On Tactical Configurations and Error-Correcting Codes* - CORE

JOURNAL OF COMBINATORIAL THEORY 2, 243-257 (1967)

On Tactical Configurations and Error-Correcting Codes*

E. F. ASSMUS, JR. "t AND H. F. MATTSON, JR.

Applied Research Laboratory, Sylvania Electronic Systems,

Waltham, Massachussetts

Communicated by Marshall Hall

1. INTRODUCTION

The study of error-correcting codes began in 1948 when Shannon's Fundamental Theorem pointed out their importance. In 1956 Paige [37] found a close connection between an error-correcting code and a classical combinatorial design. One purpose of this paper is to prove several results casting further light on the connection between error-correcting codes and combinatorial designs.

I t was Steiner [1] in 1852 who gave a succinct statement to the combinatorial problems with which we are concerned. These problems have never been fully solved, 1 and our other purpose is to point out previously unrecognized difficulties of the problems. Briefly, in stating the problem, Steiner drew out some questionable necessary conditions. The problem as he stated it seems to be much more difficult than appar- ently he supposed, as we explain in Parts 5 and 6.

Theorems 1 and 2, proved here, were stated without proof in [39], where the interested reader will find a brief history of these problems. Propositions 1 and 2 appeared without p roof in [38]; a full discussion of the Golay (11,6) code and its connection with the Mathieu groups MIX

and M12 can be found there. This code, together with the (23, 12) Golay

* The work reported herein was partially supported by Air Force Contract No. AF19(604)-8516.

Lehigh University, Bethlehem, Pennsylvania. t The solution in [9] is unjustifiably claimed to be complete.

243

244 ASSMUS AND MATTSON

code, is an example o f a so-called quadratic-residue code. Quadratic- residue codes fo rm a doubly infinite class of codes with interesting design properties; the authors will discuss these in a for thcoming paper.

Theorem 1 has several interesting corollaries which are included in Part 4 as remarks and examples.

The bibliography, though incomplete, does contain several not often cited earlier papers on combinator ia l problems and a large part o f the

literature on perfect codes. We have commented briefly on some of the entries and cited the review in Mathematical Reviews where known.

2. TACTICAL CONFIGURATIONS AND CODES

A tactical configuration on a set S o f cardinality n is a collection 2 o f subsets o f S such that each member o f ~ has cardinali ty d, and such

that every subset o f S o f cardinality t is contained in precisely ;t distinct members of ~ . The positive integers 2, t, d, and n are the parameters o f 2 , and to avoid trivial configurations we usually take 0 < t < d < n.

We shall speak of such a tactical configurat ion as a 2; t - - d - - n configuration or as a tactical configurat ion of type 2; t - - d - - n. This combinatorial not ion encompasses (a) finite affine planes (take 2 = 1, t = 2, d = m, n ~ m2), (b) finite projective planes (take 2 = 1, t = 2, d = m

+ 1, n = m 2 + m § 1), (c) balanced incomplete block designs (take t = 2), etc. For details we refer the reader to Ryser [8].

A Steiner system on S is a 1 ; t - - d - - n configurat ion; that is, every subset of t elements o f S is contained is precisely one of the chosen subsets in 2 (of d elements each).

A code is mos t generally defined as a pair (A, S) where A is a (non-

empty) set (to be thought o f as " w o r d s " ) and S is a (non-empty) finite set of functions defined on A with values in some "a lphabet" set F containing at least two elements. The functions in S are subject to the re-

striction that if a, b e A, then (a)f = (b)f for all f ~ S implies a = b. That is, the functions must distinguish the points o f A. More concretely,

one may order the functions o f S as f 0 , ...,fn-1 and then consider the concrete realization of (A, S), namely, the set of all ordered n-tuples over F of the form

((a)fo, (a)fa . . . . . (a)f,_a), a e A (1)

The f in S are the coordinate functions of the code.

ON TACTICAL CONFIGURATIONS AND ERROR-CORRECTING CODES 245

A most important special case arises when F is a field, A is a vector space of finite dimension k over F, and S is a set of linear functionals

on A. We then call (A, S) an (n, k) code over F. In the present paper we shall also find the non-linear case useful.

