ for all $j != 0$.
.Bop
.EQ
sum from i f sub i sup -1 (bQ) sub i (cQ) sub i =
sum from {i,j,k} f sub i sup -1 b sub j Q sub ji c sub k Q sub ki =
sum from {i,j,k} n sub k sup -1 b sub j c sub k Q sub ji P sub ik =
n sum from j n sub j sup -1 b sub j c sub j .
.EN
Parts (i) and (ii) follow immediately.
.Eop
(Note that if $b$ and $c$ are the inner distributions of sets $Y$ and $Z$,
then $(bQ) sub 0 = vb Y vb$, $(cQ) sub 0 = vb Z vb$, and we obtain
Proposition 2.5.3 from part (i).)
Now add at the end of \(sc2.8:
.B "Tight designs"
.LP
The following proposition is a dual to Lloyd's Theorem 2.5.4 and MacWilliams'
Inequality 2.5.5 (i).
.LP
.B "2.8.4. Proposition."
.I
Let $Y$ be a $t$-design in a $Q$-polynomial association scheme $(X, fs R )$,
and put $s = [t/2]$. Then
.IP (i)
.I
$vb Y vb >= sum from i=0 to s f sub i$, and
.IP (ii)
.I
at least $s$ nonidentity relations occur between the elements of $Y$.
.IP (iii)
.I
Equivalent are: \fR(a)\fP equality holds in (i); \fR(b)\fP equality holds in (ii);
\fR(c)\fP $space 0 { sum from i=0 to s Q sub ji = 0 }$ for the nonidentity relations $j$
occurring in $Y$.
.Bop
(i) Put $I = "{"0,1, ... , s"}"$ and $m = sum sub { i mo I } f sub i$ and
$c sub j = m sup -2 n sub j ( sum sub { i mo I } Q sub ji ) sup 2$. Then
$c sub 0 = 1$ and $c sub j >= 0$ and
.EQ
(cQ) sub k = m sup -2 sum from { g,h mo I } n sub j Q sub jg Q sub jh Q sub jk =
n m sup -2 f sub k sum from { g,h mo I } q sub gh sup k >= 0 ,
.EN
so that in particular $(cQ) sub 0 = n m sup -1$.
.br
Since $(X, fs R )$ is $Q$-polynomial, we have $(cQ) sub k = 0$ for $k > 2s$.
Now if we let $b$ be the inner distribution of $Y$, and apply 2.5.3A (ii),
we find (i), and the equivalence of (iiia) and (iiic).
.LP
(ii) We have $Q sub jk = q sub k ( z sub j )$ for certain numbers $z sub j$ and
polynomials $q sub k (z)$ of degree $k$. Now assume that $b sub j != 0$ for at
most $s$ nonzero values of $j$, and let $f(z) = sum from k=0 to s c sub k q sub k (z)$
be a polynomial of degree at most $s$ such that $f(z sub j ) = 0$ whenever
$b sub j != 0$ and $j != 0$. We may take $c sub 0 = 1$.
We have $f(z sub j ) = sum from k=0 to s c sub k Q sub jk$, and in particular
.EQ I
f(z sub 0 ) = sum from k=0 to s c sub k f sub k .
.EN
Also
.EQ I
f(z sub 0 ) = sum from j f(z sub j ) b sub j = sum from k=0 to s c sub k (bQ) sub k
= vb Y vb .
.EN
Finally,
.EQ I
f(z sub 0 ) sup 2 = mark sum from j f(z sub j ) sup 2 b sub j =
sum from j sum from k,l=0 to s b sub j c sub k c sub l Q sub jk Q sub jl =
sum from i,j sum from k,l=0 to s b sub j c sub k c sub l q sub kl sup i Q sub ji =
.EN
.EQ I
= lineup
sum from j sum from k,l=0 to s c sub k c sub l q sub kl sup i (bQ) sub i =
sum from k=0 to s c sub k supr(2) f sub k vb Y vb .
.EN
Combining these three equations and using (i), we find
$sum from k=0 to s (c sub k - 1) sup 2 f sub k <= 0$, so that $c sub k = 1$ for all $k$.
This proves (ii) and (iiib)$implies$(iiia).
But clearly (iiic) implies that at most $s$ nonidentity relations occur,
so (iiic)$implies$(iiib), finishing the proof.
.Eop
A $t$-design in a $Q$-polynomial association scheme $(X, fs R )$ is called
\fItight\fP when it satisfies the three equivalent conditions of
Proposition 2.8.4.
.LP
The concept of $t$-design in a cometric association scheme has been
generalized by
.SC Delsarte, Goethals & Seidel
.[[
Delsarte Goethals Seidel 1977
.]]
to spherical designs in Euclidean spaces, cf. \(sc2.11.
The concept of Delsarte spaces
.SC (Neumaier
.[[
Neumaier 1981 terms distances
.]],
.SC Blokhuis
.[[
Blokhuis thesis 1984
.]])
provides a common setting for designs in $Q$-polynomial schemes,
spherical designs and designs in real, complex, quaternion and
octonion projective spaces
.SC (Hoggar
.[[
Hoggar 1982 projective spaces
.]
.[
Hoggar 1984 parameters
.]],
.SC Bannai & Hoggar
.[[
Bannai Hoggar squarefree 1988
.]]).
All results on cometric schemes generalize to Delsarte spaces.
For a recent survey, see
.SC Hoggar
.[[
Hoggar 1990 delsarte spaces
.]].
CORRECTION.
On p. 56, line 8, change `theorem' into `proposition'.
ADDITION.
The McFarland difference set
.SC (McFarland
.[[
McFarland 1973 family difference sets
.]])
provides a counterexample to the conjecture on p. 68.
[Indeed, a difference set with multiplier $-1$ gives a
translation strongly regular graph. The McFarland difference set
with parameters $(4000,775,150)$ in
$Zz sub 5 supr(3) times Zz sub 2 supr(5)$ has multiplier $-1$
and gives rise to strongly regular graphs with parameters
$(v,k, lam , mu ) = (4000,774,148,150)$ and
$(4000,775,150,150)$; their duals have parameters
$(4000,1935,910,960)$ and $(4000,1984,1008,960)$.
For difference sets with multiplier $-1$, see, e.g.,
.SC Jungnickel
.[[
Jungnickel 1982 difference sets
.]],
.SC Ghinelli-Smit
.[[
Ghinelli-Smit new result difference sets 1987
.]],
.SC Pott
.[[
Pott 1989 abelian difference sets
.]],
.SC Ma
.[[
Ma association schemes schur rings partial difference sets 1989
.]],
.SC Arasu, Jungnickel & Pott
.[[
Arasu Jungnickel Pott 1990 divisible difference
.]].]