'*
This Pesearch lA1a8 done lA1hil.e the author lA1aB lA1ith the Department of
University of North Carolina at Chapel. Hiz.z~ and lA1aB supported
by the Army Research Office~ DuPham~ under Grant No. DA-ARO-D-31-124-G910
and AFOSR-68-1406.
Stati8tics~
0 7 8 9
6 1 7 8
5 0 2 7
'+ 6 1 3
0 6 5 '+ 9
7 1 0 6 5
8 7 2 1 0
9 8 7 3 2
9 8 7
2 9 8
'+ 3 9
3 '+ 5
'+ 5 6 0
6 0 1 2
1
3
5
2
'+
7
8
6
1
3
8
9
6
1
3
5
7
0
2
4
1
9
8
7
7 1 2 3
8 2 3 '+
9 3 '+ 5
0 '+ 5 6
2 5 6 0
'+ 6 0 1
6 0 1 2
1 7 8 9
3 8 9 7
5 9 7 8
3
2
9
8
5
4
3
9
8
7
6
0
2 4 6
3 5 0
'+ 6 1
5 0 2
6 1 3
0 2 '+
1 3 5
7 8 9
1 9 7 8
2 8 9 7
COMBINATORIAL
MATHEMATICS
YEA R
February 1969 - June 1970
ON t-DESIGNS AND THRESHOLD DECODING
by
J. M. Goethals"
MBLE Research Laboratol"!J
Bl'U88e '[,8 ~ Be'Lgium
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 600.29
June, 1970
ON t-DESIGNS AND THRESHOLD DECODING
by
J. M. Goethals.
MBLE Research Laboratory
Brrussels~ BeZgium
l.
INTRODUCTION.
In this paper. it is shown how some combinatorial properties of t-designs
can be used to devise a very simple method of threshold decoding. which is
applied to the (24.12) and (48,24) binary extended. quadratic-residue codes.
and
A basic role in this respect is played by the intersection numbers
the occurrence numbers of j-tuples yS which are defined in the first section.
j
2.
INTERSECTION Nlfw1BERS OF t'-DESIGNS.
A tactical configuration
(~t;v,k.t)
or a t-design is a set
elements and a collection of k-subsets of
t-subset of
number
V is contained in exactly
V.
~t
(k-j)
.
~O
( k )
j
=
~j
(k-j)
t-j
..
*
v
called blocks, such that every
blocks.
of blocks containing any j-subset of
~j
V of
For
j
= O,l, •••• t,
the
V is a constant and we have
Aj +l (v-j)
~j ( vj )
j)
At (vt-j
This research was done while the author was with the Department of
Statistics, University of North Carolina at Chapel Hill, and was supported by
the Army Research Office. Durham» under Grant No. DA-ARO-D-3l-l24-G9l0 and
AFOSR-68-1406.
2
where
AO is the total number of blocks.
may be proved by considering the
(At;v-l,k-l,t-l)
This well-known result ([ 1] , [5] )
derived tactical configurations
which are defined by the blocks of
any given element of
V.
At· 1
containing
The remaining blocks constitute the so-called
residual tactical configurations (At_I-At;v-I,k,t).
with
(At;v,k,t)
Tactical configurations
are called Steiner 8Ystems.
Mendelsohn [5] defined the intersection r&U11iJeZ'8 X. to be the number of
'Z.
blocks which intersect a given block in exactly
i
elements, and, by counting
in two different ways the number of j-tuples occurlng, easily obtained the
following equations
(1)
for
j " O,l,2, ••• ,t.
When the intersection numbers
Xi
for a given block
satisfy
Xt+l
..
Xt+2
=
...
..
~
..
0,
which is certainly the case for Steiner systems
(At-I),
the set of
(t+l)
equations (1) has an unique solution given by
(2)
for
j " 0,1,2 •• ~.,t.
For example, the well-known Steiner system
the intersection numbers
(see [4], lemma 5.1).