I f (A, S) is a code and a, b ~ A, the distance between a and b is de-

fined as the number of functions f in S such that (a)f~;~ (b)f . In the concrete realization, the distance between a and b is the number of coordinate-places where they differ. The weight of a code word is its distance from (0, 0 .... ,0) when F is a field; in other words, the weight

is then simply the number of non-0 coordinates. I f the minimum distance between any two distinct code words of A is d, then the code is capable of correcting any e or fewer errors, where e ---- [ (d - - 1)/2].

This means that, if during transmission of the symbols (a)fo, ..., (a)~_l , e or fewer of them are changed, then we can in principle correct them. The reason is that the spheres of radius e about the set of all code-points

((b)fo .. . . . (b)f~_l), b ~ A, are disjoint. Thus on changing e or fewer coordinates of the center point of such a sphere, we obtain a point still in that sphere.

A code is called perfect if this set of (disjoint) spheres of radius e about the code-points entirely exhausts the containing space F ~ ---- Fx . . . xF(n times). In this case d is necessarily odd.

3. RELATIONS BETWEEN CODES AND TACTICAL CONFIGURATIONS

Let (A, S) be a linear (n, k) code over the field F = GF(q). Let d be the minimum distance between code-vectors; in this linear case, d is

also the minimum non-0 weight in A because the distance function is invariant under translation. Let ~ be the collection of all d-sets 2 D c S

such that there is a minimum-weight vector of A with its d non-0 coordinates just those of D. Then we have:

THEOREM 1. The finear code (A, S) of minimum distance d ~- 2e -k 1 is perfect i f and only i f . ~ is a tactical configuration on S of type 2 = = ( q - - 1)'; (eq- 1 ) - - d - - n .

PROOF. Let the code be perfect. An (e q- 1)-set can be filled with non-0 coordinate-values in ( q - 1) e+l ways. Each such choice yields a vector

2 A j-set is simply a set of cardinality j. We shall use this terminology throughout.

246 ASSMUS A N D MATTSON

of weight e -+- 1 in F n, which must be at distance e or less from some code-vector (in the concrete realization). This code-vector must be of weight d, and it must agree with the original vector of weight e + 1 on all of the non-0 coordinates of the latter. Therefore the code-vectors arising in this way are all distinct; their sets D of non-0 coordinates, however, are (q - - 1)~ in number because scalar multiplication preserves these relationships. (Notice that we also use the linearity to prove that a d-set can "hold" at most one code-vector up to scalar multiplication.)

For the converse, we need only show that any point of F 'n is within distance e of some code-point. I f not, let x be a point of F n of smallest weight which is not within distance e of any code-vector. Then x has weight at least e + 1 ; let E be a set of e + 1 non-0 coordinates of x. By assumption E is contained in precisely (q - - 1) ~ sets D in 2 . On scalar multiplying we obtain, f rom each such D, q - - 1 code-vectors of weight d giving a total of (q - - 1)~+1 distinct code-vectors of weight d with the associated sets D containing E. These code-vectors are necessarily all different from each other on E; otherwise there would be a non-0 code- vector of weight less than d (by linearity). Therefore one of these code- vectors, say a, agrees precisely with x on E, so that on subtraction we reduce the weight of x by at least 1, contradicting our choice of x, since if x - - a is in the sphere of radius e about b (in A), then x is in the sphere of radius e about a + b. Q.E.D.

REMARK. When q = 2, then half of Theorem i holds even for non-linear codes (containing 0). That is, a per fec t code over GF(2) o f minimum dis-

tance d ~ 2e ~ 1 yields a Steiner system o f type (e + l ) - - d - - n in

the above way (provided (0, 0 . . . . , 0 ) is in the concrete realization). The proof is even simpler than before and will be omitted.

Notice that without linearity the converse cannot hold in general, because the code consisting of the weight-d vectors and 0 from a perfect linear code would still yield the proper configuration but of course would no longer be perfect in general.

W e now prove the following:

PROPOSITION 1. Let (A, S) be a not necessarily linear perfect code

over GF(2) containing 0. Set f ~ ---- ~y~s f and S~ = S u f~o. Let 2~o be the sets of non-zero coordinates of the minimum-weight vectors of (A, S~). T h e n 2 ~ is a Steiner system on S~ of type (e + 2) - - (d + 1) --(n + 1).


