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Designs and Coding Theory
Annals of Discrete Mathematics 17 (1983) 319-326 Publishing Company
0 North-Holland
DESIGNS AND CODING THEORY Marshall HALL Jr.* California Institute of Technology, USA
1. Introduction Coding theory has been valuable in communications and in various areas of engineering. Recently it has become a valuable tool for investigating block designs. In 1973, MacWilliams, Sloane, and Thompson [5] used the binary code of the plane of order 10 to investigate its properties. It has been shown by Anstee, Hall and Thompson [ l ] that a plane of order 10 cannot have a collineation of order 5 . It has been shown by Z . Janko (personal communication) that there is no collineation of order 3. Together with earlier results it now follows that a plane of order 10 can have only the identity collineation. This leaves the code as the main tool for investigating a plane of order 10 or showing that it does not exist. Section 2 gives some general results on the code of a design. Section 3 discusses some of the applications, in particular the plane of order 10, construction of a (41, 16, 6) design and very recent work on the code of a plane of order 12 over GF(3). 2. The code of a design. Some general results
Let A be the incidence matrix of u, b, r, k , A incomplete block design D so that A is a u by b matrix. Then it is well known (see (2.1)) that A A ' = B = ( r - A ) ] + hJ and det B = ( r - A)"-'rk. By definition, the code C of the design D over a finite field F , = GF(q) is the subspace of V = Ff;spanned by the rows of A regarded as vectors in Ff;.Let q = p ' . p a prime. If p r - A then B is of rank 1 or 0 (the latter in case p also divide r and A ) over GF(q). This is a very useful case:
I
Theorem 2.1. If C is the code of D a ( v , b, r, k , A ) design over F,. q = p ' and p r - A , then C n C' is of codimension at most one in C.
1
Proof. Let ri, r,, r, be any three rows of A. Then * This research was supported in part by NSF Grant No. MCS 7821599. 319
320
M.Hall, Jr.
(r,, r , ) = r = (r,, r , ) = A (mod k ) , and (r, - r,, rr) = 0 (mod k ) .
Hence r, - r, E C'. It is particularly useful, for a symmetric design, if p divides r - A = k - A to exactly the first power. In this case we can determine the exact dimension of C and C'. Theorem 2.2. Let D be a symmetric ( u , k , A ) design and let C be its code over Fq where q = p' and the prime p divides k - A to exactly the first power. Then if p % k , p kA, d i m C ' Z 2u - 1 dim C =and C 3 C'. 2 ' u-1 dim C =2 '
u+l dim C' = 2
and
C C C'.
Proof. Note that u must be odd since if u is even k - A is a square. Let A be the incidence matrix of D so that from (2.1)
det A
= &k(k -
Add columns 1 , 2 , . . . , u - 1 to the last column of A and then subtract the last row from the others. This gives
with det B2= ? ( k -A)"')''. diagonal form. This gives
With elementary operations we can put B, into
32 1
Designs and coding theory
U, V, unimodular, b l b 2 . b,-, = ? ( k - A ) ( v - 1)/2. Thus at most ( u - 1)/2 of br . * b,-I are multiples of p . Hence over GF(p) rank A, = rank A 3 ( v - 1)/2. Hence dim C 3 ( u - 1)/2 and if p Y k,dim C 3 ( v + 1)/2. Thus in the case p ,'j k , k Y A we have dim C 3 ( u + 1)/2 and from Theorem 2.1 dim C' 3 (v - 1)/2, but as dim C + dim C' = v these inequalities are in fact equalities. If p k, p A, then C C C' and the result follows.
I
I
A knowledge of the dimension of C is of course particularly valuable for use of the MacWilliams Identity. MacWilliams Identity. Zf
is the weight enumerator of a code C over Fq and C is of dimension s, then 1 Wc1( x , y ) = 7w c ( x
4
+ (q - l ) y , x
- y ).
