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Families of Codes with Few Distinct Weights From
Singular and Non-Singular Hermitian Varieties and Quadrics in
Projective Geometries and Hadamard Difference Sets and Designs
Associated with Two-Weight Codes
by
I.M. Chakravarti
O.
Summary.
In this paper, we present several doubly infinite families of linear projective
codes with two-, three- and four distinct non-zero Hamming weights together with the
frequency distributions of their weights.
The codes have been defined as linear spaces of coordinate vectors of points on
certain projective
sets described
non-degenerate and singular -
in
terms
of
Hermitian and quadratic
in projective spaces.
been derived by considering the geometry of
hyperplanes in relevant projective spaces.
forms
-
The weight-distributions have
intersections of projective sets by
Results from Bose and Chakravarti (1966)
and Chakravarti (1971) on the Hermitian geometry and Bose (1964) , Primrose (1951) and
Ray-chaudhuri
(1959,1962) have been used in the enumeration of weights and their
frequencies.
The paper has been organized as follows.
Preliminary definitions, concepts and
resul ts on Hermi tian geometry [from Bose and Chakravarti
(1966)
and Chakravarti
(1971)] are given in Section 1.
Two families of two-weight codes
~(VN-l) and ~(VN-l) over GF{s2) and associated
families ~'(VN-l) and ~'(VN-l) over GF{s) together with their weight-distributions are
given in Section 2.
Here V - denotes a non-degenerate Hermitian variety in PG{N,s2)
N 1
and V is its complement and a code ~(S) is defined as the linear space of the
N 1
coordinate vectors of the points in the projective set S.
- 2 -
The eigenvalues of the adjacency matrix A = B -B (B is the incidence matrix of
i
2 1
the ith associates i=1,2) of the strongly regular graph (two-class association scheme)
on s2(N+1) vertices defined by the two-weight code ~'(VN-1) over GF(s) and the (P~k)
parameters of the two-class association scheme are given in Section 3.
In Section 4, we show that for s=2, B (the association matrix of the second
2
associates) of the two-class association scheme of Section 3, is the incidence matrix
2N 1
2N + (_2)N
·
. h parameters v =
22(N+1)
a symmetr i c BIB d eSlgn
WIt
, k__2 + + (_2)N, ~A -_ 2
.
2(N+1)
and 2B -J is a Hadamard matriX of order 2
. Similarly, I+B is the incidence
1
2
2N
matrix of a symmetric BIB design with parameters v = 22(N+1), k=2 + 1 - (_2)N,
2N
2(I+B ) - J is a Hadamard matrix of order 2 2N+2. Further, it
1
2N+1
N
2N
N
is shown that the 2
+ (-2)
codewords each of weight (2
- (-2) ), which are
A = 2
- (_2)N
and
non-adjacent to the null codeword form a Hadamard difference set (Menon 1960, Mann
1965) with parameters v = 2
2N 1
(2 + _ (_2)N -1)
2N+2
2N+1
N
2N
N
, k = 2
+ (-2) , A = 2
+ (-2)
and the
codewords each of weight 2 2N together with the null codeword form
a Hadamard difference set with parameters v = 2 2N+2, k = 22N+ 1 - {_2)N,
A = 2 2N - (_2)N, for integer N.
In Section 5, a family of four-weight linear codes and the associated weightdistributions are derived.
A code here is defined as the linear sPan of a projective
set which is the intersection of a non-degenerate Hermitian variety and the complement
of one of the secant hyperplanes.
In Section 6 and 7, we consider codes which are linear sPans of projective sets
defined in terms of degenerate Hermi tian and quadratic forms in projective spaces.
The motivation here is to explore how the code parameters behave when the basic
projective set is not purely a subspace nor a non-degenerate Hermi tian or quadric
variety but an analgam of the two, which still. admi ts a geometric description (and
algebraic equations).
~
- 3 o
the basic projective set is a degenerate Hermi tian variety V N2
2
which is the intersection of a non-degenerate Hermitian variety VN- in PG(N.s ) with
1
o
one of its tangent hyperplanes. The code ~(VN-2) which is the linear space generated
In section 6.
by the coordinate vectors of the points of ~-2' is shown to be a tri-weight code.
Its weight-distribution as a code over GF(s2) as well as that of its sister code over
GF(s) are given.
In section 7. a degenerate quadric ~-1 which is the intersection of a non-
ON
in PG(N.s) with one of its tangent hyperplanes. is taken as the
basic projective set.
The code ~(~-1) which is the linear space of the coordinate
degenerate quadric
vectors of the points of ~-1' is shown to be a tri-weight code both for odd and even
N.
The frequency distributions of the weights are given for both odd and even N.
odd N. both the cases elliptic and hyperbolic have been considered.
For
These families
supplement those obtained by Wolfmann (1975) from non-degenerate quadrics.
1.
Introduction
The geometry of Hermitian varieties in finite dimensional projective spaces have
been studied by Jordan (1870). Dickson (1901). Dieudonn6 (1971). and recently. among
others by Bose (1963.
Chakravarti (1971).
1971). Segre (1965.1967). Bose and Chakravarti (1966) and
In this paper. however. we have used results given in the last
two articles.
If h is any element of a Galois field GF(s2). where s is a prime or a power of a
prime. then h=h is defined to be conjugate to h.
square matrix H = (h
Hermitian if h
row-vectors
~T
ij
= h
ij
ji
).
i. j
= O.1. ...• N.
for all
i. j.
2
Since h =h. h is conjugate to h.
with elements from GF(s2)
The set of all points
= (xO.x1 •.... ~) satisfy the equation
Hermi tian variety V - .
N 1
if
H
is Hermi tian and x
~T
(s)
H
~(s)
is
A
is called
in PG(N. s2) whose
= 0 are said to form a
the column vector whose
- 4 -
s ...• ~).
s
transpose is (x s ' xl'
The variety VN- l is said to be non-degenerate if H A
~
O
has rank N+l. The Hermitian form xT H x{s) where H is of order N+l and rank r can be
reduced to the canonical form YoYO + ... + YrYr by a suitable non-singular linear
transformation ! = Ay".