(1;24,8,5)
gives
XO • 30, Xl • 0, X2 • 448, X3 .. 0, X4
for any block,
= 280,
X5 - 0,
3
By considering a given s-tuple of elements and defining the more general
inte:rsection nwnbers ~1s-tuple in exactly
i
to be the number of blocks which intersect the given
elements, one easily obtains the following equations
(3)
for
j . 0,1,2, ••• ,min(s,t).
There are indeed
(;)
elements in the given s-tuple, each of which occurs
which intersectsthe given s-tuple in
distinct j-tuples.
i
distinct j-tuples of
times, and each block
A
j
( i )
j
elements contributes
Hence both members of (3) are equal.
For
s
~
t,
there
is an unique solution given by
(4)
for
( ~)
j " 0,1,2, ••• ,s.
Obviously
s .. A ,
s
s
X
and for
s
~
t
each one of the
j-tuples that are contained in the given s-tuple must occur the same
number of times, say
v;
times, that is
(5)
Then, using the fact that
(s)(i)
i
j
( s )(s-j)
j
i-j'
the set of equations (3) can be given the form
(6)
4
in which both numbers count the number of times a given j-tuple occurs.
first member of (6) can be interpreted in the following way:
The
there are
i-tuples which are contained in the given s-tuple and which contain a given
j-tuple. and each of these i-tuples occurs Yis times. Again. since s < t
there is an unique solution to (6). which is given by
(7)
These numbers y~.
which could be called oaeurrence nunibers of j-tup1es. are
given below (table 1) for the
Table 1:
j
s
0
0
759
506
330
210
130
1
2
3
4
5
78
(1;24.8.5)
Steiner system.
.Yj
OCCUl'renCe nunibel's
of
1
2
3
4
-
-
- -
17
-
56
40
21
16
5
-
28
12
4
1
253
176
120
80
52
-
(1;24~8~5)
5
-
-
It is observed that one has
(8)
which is a quite general property. as we now prove. in two different ways.
Firstly, using (7). one has
yS+l + ys+1
j
j+1
a
atl
i:j
)l ~
i-j - 1J i
(_l)i-j T(s+l-j) _ (s-j
[i-j
where the expression under brackets simply reduces to
s-j
(i-j)'
which proves
5
the result.
s+l
Y+
j l
But, by dropping one element from the given
(S+l)-tuple,
can be interpreted as the occurrence number of j-tuples in the derived
design, and similarly
yr
l
is the occurrence number of
j-tuples in the
residual design, and having this in mind gives (8) an obvious interpretation
since the original design is the union of the derived and residual designs.
3.
THRESKlLD DECODING WITH t"'DESIGNS.
Let us consider the binary code of length
incidence vectors of at-design
(At;v,k,t).
v
which is orthogonal to the
For example. the extended Golay
binary code of length 24 can be defined to be orthogonal to the incidence
vectors of the
(1;24,8,5)
Steiner system (see [1] and [3]).
A decoding
method for that particular code was described in [3]. which will be formalized
and extended to some other codes here.
parity cheCk set those
exceed
(v-l)
t-l.
incidence vectors of the t-design which contains
V and which are known to form a
a given element of
remaining
Al
The basic idea consists in taking as
elements of
V.
Then. i f the number
(t-l)-design on the
e.
of errors does not
the number of parity check failures can be calculated exactly by
x1
means of the intersection numbers
intersection numbers of the
of the
(1;23.7,4).
(t-l)-design.
which are easily obtained from
table 1, are as follows (table 2):
Intersection numbers ~ of (1;23,7,4)
Table 2:
For example, the
J
~
0
1
2
3
4
0
253
-
-
-
1
176
77
-
-
-
2
120
112
21
3
80
120
48
5
-
4
52
112
72
16
1
- -
6
Now.
suppose two errors occured. none of which affects the digit corresponding
to the fixed element of
exactly equal to
V.
Then the number of parity check failures will be
xi· 112.
In general. when
e.
errors occur.·
none of
which affects the fixed digit, the number of parity check failures will be
given by the SUDDDation of the intersection numbers
e..
example. for
3, it will be
120 + 5 • 125.
error affects the fixed digit and if in addition
e.