A collineation group G of a design D will in a natural way act on the code C of D. We assume here that I G 1, the order of G, is not zero in Fq, the field of the code. Thus if q = p', we assume that the prime p does not divide 1 G 1. Let C , ,. . . , c h be the orbits of G on the b coordinates of V. Let the orbit C, have length m,,and define a vector as that vector of V which has l/dG as its entry for every coordinate belonging to C, and zero for every other coordinate. As m, divides ( G1, m, is not zero in Fq and 1 / 6 exists either in Fq or in the extension Fq2. Then, since distinct orbits C,, C, are disjoint, we have
_ -
(C,,C,)=l,
_ _
(C,,c,)=O,
izj.
c, -
Hence . . . , c h are an orthonormal basis for a subspace W of V (possibly over Fq-.). We wish to consider the orbit space W and in some way C under the action of G as lying in W. For this we define an operator 8,
Here 8 is an idempotent operator and W = V8 is the subspace of V with the orthonormal basis . . , Ch.Also if xi y are vectors of V lying in W, then (x, y ) " = ( x , y ) W, the first being the ordinary inner product, - the second the inner . . , c h . The following is a product with respect to the orthonormal bases theorem of Hayden generalizing a result of Thompson [2].
c,.
c,.
M.Hall, Jr.
322
Theorem 2.3. Let V be an m-dimensional vector space over a field F. Assume that G is a permutation group on the coordinates of V and that 1 G 1 has an inverse in F. Suppose that C C V is a G module then with the operator 8 defined for v E V by
putting W = VO and H = C8 we have (C8)',= H L = (CL)8.
Thompson [ l ] proved the special case in which C = C'
3. Some applications of codes to designs
An important application of codes to designs was made in 1973 by MacWilliams, Sloane, and Thompson [5]. If it exists, a projective plane of order 10 is a symmetric (111, 11, 1) design. From Theorem 2.2, the code over F z , the binary code C,is of dimension 56 and C' consists of the words of even weight (in fact of weights multiples of 4) in C.From this and the MacWilliams Identity the entire weight distribution is determined once the number of words of weights 12, 15, and 16, viz., A , * ,A l s , and A16,are known. In this paper it was shown that Ais=O. If Wls is a word of weight 15, these being 15 points of the plane, every line of the plane intersects Wls in an odd number of points. With appropriate numbering of the points 1 to 15 we must have the following configuration of the 15 points on 21 lines, 1 2 1 6
2 6
3 7 4 8 5 9
3 7 10 10 11 12
5,
4 8 11 13
12, 14.
13
IS,
14
15.
9,
1 1 1 2 2 2
10 15, 11 14, 12 13, 7 15, 8 14, 9 13,
3 3 3 4
6
IS.
8
12. 11, 14, 12,
9 6 4 7 4 9
6 13, 5 7 If, 5 8 10.
5
(3.1)
10,
A computer search showed that this configuration cannot be completed to a full plane, so that the conclusion is that A I s= 0. It has been announced by M. El Ghabbach that he has shown that A l h= 0. A word of weight 12 would be an 'oval' of points such that every line intersects the oval in two points or more. The code C of the plane ii-of order 10 can be extended by adjoining a further coordinate x and taking it as a 'parity check', taking 1 in position x if the word in C has odd weight and 0 in position x if the word in C has even weight. The larger code C ,in V t l 2has dimension 56 and is self dual, i.e., C: = C,.
Designs and coding theory
323
If r has a collineation u of order 5, it is easily shown that the only case which requires consideration as that in which u fixes exactly one point P and exactly one line L. Numbering the points 1,. . . ,110,P we have u =(1-**5)(6.**10)-.*(101*** 105)(106*.* 11O)(P)(x),
and the fixed line L in the code CI has the form
(3.2)
\
L = (101,. . . ,105)(106,.. ., llO)(P)(x).
(3.3) Here Theorem 2.3 applies, since the orbit space W of CI is of dimension 24 consisting of 22 five cycles and fixed P and x. Since C: = C , from the theorem, it follows that with H = CIB,the words of C , fixed by u,then H ' = H in W. But Pless and Sloane [6]have determined all self dual binary codes in dimension 24. The Golay code G,,contains no words of weight 4 but in our code H the line L is a word of weight 4.From their listing H must contain at least 5 further words of weight 4 disjoint from L. Let one of these be (1* * *5)(6... 10)(11 *
* *
15)(16..-20).
This corresponds to a word of weight 20 in C fixed by u.One possibility involves the configuration of 20 lines I 2 3 4
2 3
4 5 5 1
6 7 8 9 10
I1
12,
3 5
12 13, 13 14,
4
1
14
2
15
15, 11,
8, 9, 10, 6, 7.