The equation of a non-degenerate Hermi tian variety VN- l in
2
.
s+l
s+l
s+l
PG{N. s ) can then be taken in the canonIcal form X
+ xl
+ ... + ~
= O.
o
in PG{N.s2 ) with equation !T H !(s) = O.
Consider a Hermitian variety V A
N l
2
point C in PG{N.s ) with row-vector ~T = (cO.cl ..... ~) is called a singular point of
V - if ~T H = QT or equivalently. H £(s) = Q.
N l
is called a regular point of V - .
N l
A point of V - which is not singular
N l
Thus a non-singular point is ei ther a regular
point of V - or a point not on V - . in which case it is called an external point of
N l
N l
2
PG{N.s ). with respect to V - . It is clear that a non-degenerate VN- cannot possess
N l
l
a singular point.
On the other hand. if V -
N l
singular points of VN- l constitute a
is degenerate and rank H = r
(N-~)-flat
Let C be a point with row vector £.
< N+l.
the
called the singular space of VN- l ·
e
Then the polar space of C with respect to
the Hermitian variety V - with equation ~T H ~(s)
N l
2
points of PG{N.s ) which satisfy ~T H £(s) = O.
= O.
is defined to be the set of
When C is a singular point of V - . the polar space of C is the whole space
N l
2
PG{N.s ).
When. however. C is either a regular point of V - or an external point.
N l
T H (s) = 0 is the equation of a hyperplane which is called the polar hyperplane of
£
~
C wi th respect to V - l .
N
Let C and D be two points of PG{N. s2) .
If
the polar
hyperplane of C passes through D. then the polar hyperplane of D passes through C.
Two such points C and D are said to be con1ugates to each other with respect to V - .
N l
Thus the points lying in the polar hyperplane of C are all
conjugates to C.
the points which are
If C is a regular point of V - . the polar hyperplane of C passes
N l
through C; C is thus self-conjugate.
tangent hyperplane to VN- l at C.
In this case. the polar hyperplane is called the
e
- 5 -
When V - is non-degenerate, there is no singular point.
N1
To every point, there
corresponds a unique polar hyperplane, and at every point of V - , there is a unique
N1
tangent hyPerplane.
If C is an external point, its polar hyPerplane will be called a
secant hyperplane.
The number of points in a non-degenerate Hermitian variety V - in PG(N, s 2 ) is
N1
¢(N,s2) = (sN+l _ (_I)N+l)(sN _ (_I)N)/(s2_ 1 ).
A polar hyPerplane
~N-l
of an external point
~
(also called a secant hyPerplane)
in PG(N,s2) intersects a non-degenerate Hermitian variety V - , in a non-degenerate
N1
Hermitian variety V - of rank N. It has {sN - (_I)N)(sN-l - (_I)N-l)/{s2_ 1 ) points.
N2
A tangent hyperplane
~N-l
to a non-degenerate VN- 1 at a point C, intersects VN- 1
in a degenerate VN- of rank N-l.
2
The singular space of V - consists of the single
N2
point C.
The number of points in a degenerate Hermitian variety V - of rank r < N+l in
N1
PG(N,s2) is (s2_ 1 ) f(N-r,s2) ¢(r-l,s2) + f(N-r,s2) + ¢(r-l,s2), where f(k,s2) =
(s2(k+l)_I/(s2_ 1 ).
Thus t h e number
·
0 fpoInts
in a d egenerate VN-2
0f
r ank N- 1 , is
(s2_ 1 ) f{0,s2) ¢(N-2,s2) + f(0,s2) + ¢(N-2,s2)
= 1 + (sN-l _ (_I)N-l)(sN-2 _ (_I)N-2)s2/(s2_ 1 ).
2.
Two-weight codes from non-degenerate Hermitian varieties in projective spaces.
Consider the code ~(VN-l) over GF{s2), which is the linear space generated by the
coordinate vectors of
PG(N,s2).
the points on a non-degenerate Hermi tian variety V in
N1
This variety has (sN+l) - (_I)N+l)(sN - (_I)N)/(s2_ 1 ) = n (say) points.
Thus a matrix G = (g .. ) i=O,l, ... ,N,
IJ
j=l, ... ,n, whose columns are the coordinate
vectors of the n points on VN- 1 , is a generator matrix of the code
projective linear code (n, k=N+l).
~(VN-l)
Now, a tangent hyperplane meets V - at
N 1
which is a
- 6 -
1 + {sN-l - {_I)N-l}{sN-2 - (_I)N-2)s2/{s2_ 1 ) points and a secant hyperplane meets
N
N N-l
N-l
2
V - at {s - (-I) ){s
- (-I)
)/(s -1) points; hence this code
N1
two distinct non-zero weights wI and w2 (say). where
~(VN-l)
has only
wI = {sN+l - {_I)N+l){sN - (_I)N)/{s2_ 1 )
_ {sN-l _ {_I)N-l){sN-2 _ {_I)N-2)s2/{s2_ 1 )_1
w
2
= {sN+l
~ {_I)N+l){sN - (_I)N)/{s2_ 1 )
_ {sN _ {_I)N){sN-l _ {_I)N-l)/s2_ 1
The frequency f
wI
= s2N_ 1
= s2N-l
+ {_s)N-I.
of the code-words with weight wI is equal to the number of
tangent hyperplanes (same as the number of points on V - ) mul tip l i ed by (s2_1).
N1
frequency f
w
2
The
of codewords of weight w is the number of secant hyperplanes (same as
2
the number of external points in PG{N.s2 » multiplied by (s2_ 1).