X
with odd
j
j.
For
If, on the other hand. one
e-
1
other digits are
corrupted by error. the number of parity check failures will be given by
even j
since the error in the fixed position contributes to all.
For example,
for
e·
3,
it will be
253-112· 141.
xg
parity checks.
The results concerning
the above example are summarized in table 3.
nunibe:tt of paztity cheek failures; 0000 (24,,12)
Table 3:
-
Number·
of· errors
Fixed digit
in error
not in error
1
253
77
2
176
112
3
141
125
4
128
128
It is observed that, providec! the number of errors does not exceed
3,
it can
be decided whether or not the fixed digit was in error, according as the
number of parity check failures is greater or less than
has minimum distance
8,
128.
Since the rode
this decoding method uses the full error-correcting
7
ability of the code.
MDreover, uncorrectable errors
(e.. 4) can be
detected when the number of parity cheek failures is exactly equal to
The same reasClll1Dg will
residue binary code. which
DOW
caD
be applied to the
(8;48,12,5),
as was shown by AsSIDUS
UsiDs the relations (7), the occurrence mabers are easily
calculated (table 4). frca which the intersection numbers
design
(8;47,11,4)
are obtained (table 5).
errors does not exceed
4,
j
0
the number of parity check fa1lures can be
1
2
4
3
172.96.
1
12972
4324
2
9660
3312
1012
3
7140
2520
792
220
4
5236
1904
616
176
5
3808
1428
476
140
36
3
4
0
5
- .- - - -
0
j
1
x~ of the derived
Then, provided the number of
calculated exactly in each ease (table 6).
8
extended quadratic
be clefined to be orthogou1 to the inddence
vectors of a 5-design with paraaaeters
and Mattson (2J.
(48,24)
128.
-
2
- 44
-
-
8
- -
- -
0
4324
1
3312
1012
2
2520
1584
220
3
1904
1848
522
44
-
4
1428
1904
840
144
8
- - --
8
8
Table 6:
number of pOX!ity check failures; code
Number
Fixed digit
of errors
in error
not in error
1
4324
1012
2
3312
1584
3
2740
1892
4
2426
2048
Since the code can actually correct up to
further investigation.
5
4324 - 2048 .. 2276.
fixed digit is not affected by error, aDd let
X:. 8.
which obviously cannot exceed
one easily gets
X~.
errors, the case
e· 5
deserves
If the fixed digit is in error, the number of parity
check failures will then be equal to
~,
(48~24)
x,
X~
.. 4o-5x,
X~
Suppose now the
X be the intersection number
Then, from the set of equations (3),
• 280+l0x,
X~
.. l120-l0X,
xi .. l820+5x,
~ .. 1064-x, from which it follows that the number of parity check failures is
equal to
2100
+ l6x, w}l1ch cannot exceed 2228. Hence the code can be
threshold decoded up to
5
errors, with a set of
threshold comprised in the range
2228 ••• 2276.
4324
parity checks and a
9
REFERENCES
[1] ASSMUS, E.F.,Jr. and MATTSON, H.F., On tactical configurations and errorcorrecting codes, J. CorriJinatoria't
TheOl'!J~ ~
(1967), 243-257.
[2] ASSMUS, E.F.,Jr. and MATTSON, H.F •• New 5-designs, J. CombinatoriaZ
Theor,y~
£ (1969), 122-151.
[3] GOETHALS, J.M.. On the Golay perfect binary code, J. CombinatoriaZ TheOl1l~
(to appear).
[4] GOETHALS, J.M. and SEIDEL, J.J.,
combinatorial designs,
Strongly regular graphs derived from
Canadian J. Math. ~ (to appear).
[5] MENDELSOHN, N.S.,
Some applications of intersection numbers to problems
in t-designs, presented at TheSeeond 'Chape'Z Titn Conference on
Combinatorial Mathemat"ic8
its AppUcations (May 1970).
ana