Uniformly packed codes

Uniformly packed codes

Citation for published version (APA):

Tilborg, van, H. C. A. (1976). Uniformly packed codes. Technische Hogeschool Eindhoven.

https://doi.org/10.6100/IR162111

DOI:

10.6100/IR162111

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Published: 01/01/1976

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UNIFORMLY PACKED CODES

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. IR. G. VOSSERS, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN, IN HET OPENBAAR TE VERDEDIGEN OP

DINSDAG 1 JUNI 1976 TE 16.00 UUR

DOOR

HENRICUS CAROLUS ADRIANUS VAN TILBORG

GEBOREN TE TILBURG

0

VIT PROEFSCHRIFT IS GOEVGEKEURV

VOOR VE PROMOTOREN

PROF.VR. J.H. VAN LINT

a.an

M<Vrij

CONTENTS

NOTATIONS

CHAPTER 1. INTROVUCTION

CHAPTER 2. THE GROUP ALGEBRA APPROACH TO COVES

6

2. 1. GMup

aigebtr.a.

and the

c.haJLactewuc.

poR.ynomi..a£.

0

n

a c.ode

6

2. 2.

KMWtc.houk. poR.yno miA.1..6

12

2. 3.

The wught and

c:U.6to.nc.e

enumetr.a.toJt on a c.ode

18

2.4.

Regula.Jrdy in c.odu; duigM

22

CHAPTER

3.

PERFECT ANV (STRONGLY) UNIFORMLY PACKEV COVES

26

CHAPTER 4. CONSTRUCTIONS OF UNIFORMLY PACKEV COVES

34

4.1.

Singte

eJtMJt

c.oJtJtectittg,

uni!Jo~R.y

packed c.ode6

34

4. 2.

Mui.Upte

eJtMJt

c.oJtJtec.Ung. uni!Joll.lnty packed c.ode6

44

CHAPTER

5.

CLASSIFICATION ANV NON-EXISTENCE THEOREMS

48

TABLE 4.

1.

1.

6 7

TABLE 4.2.1.

69

INVEX

70

REFERENCES

71

ACKNOWLEVGEMENTS

74

SAMENVATTING

75

CURRICULUM VITAE

76

~

A(z) Bk B(~1k) IC

a: (V)

c

cext ccompl c.L ck ck (p(x)) Cm(q)

c

(e) q d d(~,l_) e ex(k) Fc(x) f(n,q,e) g(n,q,e) Hm(q) m(~) m n N(C) p Pk (n,x) PG(m,q) p2 (n) p3(n) q Q(x) r (2. 3.2) distance enumerator (2 .1.10) ( 1. 1) complex numbers group algebra code extended code complementary code dual code (2. 3.6)

NOTATIONS

k coefficient of x in p(x) Hamming code pag. 53 minimum distance

distance of vectors ~ and l.

error correcting capability (5. 1) characteristic polynomial (5.8) pag. 53 Hamming matrix (1. 7) average of m (~) length of code (2.1.11) prime number Krawtchouk polynomial projective geometry (3. 13) (3. 14) prime power (3. 5) external distance all word of weight s k

Tr T(k) V(n,q) w(~) wk W(z) yk trace (5.1)

n-dimensional vectorspace over GF(q) weight of vector x

(2. 3 .1)

weight enumerator all words of weight k

coefficients of Krawtchouk expansion group of monomial transformations page. 36

{j. (~) (ai - aj

I

1 5 i < j s n)

\ theorem 1.3

~ theorem 1.3

p true external distance

p(~) distance of x to the code Pi eigenvalues of a graph

t

summation in the group algebra

X character

xu

(2.1. 5)

lxJ

r

x

1

max{k € r; min{k € Z k 5 x} k <: x}

CHAPTER 1

INTROVUCTION

Let V(n,q) bean-dimensional vector space over GF(q). The

weight

w(~) of a vector x E V(n,q) is the number of its nonzero coordinates, the

(Hamming)

distance

d(~,~) of two vectors ~ and~ in V(n,q) is the weight of their difference, i.e. i t is the number of coordinates, where they differ.