5
1 3 2 4
11. 12, 13, 14, 15,
13. 14, 15, 11, 12,
16 17, 17 18, 18 19, 19 20, 20 16,
8 Y 16 18, 9 10 17 19. 10 6 18 20, 6 7 19 16. 7 8 20 17.
(3.4)
There will also be 70 lines containing exactly two of the points and 21 lines containing no points. A computer search showed that (3.4)cannot be completed to a full plane. Thus is up to isomorphism one of 92 starts to be considered and computer search showed that none of these can be completed. The conclusion, published in [l], is that a plane of order 10 cannot have a collineation of order 5 . For a (41,16,6)symmetric design a similar assumption led to a very different conclusion. Assuming a collineation a of order 5 and fixing exactly one point and one block led to the construction of the design. Let the fixed point be x and the fixed block be Bo. Then a = (x)(l,.. . ,5)(6,.. . , 10)(1I,.. . ,15)(16,. . . ,20)(21,.. . ,25)
(26... . ,30)(31,.. . ,35)(36,.. . ,40)
(3.5)
on points and similarly on blocks. It is possible to find a table showing how many points of each cycle x, C , ,. . . ,C, as above lie on a representative block. One of 15 possibilities is
M. Hall, Jr.
324
The structure of this table suggests the possibility of a further collineation /3 of order 3 such that on point cycles
p
= (x)(C,,
C,)(C,, cs,C,)(C,)(CX).
c2,
In full on points
p
= (X)(1 6 11)(2 7 12)(3 8 13)(49 14)(5 10 15)
(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30) (31)(32) (33) (34) (35)(36)(37)(38)(39) (40).
(3.7a)
With a little trial this led to the full design D. Representative blocks are: B,,
B,
B, B,, B,, B2,
B,, B,, B,,
,y ,y ,y ,y 6
3 1 1 1
1 2 4 6 3 5 1 2 7 9 4 5 2 4 3 6 2 6
3 7 Y 8 13 11 8 8 7
4 13 11 10 I4 12 9
11 11
5 15 12 14 15 14 10 13 12
6 16 17 16 18
16 17 18 17
7 21 18 22 19 23 20 19 20
8 27 19 23 22
24 21 20 22
9
I0
28 20 24 25 27 28 23 25
29 21 25
26 30 29 24 27
11 30 26 26 31 31 31 25 30
I2 31 31 31 32 32 32 28 36
13 32 32 32 34 34 34 29 38
14 36 36 36 39 39 3Y 30 39
15 38 38 38 40
30 -10
(3.7b)
32
40
This construction is given in detail in [2]. For the projective plane of order 12, or symmetric (157, 13, 1) design, Theorem 2.2 applies for p = 3 but not for p = 2. Here for the code C of the plane over GF(3) we have dim C = 79, dim C' = 78 and C' consists of the words =0 whose weights are multiples of 3. All wieghts are = 0, 1 (mod 3) so that A3r+2 for all s. The MacWilliams Identity is applicable but its direct application involves heavy calculations with very large numbers.
Designs and coding rheory
325
By considerations of invariants of a finite group of order 2592 Mallows and Sloane [4] have found an expression for the weight enumerator W c ( x ,y ) initially in terms of 27 parameters. Let us write
44=
(X 3
+ gY
31,
(p4
= (x -
3),
713
=Xy
h(X
-
3,
(2x
+
'). (3.8)
Their formula is wc-(X,y
) = X~I($;,
(pi)
+ TnPz($;, (pi).
(3.9)
Here Pl(u, u ) is a homogeneous polynomial of degree 13, Pz(u,u ) of degree 12, thus giving initially 14 coefficients in PI and 13 in P 2 . Showing that A " = 1, A 1 3= 314, and A l = A, = A, = Ah= A 7 = A, = A l o =A12 = A I S= Alh=A19= A z l= Azz= 0 reduces the number of parameters from 27 to 12 which may be taken as A I s ,Az4,Azc, A,,, A28, A3",'A31, A33, ,434, A36, A37, A N . These calculations will be published in a joint paper by the author and Wilkinson [3]. A word of weight 18, Wls, will correspond to an interesting configuration. For any line L we will have (Wig, L ) = 0 (mod 3). Let there be Ci lines with exactly i points in common with Wlx.Note that CI = 0 since a line L with precisely one point in common would have ( W18.L ) = ? 1. Then Co+ Cz+ C,+ C,+ 2C2 f 3C3 + 4C4
+
C s + * * * + Ci,=157
SCs + . *
+ 13C13 = 234 = 13.18
(3.10)
The first of these counts the total number of lines, the second the incidences of the 18 points on lines, and the third the incidences of pairs of the 18 points. Subtracting the third from the second gives
C2 - 2C4 - 5C5 + *
* *
- 65C13 = 81.