Thus
= {sN+l _ {_I)N+l){sN _ (_I)N).
f
WI
f
w2
= (s2{N+l)_I)
_ {sN+l _ {_I)N+l){sN _ (_I)N)
= (s_I){s2N+l + (_s)N)*
2
Let VN- denote the set of external points of PG{N.s ) with respect to VN- 1 .
1
Thus V - is the complement of V - .
N 1
N1
Considering the intersections of V _ 1 by tangent
N
and secant hyperplanes. we get another sequence ~(VN-l) of two-weight linear
..
proJectIve
cod
es ·In s 2 symb 0 ls.i t
wh parameters
n = {s2N+l
fWI
+ (-s)N)/{s+I). k
= N+l.
WI
= {sN+l - {_I)N+l){sN _ (_I)N). fw2
= s2N
w = s2N _ s2N-l _ {_s)N-I.
2
= (s_I){s2N+l + (_s)N).
- s2N-l
*While presenting this resul t at ~ Colloque International sur la Th60rie des
Graphes et Combinatoire. Marseille-Luminy. June 1986. the author learnt that
Calderbank and Kantor (1986) have also obtained this family using the rank three
representation of unitary groups.
e
- 7 -
Now a
projective linear (n,k) code over GF(sr) wi th weights wi
i=I, ... , t,
determines a projective linear (n' ,k') code over GF(s) with weights
w: i=I, ... ,t, n' = n(sr_1)/(s-I), k' = kr, w: = sr-l wi . i=I, ... ,t, (Delsarte, 1972).
1
1
Hence the code
2
~(VN-l)
over GF(s ) determines a sister code
~I(VN-l)
over GF(s) with
parameters
N
,
2N
- (-1) )/(s-I), k=2(N+l), wI = S ; w2 =
f , = (sN+l - (_I)N+l)(sN - (_I)N), f . = (s_I)(s2N+l + (-s) N).
wI
w2
n
,
= (s
N+l
- (-1)
N+l
)(s
N
I
-
2
-
Similar ly. the code ~(VN-l) over GF( s ) determines a code ~'(VN-l) over GF( s) wi th
2N+1
N -I
-,
2N+1
2N - I
2N+1
2N
N
parameters n' = s
+ (-s) . k = 2(N+l), wI = s
- s . w2 = s
-s
+ (-s) .
(_I)N+l(sN_(_I)N), f
The family
derived
by
~'(VN-l)
Wolfmann
I
w2
= (s_I)(s2N+l+(_s)N).
of two-weight codes over GF(s) is a subfamily of the one
(1978)
from
non-degenerate
quadrics.
This
is
because a
non-degenerate Hermi tian form over PG(N, s2) becomes a non-degenerate quadric over
PG(2N+l. s) (Dickson. 1958. p. 144).
It is elliptic if N is even and hyperbolic if N
is odd (see. for instance. Heft. 1971).
3.
Strongly regular graphs of Latin square and negative Latin square types, from
two-weight codes.
Delsarte (1972) has shown that a two-weight projective (n.k) code over GF(s)
determines a strongly regular graph on v = sk vertices (a two-class association
scheme) and that the eigenvalues Pi of the adjacency matrix A = B2 -B of the graph (B
1
i
is the association matrix of the ith associates. i=I.2.:
see. for instance. Bose and
Mesner 1959) are given by
(w -w 1 )PO = 2m v/s - (w +w ){v-l),
2
1 2
(w2-w )P 1 = w +w - (1+(-I)i)v/s. i=I,2,
1
1 2
with v = sk, m = n{s-I), WI and w2 are the two distinct non-zero weights of codewords.
One can calculate the parameters P~k' i.j.k = 1.2. of the two-class association scheme
- 8 -
from PI' P , n and n , where n is the number of codewords of
2
i
2
1
r.
2
2
Writing {P +P )/2 = ~, {P -p)/2 = v~ and A = ~ + 2p+1,
1
1 2
2
1
1
11
{P+~)/2 = P12 and {P-~)/2 = P12·
Then P11 = n 1-1-P12' P22 =
weight wi' i=l,2.
~
one can show that
12
2
n 2 - P12 , P11 = n 1 - P12'
P~ = ~-1-P~2 (see, for instance, Bose and Mesner 1959).
Thus the graph on s2{N+1) vertices corresponding to the code ~'(VN-1) over GF(s)
is strongly regular and the eigenvalues of its adjacency matrix A = B2-B , are
1
2N+1
N
N+1
N+1 N
N
N
Po = {s
+ (-s) )(s-l) - {s
- (-I)
)(s - (-I) ) = n2-n 1 , PI = 1-2{-s)
N
and P = 1+2{s-1){-s).
As a two-class association scheme its parameters are
2
N+1
N+1
N
N
2N+1
N
1
2N
n 1 = {s
- (-I)
)(s - (-I) ), n2 = (s-l){s
+ (-s) ), P12 = {s-l)s ,
2N
N
1
2N
N
2
2N
N
2
P12 = {s-l)s
- (s-2){-s) - I, P11 = s -(s-l){-s) -2, P11 = s -(-s) ,
1
2N
2
N
2
2N
2
N
P22 = s (s-l) + (-s) (s-l), P22 = s (s-l) + (-s) (2s-3).
For N=2, this family gives the two-class negative Latin square association scheme
NL (s3) with g = s2_1, given by Mesner (1967, pp. 579-580).
g
Mesner considered two
points on an EG{3, s2) to be first associates if the line joining the two points,
shared a point with a non-degenerate Hermi tian curve in the ideal plane of the
PG(3,s2)
in which EG{3,s2)
is embedded.
Thus at
first
generalizes Mesner's family based on the Hermitian curve.
sight our construction
However, it is to be noted
that for every odd N, our family is the same as the hyperbolic family which is called
.
. the same as
pseudo-Latin square Lg (s N+1 ), g = s N+1 and for every even N, our famIly
IS
the elliptic family which Mesner called the negative Latin square NL (sN+1), g = sN_ 1 ,
g
(Mesner 1967, p. 578).
See also Hubaut (1975, pp. 374-379, C. 12± ).