A q-ary

code

C of length n is a subset of V(n,q), with

lei

~ 2. The elements of C are called

code wordS.

The following parameters will play an important role.

The

minimum distanae

d of a code Cis the minimum value of d(~

₁

,~

₂

>,

~1'~2 E

e,

~1 ~ ~2'

t . ab ' 1 't [d-21].

The

error correc

~ng

cap

~~~ y e of a code

e

is In the follow-ing we shall often say that C is an e-error-correctfollow-ing code. We define

( 1.1) d(~,~) = k}i, ~ E V(n,q), 0 ~ k ~ n , i.e. B(~,k) is the number of code words at distance k from x. Let

p(~) := min{O ~ k ~ n

I

B(~,k) ~ O} , i.e. the distance of x to the code C.

The

true external distance

p of a code is the maximum value of p(~), X E V(n,q),

It follows from the definitions of e and p, that the spheres with dius e around the code words are disjoint, while the spheres with ra-dius p around the code words cover the vector space. Hence

(1. 2) e

lei L

t><q-1)i ~ q n i=O ~ p (~) (q- l)i

lei L

~ q n i=O ~ (1. 3)

Inequality (1.3) implies that codes with a true external distance P close to e. must have many code words. This property makes these codes an interesting object to study from a coding point Of view. Later on we shall show that also from a mathematical point of view a number of these codes are worth to be studied.

The extremal case is of course e

=

p. In this case the spheres of ra-dius e around the code words form a partitioning of V(n,q). These codes are called perfect. If

e

is a perfect code, then

e

( 1. 4)

I

i=O

n q

Perfect codes have been studied extensively. It was shown by van Lint (1971a) and Tietavainen (1973) that there are no perfect codes, except for

i) the repetition code, n

=

2e + 1, q

=

2,

lei

=

2;

Hamni

n

g

code, (qm -1)/(q -1), 1,

lei

n-m

ii) the n = e = q

iii) the two Go~ay codes, n = 11, q = 3, e 2,

le

i

= 36

,

andn q = 2, e = 3,

le

i

=

212.

For a description of these codes we refer to van Lint (1971b) and chapter IV.

23,

Around the time that these results on perfect codes were established, various people started to study the case, where p

=

e +1. Before we go into details, we prove a lemma.

LEMMA 1.1. Let e be an e-error-correcting code in V(n,q). Then for any ~ E V(n,q) ( 1. 5) p (~) = e ( 1. 6) p (~) e + 1 ~ (n-e)(q-1) B (~,e + 1) e + ₁ B(_x,e + 1) ~ n(q - 1) e + 1

PROOF. Let p(~) =e. w.l.o.g. ~=£(otherwise translate the code over -~).Apparently there is a code word~ of weight e. Let the support vector of a vector ~ be defined as the vector with ones at the nonzero coordinates of ~ and zeros elsewhere. Since the minimum distance of

e

is at least 2e + 1, i t follows that code words of weight e + 1 have a support vector disjoint from the support vector of ~· For the same reason, the support vectors of the code words of weight e + 1, have in-ner product at most one, in which case the corresponding entry of the code words must be different. This proves (1.5). The proof of (1.6)

Let us restrict ourselves to the binary case q with p(~ ~ e, we define

2. For any x E V(n,2)

( 1. 7)