(3.1 1)
It now follows that C23 81. But a line L with exactly 2 points of Wls must have P, Q in Wlx,with the P coordinate + 1 and the Q coordinate - 1. If of the 18 points there are m with coordinate + 1 and n with coordinate - 1 then Czs mn. But as m + n = 18 it follows that C 2 681. Thus C, = 81, rn = n = 9 and from (3.1 1) C, = C, = . . = C1,= 0. Now from (3.10) Co= 52, Cz= 81, C, = 24. A line L with three points of Wlxmust have all three + 1 or all three - 1 in Wlx. Thus the 9 points which are + 1 lie three at a time on lines to form a Steiner triple system and the same holds for the 9 points which are - 1 in W18. Mallows and Sloane also found a much more complicated formula for the complete weight enumerator of the plane of order 12 which initially has 196 parame ters.
326
M. Hall, Jr.
References [ I ] R.P. Anstee, M. Hall Jr. and J.G. Thompson, Planes of order 10 do not have a collineation of order 5, J. Comb. Theory, Ser. A 29 (1980) 39-53. [2] W.G. Bridges, M. Hall Jr. and J.L. Hayden, Codes and designs, to appear in J. Comb. Theory, Ser. A. [3] M. Hall and J. Wilkinson, The ternary code for the plane of order 12, to appear. [4] C.L. Mallows and N.J.A. Sloane, Weight enumerators of self-orthogonal codes over GF(3), to appear. [5] F.J. MacWilliams, N.J.A. Sloane and I.G. Thompson, On the existence of a projective plane of order 10, J. Comb. Theory, Ser. A 14 (1973) 6 6 7 8 161 V. Pless and M.J.A. Sloane, On the classification and enumeration of self dual codes, J . Comb. Theory, Ser. A 18 (1975) 313-335. 171 2. Janko, Personal communication.
0 North-Holland
DESIGNS AND CODING THEORY Marshall HALL Jr.* California Institute of Technology, USA
1. Introduction Coding theory has been valuable in communications and in various areas of engineering. Recently it has become a valuable tool for investigating block designs. In 1973, MacWilliams, Sloane, and Thompson [5] used the binary code of the plane of order 10 to investigate its properties. It has been shown by Anstee, Hall and Thompson [ l ] that a plane of order 10 cannot have a collineation of order 5 . It has been shown by Z . Janko (personal communication) that there is no collineation of order 3. Together with earlier results it now follows that a plane of order 10 can have only the identity collineation. This leaves the code as the main tool for investigating a plane of order 10 or showing that it does not exist. Section 2 gives some general results on the code of a design. Section 3 discusses some of the applications, in particular the plane of order 10, construction of a (41, 16, 6) design and very recent work on the code of a plane of order 12 over GF(3). 2. The code of a design. Some general results
Let A be the incidence matrix of u, b, r, k , A incomplete block design D so that A is a u by b matrix. Then it is well known (see (2.1)) that A A ' = B = ( r - A ) ] + hJ and det B = ( r - A)"-'rk. By definition, the code C of the design D over a finite field F , = GF(q) is the subspace of V = Ff;spanned by the rows of A regarded as vectors in Ff;.Let q = p ' . p a prime. If p r - A then B is of rank 1 or 0 (the latter in case p also divide r and A ) over GF(q). This is a very useful case:
I
Theorem 2.1. If C is the code of D a ( v , b, r, k , A ) design over F,. q = p ' and p r - A , then C n C' is of codimension at most one in C.
1
Proof. Let ri, r,, r, be any three rows of A. Then * This research was supported in part by NSF Grant No. MCS 7821599. 319
320
M.Hall, Jr.
(r,, r , ) = r = (r,, r , ) = A (mod k ) , and (r, - r,, rr) = 0 (mod k ) .