Similarly, one can verify that the eigenvalues of the adjacency matrix
A = B2-B 1
. ht
o f the s t rong I y regu I ar graph on v = s 2{N+1).
vertIces correspond·Ing to t h e two-weIg
projective code
N
P2 = 1-2(-s).
-
~'(VN-1)
-
W2 - Awi '
over GF{s) are Po = A
-
PI = 1+2(s-1){-s)
N
and
Thus it is clear that the sequence of two-weights codes ~'(VN) over
GF{s) gives rise to essentially the same two sequences of (Latin square type Lg(n) and
~
- 9 -
negative Latin square type NL (n»
two-class association schemes, as derived from the
g
sequence of codes f(l' (VN- 1 ) .
4.
Hadamard difference sets, Hadamard matrices and symmetric BIB designs from the
association schemes.
For s=2, the parameters of the two-class association scheme corresponding to the
N
two-weight binary projective code f(l'(V _ ) are n = (2 +1 _ (_I)N+l)(2N_(_I)N),
1
N1
2N+1
N 1
2N
2
2N
1
2N
N
2
2N
N
~ = 2
+ (-2) , P12 = 2 , P12 = 2
- 1, Pll = 2 -(-2) -2, P 11 = 2 -(-2) ,
1
2N
N
2
_2N
N
P
= 2 +(-2) , P
= Z-"+(-2) .
22
22
.
1
2
2N
N
The equalIty P22 = P
= 2 +(-2) implies that the matrix B - the association
22
2
matrix of the second associates (non-adjacent vertices) is the incidence matrix of a
SYmmetric BIB design with parameters v = 22(N+l), k = 22N+1+(_2)N, A = 22N+(_2)N and
-J is a Hadamard matrix of order 22(N+l) which corresponds
2
difference set v = 22N+2 , k=22N+1+(_2)N, A = 22N+(_2)N.
2B
Consider
the
(22N+1+(_2)N)
codewords
each of
weight
non-adjacent to (second associates of) the null codeword.
codeword of weight 2
2N
to
the Hadamard
(22N _(_2)N)
which are
The null codeword and a
are adjacent to (first associates of) each other and among the
non-adjacent vertices of the null codeword, there are P~ = 2
2N
are also non-adjacent to the given codeword of weight 2
2N
+(_2)N codewords which
This implies that the
given code vector can be expressed as the differences of 22N+(-2lN pairs of code
vectors each of weight (22N_(_2)N).
Similarly, since P~2 = 22N+(_2)N if follows that
every codeword of weight 22N _(_2)N can be expressed as the differences of 22N+(_2)N
pairs of codevectors each of weight (22N _(_2)N).
Thus the codewords of weight 22N_(_2)N form a difference set with v = 22N+2,
k = 22N+1 + (_2)N, A = 22N + (_2)N.
1
2
2N
N
2N+l
N
Thus the (2
-(-2) -1) codewords each of
Again, P U +2 = P u = 2 -(-2).
weight 22N together with the null codeword form a difference set with v = 22N+2,
k
= 22N+1_(_2)N,
A = 22N _(_2)N.
- 10 -
5.
A family of four weight codes. Code defined as a linear span of a projective set
which is the intersection of a non-degenerate Hermitian variety and the complement
of one of the secant hyperplanes.
Let C;/~_1 be a hyPerplane in PG(N. s2) which is not one of the tangent hyPerplanes
of
the non-degenerate Hermitian variety VN- 1 ·
non-degenerate Hermitian variety ~-2.
intersects VN- 1 in a
Let X denote the set of points on VN- 1 which
are not on V~_2' that is. X = VN-1"~-2.
Ixi
o
Then C;/N-1
Then the number points in X is
= [(sN+1_(_1)N+1)(sN_(_1)N) - (sN_(_1)N)(sN-1_(_1)N-1)](s2_ 1 )
= s2N-1
+ (_s)N-1.
Let ce(X) be the code over GF(s2). which is the linear space generated by the
coordinate vectors of the points in X.
Then ce(X) has n = s
2N-1
+(-s)
N-1
. k = N+1.
Let C;/N-1 be another hyPerplane (distinct from C;/~_1) which is not a tangent to
VN- 1 ·
~-1
Then
intersects VN- 1 in a non-degenerate Hermitian variety VN- 2 and meets
C;/~_1 in a (N-2)-flat C;/~_2. C;/~_2 intersects V~_2 in a non-degenerate V~_3. Thus every ~
0
X·In s 2N-3 + ( -s )N-2 pOln
. t s. Le t
N- 1 meets
2N-2 2N-3
N-2
-s
-(-s)
].
w1 = IXI - IX n ~-1 I = (s+1)[s
2
Now the number of hyPerplanes in PG(N.s ). which are not tangent to V - is
N1
(s2N+2_ 1 )/(s2_ 1 )
(sN+1_(_1)N+1)/(s2_ 1 ) = (s2N+2_ s 2N+1_(_s)N+1_(_s)N)/(s2_ 1 ) =
non-tangent
<11
I
d· t·
f
N- 1 IS Inct rom
<11
I
= (s2N+1+(_s)N)/(s+1).
Thus it follows that in ce(X). there are f
each of weight w and f
1
w4
= s 2-1
o
1 +
= (s2N+1+(-s)N Hs - 1 ) _(s2_ 1 ) codewords
codewords each of weight w
4
Let C be a point of VN- 2 and let
Then the intersection of
w1
~N-1(C)
= s 2N-1 +(-s) N-1 .
be the hyperplane tangent to VN- at C.
1
~N-1(C)
and VN- 1 is a degenerate Hermitian variety V - with
N2
N-1
N-1
N-2
N-2
2
0
(s
-(-1)
)(s
-(-1)
)(s -1) points.
But ~N-1(C) meets C;/N-1 in an
(N-2)-flat ~~_2(C) which is
tangent
to V~_2 at C.