Moreover we define mas the average value of m(~). By lemma 1.1 ( 1. 8) Since

L

n +

s

e + 1 ~EV(n,2) ,p (~)~e we find that ( 1.9) m e-1 2n -

lei

L

i=O

L

I{~ E V(n,2)

I

e~d(~·E.l Se+ 1}1 cEe

The inequality that can be obtained from (1.8) and (1.9) is the spe-cialized Johnson bound (see Johnson (1962)).

One can easily rewrite (1.9) as e-1 +

.!.(

(nl

lei { L

(~) n 2n ( 1. 10) _{+ (e+1 l l}} i=O ~ m e or e +

.!. (

n ) (

E....:!:1.

(1.11)

I el { L

(~) - m)} 2n • i=O ~ m e e + 1 F (1 11) 0 1 h d h h 0 f · n+ 1 f t

rom • it ~s c ear t a t co es w ic sat~s y m

=

e+

1 are per ec. Inequality (1.8) has led to an explosion of definitions.

Goethals and Snover (1972) define nearly perfect codes as codes which satisfy

for all~ with p(~) ~ e. They give a number of examples and prove that for e s 4 there are no nearly perfe~t codes with other parameters. Recently van Lint (1974) proved that there are no nearly perfect codes for e

=

5 and Lindstrom (1975) showed the same for all e ~ 6.

semakov e.a. (1971) define as uniformly packed codes, those codes which satisfy m(~) = m for all~ with p(~) ~e. we shall call these codes

strongly un

i

formly packed.

The author (1975) has shown that the parameters of all binary strongly uniformly packed codes are already known.

Goethals and van Tilborg (1975) define a set of codes, containing the two previous cases and the q-ary case.

DEFINITION 1.2. An e-error-correcting code e in V(n,q} is called

un

i

-formly pa

cked

with parameters A and ~. iff for all x E V(n,q),

P (~)

₌

e p (~) ~ e + 1 where A < (n- e) (q- 1) e + 1 ,. B(~,e + 1) ~

REHARK. From this definition it is clear that d 2e + 2 iff A 0.

THEOREM 1.3. Let e be an e-error-correcting, uniformly packed code in V(n,q) with parameters A and~. Then

( 1. 12) !elf e-

L

l <;><q-1) n i + (1-->< J(q-1) +-< A n e 1 n J(q-ll e+l }=q. n

i=O • ~ e ~ e+l

PROOF. We compute

L

(B(~,e) + B(~,e + 1))

~,p (~)~e

in two different ways. On one hand it equals by the definition of uni-formly packed codes

e

I

i=O n i (.)(q-1) }~ ~

On the other hand it equals

L

1-{~ E V(n,q)

I

eSd(~,~) Se+l}l CEC

=lei{<:> (q- l)e +

(e~l)

(q-1)e+1}

Equation (1.12) can be rewritten as ( 1.13) e

I cl {

I

i=O n i 1n en-e (. ) (q - 1) + -( ) (q - 1) ( - - ( q - 1) - A) } ~ \.1 e e+1 n q n-e

From this expression it is clear that A = e +

1 (q- 1) is equivalent to saying that the code is perfect. For this reason definition 1.2 ex-cludes this possibility. We remark that by the definition 1.2 and the lemma 1.1 the parameters A and \.1 of a uniformly packed code satisfy

( 1. 14) ( 1. 15) O~A< (n-e)(q-1) e + 1 1 ::; \.1 ::; n (q-1) e + 1

By taking q = 2 and \.1 = A + 1 = m equation (1.12) reduces to (1.10) and the definitions imply that a code with these parameters is strong-ly uniformstrong-ly packed. By taking

A=[(n-e)(q-1)]f (n-e)(q-1) and \.1

e+1 e+1

[n (q-1)]

e + 1 '

one finds a natural generalisation of the concept of nearly perfect