Hence r, - r, E C'. It is particularly useful, for a symmetric design, if p divides r - A = k - A to exactly the first power. In this case we can determine the exact dimension of C and C'. Theorem 2.2. Let D be a symmetric ( u , k , A ) design and let C be its code over Fq where q = p' and the prime p divides k - A to exactly the first power. Then if p % k , p kA, d i m C ' Z 2u - 1 dim C =and C 3 C'. 2 ' u-1 dim C =2 '
u+l dim C' = 2
and
C C C'.
Proof. Note that u must be odd since if u is even k - A is a square. Let A be the incidence matrix of D so that from (2.1)
det A
= &k(k -
Add columns 1 , 2 , . . . , u - 1 to the last column of A and then subtract the last row from the others. This gives
with det B2= ? ( k -A)"')''. diagonal form. This gives
With elementary operations we can put B, into
32 1
Designs and coding theory
U, V, unimodular, b l b 2 . b,-, = ? ( k - A ) ( v - 1)/2. Thus at most ( u - 1)/2 of br . * b,-I are multiples of p . Hence over GF(p) rank A, = rank A 3 ( v - 1)/2. Hence dim C 3 ( u - 1)/2 and if p Y k,dim C 3 ( v + 1)/2. Thus in the case p ,'j k , k Y A we have dim C 3 ( u + 1)/2 and from Theorem 2.1 dim C' 3 (v - 1)/2, but as dim C + dim C' = v these inequalities are in fact equalities. If p k, p A, then C C C' and the result follows.
I
I
A knowledge of the dimension of C is of course particularly valuable for use of the MacWilliams Identity. MacWilliams Identity. Zf
is the weight enumerator of a code C over Fq and C is of dimension s, then 1 Wc1( x , y ) = 7w c ( x
4
+ (q - l ) y , x
- y ).
A collineation group G of a design D will in a natural way act on the code C of D. We assume here that I G 1, the order of G, is not zero in Fq, the field of the code. Thus if q = p', we assume that the prime p does not divide 1 G 1. Let C , ,. . . , c h be the orbits of G on the b coordinates of V. Let the orbit C, have length m,,and define a vector as that vector of V which has l/dG as its entry for every coordinate belonging to C, and zero for every other coordinate. As m, divides ( G1, m, is not zero in Fq and 1 / 6 exists either in Fq or in the extension Fq2. Then, since distinct orbits C,, C, are disjoint, we have
_ -
(C,,C,)=l,
_ _
(C,,c,)=O,
izj.
c, -
Hence . . . , c h are an orthonormal basis for a subspace W of V (possibly over Fq-.). We wish to consider the orbit space W and in some way C under the action of G as lying in W. For this we define an operator 8,
Here 8 is an idempotent operator and W = V8 is the subspace of V with the orthonormal basis . . , Ch.Also if xi y are vectors of V lying in W, then (x, y ) " = ( x , y ) W, the first being the ordinary inner product, - the second the inner . . , c h . The following is a product with respect to the orthonormal bases theorem of Hayden generalizing a result of Thompson [2].
c,.
c,.
M.Hall, Jr.
322
Theorem 2.3. Let V be an m-dimensional vector space over a field F. Assume that G is a permutation group on the coordinates of V and that 1 G 1 has an inverse in F. Suppose that C C V is a G module then with the operator 8 defined for v E V by
putting W = VO and H = C8 we have (C8)',= H L = (CL)8.
Thompson [ l ] proved the special case in which C = C'
3. Some applications of codes to designs
An important application of codes to designs was made in 1973 by MacWilliams, Sloane, and Thompson [5]. If it exists, a projective plane of order 10 is a symmetric (111, 11, 1) design. From Theorem 2.2, the code over F z , the binary code C,is of dimension 56 and C' consists of the words of even weight (in fact of weights multiples of 4) in C.From this and the MacWilliams Identity the entire weight distribution is determined once the number of words of weights 12, 15, and 16, viz., A , * ,A l s , and A16,are known. In this paper it was shown that Ais=O. If Wls is a word of weight 15, these being 15 points of the plane, every line of the plane intersects Wls in an odd number of points. With appropriate numbering of the points 1 to 15 we must have the following configuration of the 15 points on 21 lines, 1 2 1 6
2 6
3 7 4 8 5 9
3 7 10 10 11 12
5,
4 8 11 13
12, 14.
13
IS,
14
15.
9,
1 1 1 2 2 2
10 15, 11 14, 12 13, 7 15, 8 14, 9 13,
3 3 3 4
6
IS.