PROOF. The result is true when d = n, if one admits trivial configurations. Take d < n and observe that f ~ is not already in S, for, if it were, then the min imum weight would be even (since, by a count ing argument ,

for any f ~ S there is a minimum-weight code-point which is 0 at f ) . N o w an (e + 2)-set E f rom S~ containing f ~ is obviously contained in exactly one member o f~_~ ; if, on the other hand, E c S, and if E is not contained in any member o f 2 , then the vector in F n with l ' s at

E must lie in a sphere o f radius e about a code-point of weight d + 1 ---- 2e + 2 f rom (A, S). Q.E.D.

We also have

PROPOSITION 2. Let (A, S) be a code over GF(q), containing 0 but not necessarily linear. Suppose the min imum distance is d > 1 and that the s e t 2 o f non-0 coordinates o f weight-d code-points forms a 2;

t-d-n configuration on S. Set S ' = S - - { f } , w h e r e f is any coordinate function o f (A, S). Then (A, S ' ) is a code for which 2 ' (defined analo- gously) forms a 2; ( t - - 1 ) - - ( d - - 1 ) - - ( n - - 1) configuration on S'.

PROOF. To show that (A, S ' ) is a code we need only show that for any

two code words a and b o f A, the statement that af' = bf' holds for all f ' e S' implies that a = b. We know this is true if also af = bf, and if not then we would have the distance between a and b in (A, S) equal to 1, contrary to our assumpt ion that d > 1.

Obviously the new code has min imum distance d - - 1; the members

o f 2 ' are those of 2 which contain f , w i t h f t h e n removed, so to speak; the rest is self-evident. Q.E.D.

4. EXAMPLES AND REMARKS

1. The H-Golay Codes? Let F be GF(q) and let A be an m-dimensional

vector-space over F, m ~ 2. The set o f all non-0 F-linear functionals on A is par t ioned into subsets o f q - - 1 elements by the action o f scalar multiplication. Let S be the set o f n = ( q m _ 1 ) / ( q - 1) functionals

These codes are usually called Hamming codes, but Hamming [22] invented only the one for n = 7, q = 2. Golay [21] preceded this publication with his announcement that this (7, 4) binary code could be generalized to a perfect, single-error-correcting (n, n -- m) code as above for any prime q.


consis t ing of some one f rom each o f the above subsets. Then (A, S) is an

(n, m) code. We choose a concrete rea l iza t ion of (A, S) by order ing S,

and we define the subspace B of F ~ o r thogona l to the concrete real iza-

t ion o f (A, S) under the usual inner p roduc t as the (n, n - - m) H - G o l a y

code over GF(q). Tha t is, B is the space o f all l inear re la t ions on the

funct ionals in S. Thus the m i n i m u m dis tance d in B is at least 3. N o w we

can ei ther show easily tha t d = 3 and then count to show tha t B is perfect ,

or we can show direct ly tha t for every pa i r f , g me S there are exact ly

q - - 1 different h c S such tha t a f + f i g + y h = O for a, fl, y c F x.

In ei ther case we arr ive immedia te ly at the fo l lowing class of tact ical

conf igura t ions :

q - - 1 ; 2 _ . 3 _ q m - 1 q - - 1 q pr ime power ; m = 2, 3, 4 . . . .

( F o r q z 2, these are Steiner t r iple systems.) This conf igura t ion is of

course tha t ar is ing f rom the po in ts and lines o f project ive ( m - 1)-

d imens iona l space over F ; tha t every line consists o f q -+- 1 points is

equiva lent to the above result.

2. Vasil'ev Codes. Vasi l 'ev [31] has recent ly d iscovered a large class

o f non- l inear perfect codes over GF(2) with n = 2 m - 1 and d = 3.

Let B be any perfect code over GF(2) wi th d = 3 which conta ins 0;

then B has 2 m - - 1 coord ina te places and 2 k code-poin ts , where k = 2"

- - 1 - - m. Let z~ be any funct ion f rom B to GF(2) such tha t zl(0) ---- 0.

Let p on F n be pa r i ty (the sum of all coord ina tes ) . Then the new code C

is the set o f all words o f the fo rm (v; v q- a ; p(v) q- ~ (a ) ) of 2n -k 1

~- 2 m+l - - 1 coord ina tes , where v ~ F ~ and a ~ B. I t is easy to show tha t

tha t this code has m i n i m u m dis tance d at least 3 and tha t it consists of

2 n �9 2 n-m words ; it is therefore perfect and d = 3. C conta ins 0, bu t if

B is l inear and z~ is non- l inear , then C is non- l inear .