codes to the q-ary case. A number of nice properties of uniformly packed codes can be derived from the combinatorial definition 1. 2. However, to obtain strong results on this subject, it turns out that a different approach is necessary.

CHAPTER 2

THE GROUP ALGEBRA APPROACH TO COVES

The majority of the results in the paragraphs 2.1 and 2.4 were obtain-ed by Delsarte (1973a) and (1973b). These results appearobtain-ed more expli-citly in a paper by Goethals and van Tilborg (1975). The theorems in paragraph 2.3 were taken from an article by MacWilliams, Sloane and Goethals (1972).

2. 1.

G.!toup ai.ge.bJta. and :the. c.haJr.a.&e.J!,{,6tic. po!ynomi..ai. o6 a c.ode.

Let (~(V),~,*) be the group aLgebra of V(n,q) over~, i.e. the vector space over ~, the complex numbers, with the vectors ~ E V(n,q) as base vectors and a multiplication * defined by

(2. 1. 1)

e

(l (~) ~

*

e

s

(l,) l. = =

e

L

(l (~ 13t:t.> > ~

~EV(n,q) :i,EV(n,q) ~EV(n,q) ~-ty=~

With each set X c V(n,q) we associate the element

l

~in a:(V). ~le

X€X

shall also denote this element in a:(V) by X. In particular we shall frequently use

(2. 1. 2) Yk := {~ E V(n,q) k}

(2 .1. 3) _sk := {~ E V(n,q) W (~) !> k} 1

being respectively the set of words of weight k and the set of words of weight at most k.

According to the multiplication rule {2.1.1) and the definition (1.1) one has

(2.1. 4)

l

B(~1 k)~

~EV(n,q)

By using the group algebra one can describe properties of subsets of V(n,q) in terms of properties of elements of a:(V). For instance

(2.1.13) and theorem {3.5) will be examples of this. However, to turn this alternative description into a powerful tool, we shall need some more definitions.

Let x be any

nonprincipal aharaater

of the additive group of GF(q), i.e. xis a mapping from GF(q) into 4:\{0}, satisfying x<a+b) =x(a)X(b), X not identical to 1.

For any~ € V(n,q) we define xu: V(n,q) ~ C by

(2.1. 5) v € V(n,q) ,

n

where (u, v) is the inner product

L

ui vi in GF (q) • We now extend xu to i=l

a linear functional from C(V) to

c

by

(2.1.6) _Xu(

l

a(~)~

₌

L

a(~)Xu(~) _• - ~€V(n,q) . ~€V(n,q)

To get more familiar with characters, we shall prove some properties.

LEMMA 2 • 1. 1.

v

v

Cx

(A

*

B) ~€V(n,q) A,B€C(V) u X (A) X (B)] • u u PROOF. Let A

=

l

a(~)~ and B

t

~€V(n,q) l_€V(n,q) Then by (2.1.1) and (2.1.6) Xu(A *B)

x

<

e

~ ~€V(n,q) ~~=~

L

a (~) B (l_) ) ~)

I

I

a

<~>

8 <l.> xu

<~

+ l.> ~€V(n,q) ~+l.=~

=

I

I

a (x) 8 (v) X (x) X (y) =X (A) X (B). - .... u u u u LEMMA 2 • 1. 2 •

I

~€V(n,q) ~€V(n,q) l.€V(n,q)

_-

_-

_-

_-

₀

X (v) u -n 00 q _,u u € V(n,q) •

PROOF. For ~ = Q_, xu(~) = 1 and the assertion is trivial. For ~ f. Q_,

LEMMA 2.1.3. The linear functionals Xu'~ E V(n,q) span the vector space of all linear functionals of C(V).

PROOF. Since there are qn linear functionals X , it is sufficient to

u

show that they are linearly independent. Suppo;e

L

identically zero. Then by lemma 2.1.2 E_EV(n,q)

0

I

~EV(n,q)

I

~EV(n,q)

I

~EV(n,q) <

I

a(ulx (wllx (w) _{- u - v} -E_EV(n,q) a(~)

I

~EV(n,q) a-(~)

L

~EV(n,q)

-

-X (wlx (w) u - v -X (w) uv