Thus ~~-2 meets V~_2 in a
- 11 -
degenerate ~-3 consisting of
1 + (sN-2_{_1)N-2){sN-3_{_1)N-3)s2/{s2_ 1 ) points.
Thus
2N-3 + (-s) N-l .
Ix n ~N-l{C)1 = IVN_ 1 n ~N-l{C)1 - IV~_2 n ~~_2{C) = s
Let
W = Ixi - Ix n ~N-l{C)1 = s2N-l_ s 2N-3 = s2N-3{s2_ 1 )
2
and let f
W2
= (s2_1).
(no. of points on ~-2)
= (sN_{_1)N){sN-1_{_1)N-l).
Then in <€(X) there are f
w2 codewords each of weight w2 .
Let ~N-1{P) be the tangent hyperplane to VN- 1 at a point P which is not on ~-2 .
Then
~N-l{P)
points.
meets VN- 1 in a degenerate VN- 2 consisting of
1 + (sN-l_{_1)N-l){sN-2_{_1)N-2)s2/{s2_ 1 )
But ~N-1{P) intersects ~~-l in an (N-2)-flat ~~-2 which is not a tangent to
0 0 0
0
VN- 2 · Thus ~N-2 meets VN- 2 in a non-degenerate VN- 3 consisting of
(sN-1_{_1)N-1){sN-2_{_1)N-2)/{s2_ 1 ) points. Hence
Ix n ~N-l{p)1 = IVN_ 1 n ~N-l{P)1 - 1~-2 n ~-21
= s2N-3+{_s)N-l+{_s)N-2
2N-1
2N-3
N-2
Let w3 = IXI - Ix n ~N-1{P) = s
- s
-(-s)
.
and f
= (s2_ 1). (No. of points in X)
w3
= (s2_ 1 ){s2N-1)+{_s)N-1).
Thus in <€(X) there are f
w3 codewords each of weight w3 . Thus the linear projective
code <€(X) over GF{ s2) has four distinct non-zero weights w. wi th corresponding
1
frequencies f
Wi
i=1.2.3.4.
It follows that <€(X) determines a linear projective code
<€'{X) over GF{s) with parameters n' = n{s+1) = (s+l){s2N-l+{_s)N-l). k' = 2k = 2{N+l).
)N-1
w1, = sW 1 = ( s+ 1){ s 2N-1 -s 2N-2+(-s
• )w,2 = sW2 = s2N+2{s2_ 1 ). w3' = sW3 =
s2N_ s 2N-2+{_s)N-1
and
w = sW = s2N_{_s)N
and the associated frequencies
4
4
f
Wi
i=1 •.... 4 remain unchanged.
- 12 -
6.
A family of three-weight codes from degenerate Hermitian varieties in projective
spaces.
2
Consider the section of a non-degenerate Hermitian variety V - in PG(N.s ) with
N1
the tangent space ~N-l(C) at a point C on VN- 1 .
This section is a degenerate V~_2(C)
of rank N-l and its singular space consists of the single point C.
A (N-2)-space ~N-2 contained in ~N-2(C) but not containing C. intersects V~_2(C)
in a non-degenerate Hermitian variety VN- 3 of rank N-l.
Every point of ~_2(C) lies
on some line joining C to a point of VN- 3 . Thus the number of points on ~_2(C)
= 1 + (sN-l_(_I)N-l)(sN-2_(_I)N-2)s2/(s2_ 1 ).
Let ~(~-2) be the code over GF(s2) which is the linear space of the coordinates
of the points on Vo _ (C). This is a linear projective code with
N2
n = 1 + (sN-l_(_I)N-l)(sN-2_(_I)N-2)s2/(s2_ 1 ) and k=N.
There are [(s2)N_ 1 ] - [(s2)N-l_ 1 ]/(s2_ 1 ) = (s2)N-l
(N-2)-spaces
~-2
in
~_I(C) ~
Each such ~-2 meets V~_2(C) in a non-degenerate V - 3
N
~l
~1
~2
~2
2
0
with «s)
-(-1)
)«s)
-(-1)
)/(s -1) points. Thus in ~(VN-2) there are
which do not pass through C.
f
wI
= (s2_ 1 )(s2)N-l = s2N _ s2N-2
codewords each of weight
wI = 1 + s2(sN-l_(_I)N-l)(sN-2_(_I)N-2)/(s2_ 1 )
_ (sN-l_(_I)N-l)(sN-2_(_I)N-2)/(s2_ 1 )
= s2N-3 + (_s)N-l + (_s)N-2.
Let
*
~N-2
be a fixed (N-2)-flat contained in
~N-l(C),
not passing through C and
let V;-3 a non-degenerate variety of rank N-l. denote the intersection of ~;-2 and
o _ (C).
VN
2
o
VN_2 (C).
Let
~N-2(D)
be the tangent space to V*
N- 3 at D which is a regular point of
Then the (N-2)-space ~N_2(C.D) which is the join of the point C and ~N-3(D).
o
is the tangent space to VN_2 (C) at D and to every point on the line joining D to C.
~
- 13 The tangent space
o
~N_2(C,D)
0
intersects V _ (C) in a degenerate variety V _ (C,D) of
N2
N3
rank N-3 and its singular space is the line joining C and D.
Number of points on such V~_3(C,D) (degenerate of order 2) is
(s2_ 1 )(s2+ 1 ) (sN-3_(_1)N-3)(sN-4_(_1)N-4)/(s2_ 1 )
+ s2 + 1 + (sN-3_(_1)N-3)(sN-4_(_1)N-4)/(s2_ 1 )
= 1 + s2 + s4(s2N-7 + (_s)N-3 + (_s)N-4_ 1 )/(s2_ 1 )
Thus
w2
=1 +
N-1
N-2
N-2
2
2 N-1
s (s
-(-1)
)(s
-(-1)
)/(s -1)
-1 - s2 - s2(s2N-S + (_s)N-1 + (_s)N-2_ 1 )/(s2_ 1 ) = s 2N-3
is the weight of a codeword which corresponds to the tangent space
~N-1(C),
o_ (C),
VN
2
2
in
is equal to the number of regular points D on
*- a non-degenerate Hermi tian variety.
that is the number of points on VN
2
Thus the frequency f
f
~N_2(C,D)
The number of such
~N_2(C,D)
W2
of the codewords in cg(~-2) of weight w2 is
= (sN-1_(_1)N-1)(sN-2_(_1)N-2).