3. The Golay and Related Codes. Let z be a pr imi t ive n-th roo t of

un i ty over GF(q) -~ F (assume (n, q) : 1 of course) . Let K ~ F(z)

and let T be the t race f rom K to F. Define a set S o f l inear func t iona l

on A = F • K by S = {f0 . . . . . fn-1}, where

(Co, c)f~ : Co -k T(czi) , Co ~ F, c ~ K; i : 0, 1 . . . . . n - - 1.

Then (A, S) is a l inear (n, k) code, where k - - 1 is the degree of K over F.


There are two special cases of this construction which are known to yield perfect codes. They are

(a) q = 2 and n = 2 3 . Then k = 12 and d = 7 [25]. The count

21~. 0 + 2 3 + ( 2 2 3 ) + ( 2 3 3 ) ) = 2 ~

then shows that the code is perfect. This code yields a Steiner system of type 4-7-23 by Theorem 1, as Paige [37] observed, though he presented the code in quite a different form.

Applying Propositions 1 and 2 to this code we obtain Steiner systems of types 5-8-24 and 3-6-22. Applying Proposition 2 again one obtains a Steiner system of type 2-5-21, which is, of course, the projective plane of order 4. Presumably the fact that this plane has such unusual "exten- sions" explains its peculiarities. See [42].

(b) q = 3 a n d n = l l . T h e n k = 6 a n d d = 5 [38]. Since

3 6 . ( 1 + 2 . 1 1 + 4 . ( 1 2 ) ) = 3 1 1 ,

the code is perfect. It yields then a tactical configuration of type 4; 3-5-11.

The s e t ~ for this code also yields a Steiner system of type 4-5-11. For a given 4-set E, when assigned coordinates ~ 1, is a weight-4 vector which must lie in a sphere of radius 2 about a code-vector of weight 5 or 6. If all sixteen choices of ~ l 's yielded weight-6 vectors there would be two code-vectors of weight 6 with five non-0 coordinate places in com- mon; this could not happen because it would imply d < 5. In fact, every 4-set is contained in a unique member of 2 and in three distinct 6-sets arising from weight-6 code-vectors.

-- ~Zi=oJ~, then If to this (11,6) code we append a functional f~ = 10 the resulting code gives rise to a Steiner system of type 5-6-12. That the minimum weight in the new code is 6 and that a f ~ = 0 if a has weight 6 (a ~ A) require some proof (see [38]). Assuming these facts, it suffices to consider a 5-set E of S = f0 ..... f l0. We have just observed that any 4-subset of E is contained in 5- and 6-subsets of S belonging to code- vectors of the perfect code; the extra coordinates exhaust the seven remaining coordinates. Hence E is contained in exactly one of these, which either is or gives rise to, a weight-6 vector of the new code (cf. [38]).

4. There are no other perfect codes known. See [27, 28, 29, 30, 32, 33].


In particular there are no known perfect codes with d ~ 7. Correspond- ingly, there are no known tactical configurations with t ~ 6.

5. It is well known and easy to show that a necessary condition for the existence of a tactical configuration of type 2; t-d-n is that

should be an integer for h = 0, 1 . . . . , t - 1. Thus Theorem 1 yields necessary conditions for the existence of a perfect code over GF(q); these conditions for h = 0 and h = e in the case q = 2 have been found by Shapiro and Slotnick [28] and for all h by Lloyd [26].

6. For h = 0, the expression above is simply the number of d-sets in the configuration. Thus, the number of code-vectors of minimal weight in a linear perfect code is

n n

d = ( q - - 1 d i ]

7. Still another application of Theorem 1 allows us to show that the code-vectors of minimal weight in a linear perfect code span the code-space, for they span a subspace of the code-space with the same minimal weight vectors and hence yield the tactical configuration, whence a perfect code. But, for a perfect linear code, d and n determine the dimension k and hence they span the whole code.