-for all v E V(n,q). Hence for all~ E V(n,q) 1 a(~) 0.

a(ulx is - u

0

LEMMA 2 .1. 4. S is the only element in C(V) satisfying X (S )

=

6 qn,

n u n O,u for all ~ E V(n,q). PROOF. By lemma 2.1.2 X (S ) u n

L

X u -(v)

=

60 ,u qn ~EV(n,q) n

Assume that also X (S) = 6

0 q . "Then Xu(S- S) = 0 for all ~EV(n,q).

u ,u n

Now lemma 2. 1. 3 implies that-any linear-functional maps S - S into 0,

n

i.e.

s

=

s

_n

.

Before we continue with this subject, we need a definition.

DEFINITION 2.1.5. For a fixed value of q, the KrCIJJ)tchcuk polynomial Pk (n,-x) is defined by

k

(2 .1. 7)

_I

i=O

Clearly Pk(n,x) can be interpreted as the coefficient of uk in ( 1 - u) x (1 + (q- 1) u) n-x, if x is an integer between 0 and n.

LEMMA 2.1.6. An equivalent expression for Pk(n,x) is

(2 .1. 8)

PROOF. Let x be an integer, 0 ~ x ~ n

n

I

k=O n

I

i=O n

I

i=O k u i x i n-i (-q) (.)u {1 + (q-1)u} ~ n

{1 + (q -1)u}n-x

L

(,)(-qu) {1+(q-1)u} x i x-i ~

i=O

{1 + (q- 1)u}n-x{-qu + 1 + (q- 1)u}x ={1 + (q- 1)u}n-x(l- u)x

Comparing the coefficient of uk on both sides we find the required result for 0 ~ x ~ n, x integer. However if two polynomials of degree

at most n are equal on n +1 points, then they must be,identical.

0

By lemma 2.1.6 Pk(n,x) evidently has degree k. Therefore the

Krawt-chouk polynomials Pk(n,x), 0 ~ k ~ n, form a base in the vector space of polynomials of degree at most n. Because of the importance of Krawt-chouk polynomials we shall devote the entire next paragraph to them.

The next lemma will establish their. relation to characters.

LEMMA 2.1.7. For any~ e V(n,q) and 0 ~ k ~ n

PROOF.

a) From X (0) and

I

x

(a)

llEGF(q)

0, it follows by induction that

b)

I

x(a1+a2+ ••• +ai> a 1,a2, ••• ,aiEGF(q) a 1•a2• ••• •aifO

I

X(

'~·~.ll

~EV(n,q) ,w(~)=k (-1) i

The assertion now follows from a) and formula (2.1.7).

0

The next part of this paragraph will clarify the importance of Krawt-chouk polynomials to coding theory.

For a code

c

in V(n,q) we define the characteristic numbers Bk, 0 ::; k

s

n,

by

(2.1.10) I X (C) 12

u

In paragraph 2. 3 \'le shall treat these numbers more extensively. Let (2.1.11) N(C) := {1 S k S n

I

Bk i' O) .

The characteristic poZynomiaZ Fe of a code

c

is defined by

n

(2.1.12) :=

k

n

(1-~l • ICI kEN(C)

It follows from lemma 2.1.6 that the Krawtchouk polynomials form a base in the ring of polynomials of degree at most n. Hence each poly-nomial F(x) of degree at most n can be expanded in a unique·~ay with respect to ·this base,

n

F(x)

I

akPk(n,x) •

k=O

THEOREM 2.1.8. Let a₀,a₁, ••• ,an be the coefficients of the Krawtchouk expansion of the characteristic ~lynomial Fe of a code

c.

Then

c

sa-tisfies in It (V)

n

(2.1.13)

c

*

~ akYk k==O

s

_n

PROOF. Let u € V(n,q), w(u)

X _u (C

*

n

w. By lemma 2.1.1 and lemma 2.1.7

Xu(C)

f

akPk(n,w) = Xu(C)FC(w)

- k=O

If~

f

0 then Xu(C)

f

0 implies Bw

f

.

O,

i.e. FC(w) the right hand side equals

0. If u ==

£

1 then

The theorem now follows from lemma 2.1.4.

0

The external distance r of a code c is defined by