Let
*
~N-2
* which is not a tangent to --*
be a (N-3)-flat in SN-2'
vN- 3 ·
the join of C and
~N-3'
Since
~N-3
o _ (C) with
VN- 4 , the section of VN
2
Let
~N-2
be
meets V*
N- 3 in a non-degenerate Hermitian variety
~N-2
*- which consists of all the
is a degenerate VN
3
points on the lines joining C to the points of V - .
N4
Let w = IV~_2(C)1 - 1V;-3 1 = 1 + s2(sN-1_(_1)N-1)(sN-2_(_1)N-2)/(s2_ 1 )
3
-1 - s2 _ s2(sN-2 _(_1)N-2)(sN-3_(_1)N-3)/(s2_ 1 )
= s2N-3 + (_s)N-1.
Thus w3 is the weight of a codeword which corresponds to an (N-2)-flat
passes through C, but is not a tangent to V~_2(C),
.
In
*
~N-2
Number of such (N-2)-flats
*.
whIch
are tangents to .-*
V N3
= ((s2)N-1_ 1 )/(s2_ 1 ) _ (sN-1_(_1)N-1)(sN-2_(_1)N-2)/(s2_ 1 )
~N-2 =
Number of
~N-3
~N-2
- Number
~N-3
in
= (s2N-3+(_s)N-2)/(s+1)
Thus the frequency f
3
of codewords of weight w
3
~N-2'
which
- 14 -
= (s2_ 1 )(s2N-3+(_s)N-2)/(s+l)
= (s_I)(s2N-3+(_s)N-2).
It follows that ~(V~_2) determines a linear projective code ~'(V~_2) over GF(s)
2 N-l
with parameters n' = n(s+l) = (s+l) + {s (s
-(-1) N-l )(sN-2-(-1) N-2 )}/(s-I). k' = 2k.
,
WI = sW
I
= s
2N-2
N
N-l,
2N-2,
-(-s) - (-s)
. w2 = sW2 = s
• w3
frequencies f
7.
Wi
= sW3 = s 2N-2-(-s) N
and the
i=1.2.3 remain unchanged.
Families of three-weight codes from degenerate quadrics (cones) in projective
spaces.
Let ON be a non-degenerate quadric in Pe(N.s) and let
~N-1(P)
be the hyperplane
tangent to QN at P. Then the intersection of ~N-1(P) and ON is a cone Q~-1(P) of rank
N and order 1 and its vertex consists of the single point P.
A (N-2)-flat
~N-2
contained in
~N-1(P)
but not passing through P. intersects QN
o
The points of ON-I(P) = ~N-1(P) n QN' are then all
in a non-degenerate quadric ON-2.
the points in the lines joining P to the points of Q~-2·
1 + s
~(N-2.0)
Let
~N-l(P),
*
~N-3
points where
~(N-2.0)
* = ~N-2
* n QN-1
0
QN-2
(P).
*
non-degenerate %-3.
~N-2(P) = P
U~;-3
QN-1(P) has then
is the number of points on ON-2.
.
be a (N-3)-flat contained In·a
not passing through P.
~
Suppose that
*
which is one of the (N-2)-flats of
*
is not a tangent to
~N-2
~N-3
* intersects %-2
* (a non-degenerate quadric) in a
Then ~N-3
Then the join of P and
*
~N-3
which is a (N-2)-flat
passing through P. intersects Q~-1(P), in a cone Q~-2 of order 1
with its vertex a single point P.
Now suppose that ~;_3(R) an (N-3)-flat contained in ~-2' is a tangent to Q;-2 at
R.
*
* in a cone QN-3
0
Then ~N-3(R)
intersects QN-2
of order 1 wi th R as its vertex.
Hence the join of P and
*
~N-3(R)
which is an (N-2)-flat
*
~N-2(P.R)
(say) intersects
Q~-1(P) in a cone ~_2(P.R) of order 2 and the line joining P and R is its vertex.
~
- 15 -
Let ~(~-1(P» be the code over GF(s). which is the linear space generated by the
coordinate vectors of the points on the cone
o
~-1(P).
The weight-distributions of the
linear projective codes are next derived treating the two cases
(ii)
(i)
N=2t and
N = 2t-1 separately.
N - 2t.
In this case. ~t is a non-degenerate quadric in PG(2t.s) and ~t-1(P)
is the intersection of
Thus
(i)
o
~t-1(P)
~t
with its tangent hyPerplane
~2t-l(P)
is a cone of order 1 with P as its vertex.
at a point P of
~t·
Thus
l~t-1(P)1 = number of points on ~t-l(P)
= 1 + s ~(2t-2.0) = 1 + s(s2t-2_ 1 )/(s_1)
= (s2t-1_ 1 )/(s_1).
The code ~(~t-1(P»
o
~t-1(P)
has then the parameters n = (s
A (2t-2) flat
o
~t-l(P)
which is the linear space generated by the coordinate vectors of
~2t-2
contained in
2t-1
)/(s-I). k=2t.
~2t-1(P)
but not passing through P intersects
in a non-degenerate quadric ~t-2. Thus
2t-1
2t-2
2t-2
w1 = (s
-1)/(s-1) - (s
-I)/(s-1) = s
is the number of points of ~t-1(P) which are not on such a (2t-2)-flat.
f
W1
of such (2t-2)-flats in
f
WI
Let
~2t-1(P),
*
*
~2t-3
~2t-1(P)
is given by
= (s2t_ 1 )/(s-1) - (s2t-l_ 1 )/(s_1) = s2t-1
be a (2t-3)-flat contained in
*
~2t-2
not passing through P and assume further
*
~t-2 = ~2t-2
n
0
The number
~t-1(P).