5. CLOSED STEINER SYSTEMS

A problem going back to Steiner [1] is to find collections of subsets of a given set S of n elements with the following properties:

~ 3 is a collection of 3-sets of S forming a Steiner triple system on S (i.e., a 1; 2-3-n configuration);

3 4 is a collection of 4-sets of S such that every 3-set of S not a member of 2 3 is contained in exactly one member of ~ 4 , and no element of ~ 4 contains an element of 2 3 ;

ON T A C T I C A L CONFIGURATIONS AND ERROR-CORRECTING CODES 251

2 5 is a collection of 5-sets of S such that every 4-set of S not containing any member of ~ 3 u ~ 4 is contained in exactly one member of 2 5 ,

and no element of 2 5 contains an element of ~ 3 u 2 4 ; and so on, up

to ~ k -

Such a system is called closed if 2 k is non-empty but every k-set of S contains some member of 2 u ~u...u 2 3 ~ 4 k"

We may define the above k-th order S te iner sys t em more compactly

as follows: ~@i is a collection of non-empty/-sets of S, i ---- 3, 4 ..... k; a f r e e t-set of S is one not containing any member of ~ . . . u 2 t ; for t = 2, 3 ..... k - - 1 , every free t-set is contained in precisely one

member of ~t+~ and every proper subset of every element of ~t+~ is free. Then the system is closed if and only if there are no free k-sets.

We have followed the terminology of [9] except that there a free t-set

is one not containing any member of o@~ "''u ~ t - 1 ; they then speak of free t-sets not members of ~ t , which are our free t-sets.

We now present a theorem from [9]; the result and our proof were

discovered independently.

THEOREM 2. There ex is t s a closed k- th order S te iner s y s t em on n z 2 k a

- - 1 points , f o r k ~- 3, 4 . . . . .

PROOF. We realize the desired system by means of the H-Golay code B over GF(2) of n = 2 k-1 - - 1 coordinates. Recall that B is the linear space of all n-tuples over GF(2) which are relations on the n non-0 func-

tionals (in some fixed order) on (k - - 1)-dimensional space over GF(2). Let S be this set of functionals. We define 2 t as the collection of all

t-sets on S which are linearly dependent but which have all proper subsets linearly independent. This is the same as saying that 2 t is the collection of all sets M of non-0 positions of code-vectors of weight t such that no code-vector of smaller weight has its non-0 positions included

in M. Obviously now a free t-set is a linearly independent t-set. Now that every free t-set of S is contained in exactly one member of

=@t+l follows immediately from the definition of the code, for t ~ 2, 3, .... n - 1. (That is, a free t-set cannot "hold" a code-vector; thus the

sum of all those t functionals is a non-0 functional not one of the t we started with; these must now constitute a member of ~rt+a. )

o~k+ ~ is empty because every set of k functionals is linearly dependent;

thus there are no free k-sets. We now generalize Theorem 2, by finding the design properties of


the general H-Golay code B over GF(q) for any prime power q, where

n = ( q k - 1 1 ) / ( q - 1).

F rom Theorem 1 we know that the set 2 , of non-0 positions of weight-3 code-vectors yields a 2; 2-3-n configuration on the set S of the n functionals, where 2 = q - - 1. A free t-set is defined as any set of t functionals of S not holding a code-vector (means in this case: linearly independent!). Now, since the code is perfect with d = 3, every assignment of non-0 coordinates to the places of a free t-set E produces a code-vector at distance 1 from the resulting weight-t vector in F~(F = GF(q)); the code-vector can only be of weight t + 1. I t therefore agrees with the weight-t vector at each point of E; this means that a different (t § l)-st functional must arise f rom different assignments of coordinates to E (except for scalar multiplication throughout). Therefore every free t-set is contained in exactly ( q - 1 ) t - l = 2 t-1 different members of

2 t + l . This agrees with Theorem 1; and there are no free k-sets, as before.