(2.1.14) r :==degree of FC(x) = IN(C) I

This definition enables us to formulate a corollary of theorem (2.1.8).

COROLLARY 2.1.9. Let a₀,a₁, ••• ,ar be the coefficients the Krawtchouk

expansion of the characteristic polynomial Fe of a code c. Then for all ~ € V(n,q)

r

(2.1.15)

L

akB<.~,k) 1 . k==O

PROOF. This is a direct consequence of theorem 2.1.8 and formula

(2.1. 4).

It follows from this corollary that the true external distance P of code satisfies p s r.

2. 2. K~UWJtchouk po.tynomi..a.l-6

Krawtchouk polynomials are known to form a family of orthogonal polyno-mials. One of the classical books on orthogonal polynomials is that of Szego (1959). In chapter 3 of this book one can find various recurren-ce relations, orthogonality relations and theorems on the distribution of the zeros of these polynomials. The first half of this paragraph will be on the derivation of some classical results. In the second half we shall obtain more specific information on the zeros of Krawt-chouk polynomials. This information will prove to be important, as might be guessed from theorem 2.1.8 and formula (2.1.12).

LEMMA 2.2.1.

(2. 2. 1)

PROOF. By the definition of Krawtchouk polynomials one has

n n i i n-i i n-i

L

(i)(q-1) (1-x) (l+(q-1)x) (1-yl (l+(q-1)y)

i=O { (q - 1) ( 1 - x) ( 1 -

y)

+ ( 1 + (q- 1) x) (1 + (q - 1) y) } n n ~ n k k k q L (k) (q-1) X y ~ k=O k R.

The result is immediate after comparing the coefficients of x y on.

both sides. .

0

LEMMA 2.2.2.

k R. PROOF. Compare the coefficients of x y in

( ( q - 1 ) y (1 - x) + (1 + ( q - 1 ) x) ) n = ( ( q - 1) x ( 1 - y) + (1 + ( q - 1 ) y) ) n =

\ n k k k n-k ,· \ n k k R.

L (k)(q-1) X (1-y) (1+(q-1)y) =t.. L (k)(q-1) PR.(n,k)xy •

k k R. LEMMA 2, 2, 3. n (2.2. 3)

L

Pk(n,i)P. (n,R.)

=

6 qn i=O L k,R. PROOF. Substitute (2.2.2) in (2,2,1). THEOREM 2, 2, 4, (2.2.4) PROOF. k

L

PR.(n,x) Pk(n-1,x-1). R.=O k

L

PR. (n,x) R.=O k

I

i=O k

I

i=O

0

0

0

LEMMA 2.2.5. Krawtchouk polynomials satisfy the recurrence relation

(2.2.5) (k + 1)Pk+

1 (n,x) = {k + (q- 1) (n- k) - qx}Pk (n,x)

- (q -1) (n -k + l)Pk_

1 (n,x) •

PROOF. Differentiate the equality

\ k x n-x

L. Pk(n,x)u = (1-u) (1 + (q-1)u)

k

to u and multiply the result by (1- u) (1 + (q- l)u). One obtaines

\ k-1 x n-x

(1-u)(1+(q-l)u)L. kPk(n,x)u =(1-u) (l+(q-l)u) {-x(l+(q-1)u)+

k

(n - x) (q- 1) ( 1 - u) } {n(q-1) -xq-n(q-l)u}L Pk(n,x)uk

k

Formule (2.2.5) follows readily after comparing the coefficients of uk

on both sides.

0

Lemma 2.2.5 will JJe ~ufficient to prove a strong theorem on the zeros of Krawtchouk polynomials.

LEMMA 2.2.6.

(2. 2. 6)

(2. 2. 7) Pk(n,n)

(2.2.8) (q- 1) k-1 {(q- 1)(k)-q(k-1)} n n-1

PROOF. Substitute x 0, n resp. lin (2.1.7)-. resp. (2.1.8).

THEOREM 2.2.7. Pk(n,x) has k distinct real zeros in the interior of [O,n]. Moreover, denoting the zeros of Pk_

1(n,x) by u1,u2, ••• ,~_

1

, (ui < ui+l) 1 and the zeros of Pk(n,x) by v 1,v2 , ••• ,vk' (vi< vi+l),

we have

(2 .2. 9) and

(2.2.10) sign Pk_₁ (n,vt) (-1) R-+1 •

PROOF. The proof is by induction on k.

Since P ₀ (n, x)