. •
Then the JOIn
of P and
~2t-1(P), passing through P.
which is one of the (2t-2)-flats of
* is not a tangent to
*.
~2t-3 IS a (2t-2)-flat S2t-2{P)
~2t-3
of
~2t-2{P) meets ~t-l{P) in a cone ~t-2{P) of order 1
with P as its vertex and ~t-2{P) consists of all the points on the lines joining P to
the points of a non-degenerate
*
~t-3
which is the intersection of
Number of points on ~t-2{P) = l~t-2{P)1 = 1 + s ~(2t-3.0).
where ~(2t-3.0) is the number of points on o;t-3.
Thus
*
~2t-3
and
*
~t-2.
- 16 -
0
I
I~t-2(P)
= 1 + s(s
= 1 + s(s
t-l
t-l
+1)(s
-1)(s
t-2
t-2
.
*
-1)/(s-I). If 02t-3 is elliptic.
+1)/(s-l)
*
if 02t-3 is hyperbolic.
Then w = (s2t-l_ 1 )/(s_l) - 1 - s(st-l+ 1 )(st-2)/(s-l) = s2t-2 + st-l
2
is the number of points of
Let f
W2
0
*
which *
are not on S2t-2(P),
if 02t-3 is elliptic.
~t-l(P)
*
denote the number of such S2t-2(P)
flats.
*
If 02t-3 is hyperbolic. then
w3 = (s2t-l_ 1 )/(s_l) _ 1 _ s(st-l_ 1 )(st-2+ 1 )/(s-l) = s2t-2 _ st-l.
is the number of points on ~t-l(P) which are not an
number of such (2t-2)-flats
Let
in o;t-2.
~2t-3(R)
*
~2t-2(P).
Let f
W3
denote the
*
~2t-2(P).
be a (2t-3)-flat in
*
~2t-2
*
which is a tangent to 02t-2
at a point R
Hence ~2t-3(R) meets o;t-2 in a cone ~t-3(R) of order 1 and its vertex is
the point R.
The join of P and
~2t-3(R)
which is a (2t-2)-flat
~2t-2(P.R)
intersects
~t-l(P) in a cone ~t-2(P.R) of order 2 and its vertex is the line joining P and R. ~
0
I = (s 2-1)/(s-l) + s 2 >/I(2t-4.0) = (s+l) + s 2 (s 2t-4-1)/(s-l) =
Then
I~t-2(P.R)
2t-2
2t-l
2t-2
2t-2
(s
-1)/(s-I). Thus w = (s
-1)/(s-l) - (s
-1)/(s-l) = s
is the number
4
of points of ~t-l(P) which are not on ~2t-2(P.R).
Then the number f
*
which is equal to the number of points on 02t-2'
is given by f
Now. f w + f w
2
3
= (number
of (2t-2)-flats in
* of
(2t-3)-flats in a (2t-2)-flat S2t-2
*
~2t-l(P),
~2t-l(P),
W4
W4
= (s
passing through P) - (number of
not passing through P and tangents to
= (s2t-l -1)/(s-l)
(2t-2)-flats contained in
~2t-l(P).
Hence counting (point. (2t-2)-flat) pairs in two
ways. one gets
wI
2t-2-1)/(s-I).
2t-2
2t-2
- (s
-1)/(s-l) = s
).
o
2t-l.
2t-l
Now ~t-l (P) has (s
-1)/(s-l) pOInts and each point is on (s
-1)/(s-l)
02t-2)
f
of such flats.
= (s2t-2_ 1)/(s_l)
+ f
w
3
+ f
(s2t-2_ 1 )/(s_l) + f (s2t-2_ s t+ s t-l_ 1 )/(s_l)
w2
w2
2t-2
t
t-l
(s
+ s - s
-1)/(s-l)
- 17 = (s2t-l_ 1 )/(s_l) x (s2t-l_ 1 )/(s_I).
+f
= s2t-2. f = s2t-l. f
w2 w2
wI
w4
Solving these equations. one gets
= (s2t-2_ 1 )/(s_I).
f
f
w2
= (s2t-2
- st-l)/2. f
Thus the weight-distribution of the
w3
o
~(~t-l(P»
o
s
s
1
2t-2
2t-2
is
frequency
weight
s
= (s2t-2+ s t-l)/2.
+ s
(s_I){s2t-l + (s2t-2_ 1 )/s_l} = s2t_ s 2t-l+ s 2t-2_ 1
t-l
(s_I)(s2t-2_ s t-l)/2 = (s2t-l_ s 2t-2_ s t+ s t-l)/2
2t-2 - s t-l
(s_I)(s2t-2_ s t-l)/2
= (s2t-l_ s 2t-2_ s t+ s t-l)/2
s
2t
Thus this is a tri-weight linear projective code over GF(s) with n = (s2t-l_1)/(s_l)
and k
= 2t.
(ii)
N -
2t-1.
In this case 0_
is a non-degenerate quadric in PG(2t-l. s) and
~t-l
o
.
GSt-2(P) is the intersection of ~t-l and its tangent hyperplane ~2t-2(P) at a point
o
P. Thus GSt-2(P) is a cone of order 1 and P is its vertex. Then the number of points
on ~t~2(P) is
l~t-2(P)1 = 1 + s 'iI(2t-3.0)
=1
is elliptic;
t-l
t-2
-1)(s
+1)/(s-l)
1 + s(s
= (s
if
Thus the code
o
~(~t-2(P»
t-2
= (s
if
=
t-l
-1)/(s-l)
+ s(s
+1)(s
2t-2
- s
t
+ s
t-l
-1)/(s-I).