We are thus led to define a k-th order tactical configuration as a collection of subsets . ~ , 2 4 , ..., 2 k of a set S of n points such that 2 i consists of / -se ts of S and, for t = 2, 3, ..., k = 1,

(i) every free t-set on S is contained in exactly 2 t-~ members of 2 t + ~ ; (ii) every proper subset of every member of 2 t+a is free. A free t-set

u ~ u . . . u is one not containing any member of ~ 3 ~ 3 2 t . Such a system is called closed if 2 k is non-empty and there are no free k-sets. We have proved

THEOREM 3. Let q be a prime power and set 2 = q - 1. I f

n = ( q k - X 1 ) / ( q - 1),

then there exists a closed k-th order tactical configuration on n points. We now count the number Nt of free t-sets connected with a k-th

order tactical configuration on a set S of n points. The "multiplicity of inclusion" is 2. Obviously we have

1

We attempt to find a design on the collection of all free sets, since every


subset of a free set is free. Thus, how many free 3-sets is a given pair contained in ? The pair is in exactly 2 different triples of 2 3 ; therefore it is in exactly n - - 2 - - 2 free 3-sets. Therefore

3N 3 ( n - 2 - 2)Nz,

N3 = l n ( n _ 1 ) ( n - - 2 - - 2 ) .

Now a given free 3-set is contained in exactly 22 different members of ~ 4 ; however, each of its 3 pairs is contained in 2 members of ~ 3 - I f all these resulting fourth points are different from each other (as happens in the H-Golay codes, where "free" reduces to "linearly independent"), then the free triple is contained in exactly n - - 22 - - 3 2 - 3 free 4-sets. But in general we can only conclude that this is a lower bound. Therefore

4N4 ~> ( n - 2 2 - 3 2 - 3)N3

with equality holding in the linear situation of Theorem 3. Note that equality holds here also when 2 = 1, because no two members of 2 ~ may intersect in two points. But even for 2 = 1 equality need not hold at t = 5 .

In general, with a free t-set we mus t exclude perhaps as many as

of the remaining n - - t points; that is,

(2 @ 1) t - ( t + 1) Nt+l>~ n - - 2 1 )Nt,

with equality in the linear case previously mentioned. Therefore

PROPOSITION 3. The number Nt of free t-sets associated with a k-th order tactical configuratiin satisfies

N t > ~ n - - - t = 4, k; (2)

and equality holds when the system is realized f rom a generalized


H-Golay code as in Theorem 3. In all cases we have equality in (2) for t = 2 and 3, and also for t = 4 w h e n 2 ~ 1. 4

REMARK. The free sets associated with the linear case are the linearly independent sets f rom projective (k - - 1)-space over GF(q), as in Theo-

rem 3. We have proved that the number of such t-sets is given by equality in (2). Furthermore, the free t-sets fo rm the following design: For 2 ----- q - - 1, every pair on S is contained in exactly n - - 2 - - 2 free 3-sets; every free 3-set is contained in exactly n - - 22 - - 32 - - 3 free 4 - s e t s ; . . . ; every free t-set is contained in exactly n - - ((2 § 1) t - - l ) /2 free (t + 1)-

sets, for t = 2 , 3 . . . . , k - - 1. Since n = ((2 q- 1) e - l - 1)/2, the process stops at t ---= k - 1.

6. A. COUNTEREXAMPLE

In posing the problem o f the existence o f k-th order Steiner systems, Steiner [1] gave what he claimed are necessary conditions. The conditions are correct for k = 3 (so-called Steiner triple systems) and Kirk- man [10] proved them sufficient. The condit ions are also correct for

k = 4, and Hanani [5] has proved them sufficient. These conditions, however, may very well be incorrect for k > 4. No t Steiner [1], Netto [3, pp. 202-204], or Hanani and Schonheim [9] offer any p r o o f that these

conditions, namely, that n is odd and that

1 t--2 - - II ( n - r 1 - - 2 i) is an integer for 3 < t < k, (3) t! i=o

are in fact necessary. Notice that equality in (2) for 2 = 1 would imply (3). Steiner may well have assumed this equality, and it is asserted in

[9, p. 140, (2)]. This equali ty is false in general (as the following counterexample shows), and for complete Steiner systems it is unproved.

Suppose B is any perfect code over GF(2) containing 0 with n = 15.

Consider ~ 3 , 9 4 , and 2 5 , the weight-3, weight-4, and weight-5 code-points. It is easy to see that they furnish a fifth-order Steiner system.

4 In the case )~ ~ 1, Steiner [1] seems to have asserted equality in (2) for all t. Netto [3] quoted Steiner's assertion verbatim, and in [9] equality is asserted. As our example will show, there exist fifth-order Steiner systems where the strict inequality holds in (2).


If B is linear it is in fact dosed since any five of the coordinate functionals of the orthogonal complement are linearly dependent. Now, Vasil'ev's

construction (see Part 4.2) furnishes many non-linear perfect codes with n = 15. We next show that for some of these there are free 5-sets.