8
12. 11, 14, 12,
9 6 4 7 4 9
6 13, 5 7 If, 5 8 10.
5
(3.1)
10,
A computer search showed that this configuration cannot be completed to a full plane, so that the conclusion is that A I s= 0. It has been announced by M. El Ghabbach that he has shown that A l h= 0. A word of weight 12 would be an 'oval' of points such that every line intersects the oval in two points or more. The code C of the plane ii-of order 10 can be extended by adjoining a further coordinate x and taking it as a 'parity check', taking 1 in position x if the word in C has odd weight and 0 in position x if the word in C has even weight. The larger code C ,in V t l 2has dimension 56 and is self dual, i.e., C: = C,.
Designs and coding theory
323
If r has a collineation u of order 5, it is easily shown that the only case which requires consideration as that in which u fixes exactly one point P and exactly one line L. Numbering the points 1,. . . ,110,P we have u =(1-**5)(6.**10)-.*(101*** 105)(106*.* 11O)(P)(x),
and the fixed line L in the code CI has the form
(3.2)
\
L = (101,. . . ,105)(106,.. ., llO)(P)(x).
(3.3) Here Theorem 2.3 applies, since the orbit space W of CI is of dimension 24 consisting of 22 five cycles and fixed P and x. Since C: = C , from the theorem, it follows that with H = CIB,the words of C , fixed by u,then H ' = H in W. But Pless and Sloane [6]have determined all self dual binary codes in dimension 24. The Golay code G,,contains no words of weight 4 but in our code H the line L is a word of weight 4.From their listing H must contain at least 5 further words of weight 4 disjoint from L. Let one of these be (1* * *5)(6... 10)(11 *
* *
15)(16..-20).
This corresponds to a word of weight 20 in C fixed by u.One possibility involves the configuration of 20 lines I 2 3 4
2 3
4 5 5 1
6 7 8 9 10
I1
12,
3 5
12 13, 13 14,
4
1
14
2
15
15, 11,
8, 9, 10, 6, 7.
5
1 3 2 4
11. 12, 13, 14, 15,
13. 14, 15, 11, 12,
16 17, 17 18, 18 19, 19 20, 20 16,
8 Y 16 18, 9 10 17 19. 10 6 18 20, 6 7 19 16. 7 8 20 17.
(3.4)
There will also be 70 lines containing exactly two of the points and 21 lines containing no points. A computer search showed that (3.4)cannot be completed to a full plane. Thus is up to isomorphism one of 92 starts to be considered and computer search showed that none of these can be completed. The conclusion, published in [l], is that a plane of order 10 cannot have a collineation of order 5 . For a (41,16,6)symmetric design a similar assumption led to a very different conclusion. Assuming a collineation a of order 5 and fixing exactly one point and one block led to the construction of the design. Let the fixed point be x and the fixed block be Bo. Then a = (x)(l,.. . ,5)(6,.. . , 10)(1I,.. . ,15)(16,. . . ,20)(21,.. . ,25)
(26... . ,30)(31,.. . ,35)(36,.. . ,40)
(3.5)
on points and similarly on blocks. It is possible to find a table showing how many points of each cycle x, C , ,. . . ,C, as above lie on a representative block. One of 15 possibilities is
M. Hall, Jr.
324
The structure of this table suggests the possibility of a further collineation /3 of order 3 such that on point cycles
p
= (x)(C,,
C,)(C,, cs,C,)(C,)(CX).
c2,
In full on points
p
= (X)(1 6 11)(2 7 12)(3 8 13)(49 14)(5 10 15)
(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30) (31)(32) (33) (34) (35)(36)(37)(38)(39) (40).