~t-l
~t-l
2t-2
+ s
t
- s
t-l
-1)/(s-I).
is hyperbolic.
which is the linear space generated by the coordinate
vectors of the points of ~t-2(P) is a linear projective code over GF(s) with
n
= l~t-2(P)1
and k
= 2t-l.
- 18 -
A (2t-3)-flat
~2t-3
contained in
~2t-2{P)
but not passing through P. intersects
~
02t-l in a non-degenerate quadric 02t-3 which is elliptic (hyperbolic) if 02t-l is
Thus the intersection of such a ~2t-3 and ~t-2{P) is a
elliptic (hyperbolic).
non-degenerate 02t-3'
Hence if
02t-3 is elliptic. the number of points on
~t-2{P) which are not
~2t-3 is
WI (ellip.)
= (s2t-2
= s
On
2t-3
- st + st-l_ 1 )/{s-l) - (st-l+ 1 ){st-2_ 1 )/{s_l)
- s
t-l
+ s
t-2
the other hand if 02t-3 is hyperbolic. the number of points on
~t-2{P) which are
not on S2t-3 is
WI (hyperbol.) = (s2t-2 + st - st-l_ 1 )/{s-l) - (st-l_ 1 ){st-2_ 1 )/{s_l)
= s
The number f
*
+ s
t-l
- s
t-2
of such flats is s2t-2.
WI
~2t-3
Let
2t-3
be one of the (2t-3)-flats of
the non-degenerate
*
~t-3
~2t-2{P),
*
denote the intersection of ~2t-3
and
*
be a (2t-4)-flat of ~2t-3'
which is not a tangent to
of P and ~;t-3'
~t-4
- ~
-
2t-4
*
02t-3'
*
Let ~2t-4
0
02t-2{P),
Let ~2t-3{P) be the join
Then ~2t-3{P) intersects ~t-2{P) in a cone ~t-3{P) of order 1 with
P as its vertex and it is the join
O~
not passing through P and let
of P and a non-degenerate quadric
*
~t-4
where
n O~
~t-3'
o
Now. the number of points on 02t-3{P) is
l~t-3{P)1 = 1 + s ~(2t-4.0)
= 1 + s{s2t-4_ 1 )/{s_l)
= (s2t-3
- 1)/{s-I).
Thus if 02t-l is elliptic. the number of points on ~t-2{P) which are not on such a
~2t-3{P)
flat is
w (ellip.)
2
= (s2t-2
= s 2t-3
- st + st-l_ 1 )/{s-l) - (s2t-3_ 1 )/{s_l)
- s
t-l
~
- 19 -
If. however. ~t-1 is hyperbolic. the number of points on ~t-2(P) which are not on
~2t-3(P)
such a
flat is
w (hyperbol.) = (s2t-2 + st - st-1_ 1 )/(s-1) - (s2t-3_ 1 )/(s_1)
2
2t-3
t-1
= s
+ S
•
The number f
f
w2
= (s2t-2_ 1 )/(s_1)
(ellip.)
= s 2t-3
~t-1
If
f
w
~2t-3(P)
of such
W2
+
S
t-2
flats if
~t-1
- (st-1+ 1 )(st-2_ 1 )/(s_1)
•
is hyperbolic. the number f w of such
2
= (s2t-2_ 1 )/(s_1)
(hyperbol.)
is elliptic is given by
~2t-3(P)
flats is
- (st-1_ 1 )(st-2+ 1 )/(s-1)
2
= s 2t-3
Let
Then
~2t-4(R)
~2t-4(R)
vertex.
- s
t-2
be a (2t-4)-flat in
*
~2t-3
*
.
and a tangent to 02t-3
at the
poInt R.
*
0
R) of order 1 with the point R as its
meets 02t-3
in a cone 02t-i
The join of P and
~2t-4(R)
is a (2t-3)-flat
~2t-3(P.R)
which intersects
o
~t-2(P) in a cone 02t-3(P.R) of order 2 and its vertex is the line joining the points
P and R.
It is clear that ~t-3(P.R) is the join of the line PR and a non-degenerate
quadric ~t-5'
Number of points on ~t-3(P.R)
= (s+l)
+ s2 ~(2t-5.0)
= (s2t-3 - st + st-1_ 1 )/(s-1) if ~t-1 is elliptic
= (s2t-3
+ st - st-1 - l)/(s-l) if ~t-1 is hyperbolic.
Thus the number of points on ~t-2(P) which are not on such flats ~2t-3(P.R) is
2t-2 t t-1
2t-3 t t-1
w3 = (s
-s +s
-l)/(s-l) - (s
-s +s
-l)/(s-l)
2t-3
= s
if ~t-1 is elliptic.
2t-3
It is easy to check that w3 = s
also if ~t-1 is hyperbolic. Number f w of such
3
- 20 -
~2t-3(P,R)
(2t-3)-flats is equal to the number of points on a non degenerate quadric
~t-3·
Thus f w (ellip.)
= (s t-1 +l)(s t-2-l)/(s-l)
if
~t-1
~
is elliptic,
3
and
f w (hyperbol.) = (s
t-1
-l)(s
t-2
+l)/(s-l) if
~t-1
is hyperbolic.
3
Thus the weight-distributions of the code
~t-1
elliptic and
0_
""2t-1
s
s
s
o
~(~t-2(P»
~t-1
for the two cases -
hyperbolic are given below.
elliptic
~t-1
hyperbolic
weight
frequency
weight
frequency
0
1
0
1
2t-3 t-1 t-2
+s
-s
2t-3 t-1 t-2
-s
+s
(s_1)s2t-2
S
2t-3
(s_1)(s2t-3+ s t-2)
s 2t-3+s t-1
s 2t-3-s t-1 +s t-2-1
s 2t-3
-s
t-1
2t-3
s
(s_1)s2t-2
(s-l) (s2t-3_ s t-2)
s
2t-3 t-1 t-2
+s
-s
-1
2t-1
e
s 2t-1
.
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