Thus, the number of free 5-sets depends on how 2 4 and ~-~5 are intro-

duced. So, take the perfect code over GF(2) with n = 7 ; it consists of (0000000),

(1111111), (1101000) together with its cyclic shifts, and (0010111)

together with its cyclic shifts. Take er to be 0 except on one weight-4 vector, say (0010111). One checks easily that the following 5-sets are

free in the constructed code:

1000000; 0010111; 0

0100000; 0010111; 0

0001000; 0010111; 0

We have shown, therefore, that there are two fifth-order Steiner systems

on a set with 15 elements, one having free 5-sets, the other not.

REFERENCES

On Tactical Configurations

1. J. S~INER, Combinatorische Aufgabe, J. Reine Angew. Math. 45 (1853), 181-182.

2. E. H. MOORE, Tactical Memoranda, Amer. J, Math. 18 (1896), 264-303.

3. E. NETTO, Lehrbuch der Kombinatorik, 2nd ed., Teubner, Leipzig, 1927; reprinted by Chelsea, New York.

4. A. EMCH, Triple and Multiple Systems, Their Geometric Configurations and Groups, Trans. Amer. Math. Soc. 31 (1929), 25-42.

5. H. HANANI, On Quadruple Systems, Canad. J. Math. 12 (1960), 145-157.

6. H. HANA~, The Existence and Construction of Balanced Incomplete Block Designs, Ann. Math. Statist. 32 (1961), 361-368. (MR29 (1965), ~4161).

7. H. HANANI, On Some Tactical Configurations, Canad. J. Math. 15 (1963), 702-722 ( M R 2 8 (1964), $1136).

8. H. J. RYSER, Combinatorial Mathematics, Wiley, New York, 1963. A standard reference; contains a rather large bibliography.

9. H. HANANI AND J. SCHONHEIM, On Steiner Systems, Israel J. Math. 2 (1964), 139- 142 (MR31 (1966), g 73).


On Steiner Triple Systems

10. REV. TaOMAS KaRrd~AN, On a Problem in Combinations, Cambridge and Dublin Math. J. 2 (1847), 191-204.

11. A. CAYLEY, On the Triadic Arrangements of Seven and Fifteen Things, Philos. Mag. 37 (1850), 50-53 (Collected Mathematical Papers, I, 481-484).

12. A. CAYLEV, On a Tactical Theorem Relating to the Triads of Fifteen Things, London, Edinburgh, and Dublin Philos. Mag. and J. 25 (1863), Set. 4, 59-61 (Col- lected Mathematical Papers, V. 95-97).

13. J. J. SVLVESTER, Elementary Researches in the Analysis of Combinatorial Aggre- gation, Philos. Mag. 24 (1844), 285-296 (Collected Mathematical Papers, I, 91-102).

14. J. J. SYLVESTER, Note on the Historical Origin of the Unsymmetrical Six-Valued Function of Six Letters, Philos. Mag. 21 (1861), 369-377 (Collected Mathematical Papers, II, 264-271).

15. J. J. SYLV~SW~R, Remark on the Tactic of Nine Elements, Philos. Mag. 22 (1861), 144-147 (Collected Mathematical Papers, II, 286-289).

16. J. J. SYLV~7"ER, Note on a Nine Schoolgirls Problem, Messenger of Math. 22 (1892-1893), 159-160, and (Correction) 192 (Collected Mathematical Papers, IV, 732-733).

17. M. R~Iss, Ober eine Steinersche Combinatorische Aufgabe Welche im 45sten Bande Dieses Journals, Seite 181, Gestellt Warden ist, Crelle's J. 56 (1859), 326-344.

18. W. BtrRNSrO~, On an Application of the Theory of Groups to Kirkman's Problem, Messenger Math. 23 (1893-1894), 137-143.

19. E. H. MOORE, Concerning Triple Systems, Math. Ann. 43 (1893), 271-285.

20. M. HALL, JR., AND J. D. SWIFr, Determination of Steiner Triple Systems of Order 15, Math. Tables Aids Comput. 9 (1955), 146-152 (MR 18 (1957), p. 192).