(3.7a)
With a little trial this led to the full design D. Representative blocks are: B,,
B,
B, B,, B,, B2,
B,, B,, B,,
,y ,y ,y ,y 6
3 1 1 1
1 2 4 6 3 5 1 2 7 9 4 5 2 4 3 6 2 6
3 7 Y 8 13 11 8 8 7
4 13 11 10 I4 12 9
11 11
5 15 12 14 15 14 10 13 12
6 16 17 16 18
16 17 18 17
7 21 18 22 19 23 20 19 20
8 27 19 23 22
24 21 20 22
9
I0
28 20 24 25 27 28 23 25
29 21 25
26 30 29 24 27
11 30 26 26 31 31 31 25 30
I2 31 31 31 32 32 32 28 36
13 32 32 32 34 34 34 29 38
14 36 36 36 39 39 3Y 30 39
15 38 38 38 40
30 -10
(3.7b)
32
40
This construction is given in detail in [2]. For the projective plane of order 12, or symmetric (157, 13, 1) design, Theorem 2.2 applies for p = 3 but not for p = 2. Here for the code C of the plane over GF(3) we have dim C = 79, dim C' = 78 and C' consists of the words =0 whose weights are multiples of 3. All wieghts are = 0, 1 (mod 3) so that A3r+2 for all s. The MacWilliams Identity is applicable but its direct application involves heavy calculations with very large numbers.
Designs and coding rheory
325
By considerations of invariants of a finite group of order 2592 Mallows and Sloane [4] have found an expression for the weight enumerator W c ( x ,y ) initially in terms of 27 parameters. Let us write
44=
(X 3
+ gY
31,
(p4
= (x -
3),
713
=Xy
h(X
-
3,
(2x
+
'). (3.8)
Their formula is wc-(X,y
) = X~I($;,
(pi)
+ TnPz($;, (pi).
(3.9)
Here Pl(u, u ) is a homogeneous polynomial of degree 13, Pz(u,u ) of degree 12, thus giving initially 14 coefficients in PI and 13 in P 2 . Showing that A " = 1, A 1 3= 314, and A l = A, = A, = Ah= A 7 = A, = A l o =A12 = A I S= Alh=A19= A z l= Azz= 0 reduces the number of parameters from 27 to 12 which may be taken as A I s ,Az4,Azc, A,,, A28, A3",'A31, A33, ,434, A36, A37, A N . These calculations will be published in a joint paper by the author and Wilkinson [3]. A word of weight 18, Wls, will correspond to an interesting configuration. For any line L we will have (Wig, L ) = 0 (mod 3). Let there be Ci lines with exactly i points in common with Wlx.Note that CI = 0 since a line L with precisely one point in common would have ( W18.L ) = ? 1. Then Co+ Cz+ C,+ C,+ 2C2 f 3C3 + 4C4
+
C s + * * * + Ci,=157
SCs + . *
+ 13C13 = 234 = 13.18
(3.10)
The first of these counts the total number of lines, the second the incidences of the 18 points on lines, and the third the incidences of pairs of the 18 points. Subtracting the third from the second gives
C2 - 2C4 - 5C5 + *
* *
- 65C13 = 81.
(3.1 1)
It now follows that C23 81. But a line L with exactly 2 points of Wls must have P, Q in Wlx,with the P coordinate + 1 and the Q coordinate - 1. If of the 18 points there are m with coordinate + 1 and n with coordinate - 1 then Czs mn. But as m + n = 18 it follows that C 2 681. Thus C, = 81, rn = n = 9 and from (3.1 1) C, = C, = . . = C1,= 0. Now from (3.10) Co= 52, Cz= 81, C, = 24. A line L with three points of Wlxmust have all three + 1 or all three - 1 in Wlx. Thus the 9 points which are + 1 lie three at a time on lines to form a Steiner triple system and the same holds for the 9 points which are - 1 in W18. Mallows and Sloane also found a much more complicated formula for the complete weight enumerator of the plane of order 12 which initially has 196 parame ters.
326
M. Hall, Jr.
References [ I ] R.P. Anstee, M. Hall Jr. and J.G. Thompson, Planes of order 10 do not have a collineation of order 5, J. Comb. Theory, Ser. A 29 (1980) 39-53. [2] W.G. Bridges, M. Hall Jr. and J.L. Hayden, Codes and designs, to appear in J. Comb. Theory, Ser. A. [3] M. Hall and J. Wilkinson, The ternary code for the plane of order 12, to appear. [4] C.L. Mallows and N.J.A. Sloane, Weight enumerators of self-orthogonal codes over GF(3), to appear. [5] F.J. MacWilliams, N.J.A. Sloane and I.G. Thompson, On the existence of a projective plane of order 10, J. Comb. Theory, Ser. A 14 (1973) 6 6 7 8 161 V. Pless and M.J.A. Sloane, On the classification and enumeration of self dual codes, J . Comb. Theory, Ser. A 18 (1975) 313-335. 171 2. Janko, Personal communication.