On Error-Correcting Codes

21. M. J. E. GOLAV, Notes on Digital Coding, Proc. lnst. Radio Engrs. 37 (1949), Correspondence, 657.

22. R. W. HAMMING, Error D6tecting and Error Correcting Codes, Bell System Tech. or. 29 (1950), 147-160 (MR 12 (1951), p. 35).

23. M. J. GOLAY, Binary Coding, IRE Trans. PGIT-4 (1954), 23-28.

24. W. W. P~TERSON, Error-Correcting Codes, The M.I.T. Press Cambridge, Mass.D 1961. (MR 22 (1961), f~ 12003). The standard reference; contains a large bibliography.

25. H. F. MATTSON AND G. SOLOMON, A New Treatment of Bose-Chaudhuri Codes, J. Sac. Indust. AppL Math. 9 (1961), 654-669 (MR 24 (1962), ~ B1705). Contains the first non-exhaustive proof that the (23, 12) code of Golay is perfect.


On Perfect Codes

26. S. P. LLOYD, Binary Block Coding, Bell System Tech. J. 36 (1957), 517-535 (MR 19 (1958), p. 465).

27. M. J. E. GoLAY, Notes on the Penny-Weighing Problem, Lossless Symbol Coding with Nonprimes, etc., IRE Trans. Inform. Theory 4 (1958), 103-109 (MR 22 (1961),

13354).

28. H. S. SI~AemO AND D. L. SLOTN~CK, On the Mathematical Theory of Error- Correcting Codes, IBMJ. Res. Develop. 3 (1959), 25-34 (MR 20 (1959), :~ 5092).

29. C. ENGLEMAN, On Close-Packed Double Error-Correcting Codes on P Symbols, 1RE Trans. Inform. Theory 7 (1961), 51-52 (MR 23 (1962), ~: B3075).

30. S. JOHNSON, On Perfect Error-Correcting Codes, Memorandum RM-3403-PR The RAND Corporation, Santa Monica, Calif. (1962).

31. Ju. L. VASIL'EV, On Nongroup Close-Packed Codes, Probl. Cybernetics 8 (1962), 337-339 (MR 29 (1965), g 5661).

32. E. L. COrtEN, A Note on Perfect Double Error-Correcting Codes on q Symbols, Information and Control 7 (1964), 381-384 (MR 29 (1965), ~: 5656). Contains a bibliography on the diophantine equations arising out of this problem. See also Amer. Math. Soc. Notices, 13 (1966), 245-246.

33. M. H. McA~DREW, An Algorithm for Solving a Polynomic Congruence, and Its Application to Error-Correcting Codes, Math. Comp. 19 (1965), 68-72 (MR 30 (1965), ~ 4612).

34. V. K. LEONT'EV, On a Problem of Close-Packed Codes, Diskret. Analiz. (Russian), No. 2 (1964), 56-58 (MR 29 (1965)~ 4621).

Related Works

35. E. W~Tr, Die 5-fach Transitiven Gruppen yon Mathieu, Abh. Math. Sem. Hansi- sehen Univ. 12 (1936), 256-264.

36. E. WITT, Ober Steinersche Systeme, Abh. Math. Sem. Hansischen Univ. 12 (1936), 265-275.

37. L. J. PAIGE, A Note on the Mathieu Groups, Canad. Y. Math. 9 (1956), 15-18 (MR 18 (1957), p. 871).

38. E. F. AssuMs, JR., AND H. F. MATTSON, Perfect Codes and the Mathieu Groups, Arch. Math. 17 (1966), 121-135.

39. E. F. AssMus, JR., AND H. F. MATTSON, JR., Steiner Systems and Perfect Codes, Univ. N. Carolina, Inst. Statistics Mimeo Ser.

40. R. O. CARMICI-IAEL, Introduction to the Theory of Groups of Finite Order, Ginn. Boston, 1937; reprinted by Dover, New York, 1956.

41. R. H, BRUCK, What is a Loop?, in Studies in Modern Algebra, Vol. 2 of M.A.A. Studies in Mathematics, Prentice-Hall, Englewood Cliffs, N.J., 1963, pp. 59-99.

42. W. L. EDGE, Some Implications of the Geometry of the 21-Point Plane, Math. 2". 87 (1965), 348-362 (MR a0 (1965), $ 5209).

On Tactical Configurations and Error-Correcting Codes* - CORE

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