=

1 and P ₁ (n, x)

=

(q-1) n - qx, the assertion is true for k

=

1. Assume the assertion is true fork. Substitution of vt in

(2.2.5) yields

- (q-1) (n- k + 1)Pk_

1 (n,vt) • Hence

Since by (2.2,6) and (2.2.7) sign Pk+

1 (n,O)

=

1 and sign Pk+1 (n,n)

=

(-1)k+l, it follows immediately that Pk+

1 (n,x) has zeros w1,w2 , ••• ,wk+ 1 satisfying

Finally sign Pk(n,wt) sign of Pk(n,x) on (vt-l'vt) this define w₀

=

0 and wk+l

=

n).

(-1) R.+l (for 0

It will turn out in chapter 5, that we need more information on the position of the zeros of Krawtchouk polynomials. To obtain this

infor-k k-1 k-2 mation we shall have to know the coefficients of x , x x in Pk(n,x). In the sequel we shall denote the coefficient of xt in a po-lynomial p(x) by CR.(p(x)). LEMMA 2,2, B. (2. 2. 11) (2. 2.12) (2. 2. 13) ~ (Pk (n,x)) k ~ k! (-q)k-2 2 Ck_₂ (Pk (n,x))

=

₂₄ (k _ ₂)! [q (k- 2) (3k- 1) + + 12q(q-1)(n-k+1)(k-2) +12(q-1)2(n-k+2)(n-k+l)}.

PROOF. use expression (2.1.8) for Pk(n,x). The proof is only a straight-forward calculation. To derive (2.2.13) one needs

L

i · j 1Si<j!>k-1

k (k- 1) (k- 2) (3k- 1) 24

THEOREM 2.2.9. Let v

1 < v2 < ••• < vk be the zeros of Pk(n,x). Then

(2. 2. 14) (2. 2. 15) k

L

i=1 =~(q-

₁

)n _ (q-2)(k-1)} vi q 2 ' v.v. ~ J + 12q(q-1)(n-k+1)(k-2) +12(q-1)2(n-k+2)(n-k+1)}, 2 k (k- 1)[12n(q-1) + 12q2 + k (q2- 12q + 12) + (q2 + 12q- 12)} ,

0

(2. 2 .17) s k(k- 1) [12n(q- 1) +k(q2 -12q + 12) + (q2 + 12q- 12) }. 6q2

PROOF. (2.2.14) and (2.1.15) follow directly from lemma (2.2.8), since

k

L

i=l and

L

i<j Since

I

i<j v. ~ v.v. ~ J

(2.2.16) follows by straightforward calculati~n from (2.2.14) and (2.2.15). Finally (2.2.16) implies (2.2.17) since

For q = 2 there is an additional nice property of Krawtchouk polyno-mials expressed in the following lemma.

LEMMA 2.2.10. Let q = 2 and let v

1 < v2 < ••• < vk be the zeros of Pk(n,x). Then

(2.2.18) vi+ vk-i = n,

PROOF. By (2.1.7) Pk(n,x) (-1) Pk(n,n-x). k

For the sake of completeness we end this paragraph with two small tables of Krawtchouk polynomials.

TABLE 2.2.1. P 0 (n,xl 1 , P 1 (n,x) = n(q -1) -qx , 1 2 2 2 P 2(n,x) =~qx -q(2qn-q-2n+2)x+(q-1) n(n-1)}, 1 3 3 2 2 P 3 (n,x) = i'<-q x + 3q { (q- 1) (n- 2) +q}x + 2 2 2 - q{3(q- .1 ) n -3(q-1)(2q-3)n+2q -6q+6}x + 3 + (q-1) n(n-1)(n-2)}. TABLE 2.2.2. P 0 (n,x) 1 , P 1 (n,x) n-2x , P 2 (n,x) 2 n 2x - 2nx + < 2> P 3 (n,x) =

~-4x

3 + 6nx2 - (3n2- 3n + 2)x +n(n- 1) (n- 2)/2} ,

0

P 4 (n,x)

1 4 3 2 2 3 2 n

j{2x - 4nx + (3n - 3n + 4) x - (n - 3n + 4n) x + 3 ( 4)}.

2. 3.

The.

wugh:t

and

cU6.tance. e.nwne.Jtatolt

o6

a

code

One way of studying codes is to determine the weights and the mutual distances of the code words. It is for this reason that we spend this paragraph to this subject.

DEFINITION 2.3.1. The weight enumerator W(z) of a code cis the poly-nomial

(2. 3.1) W(z) := :=

I

z -w(c) •

CEC

DEFINITION 2.3.2. The distance enumerator A(z) of a code c is the po-lynomial

(2. 3. 2) A(z) :=

L

2 zd(_£,~) •

(_£1~) EC

It is clear from these definitions that wi is the number of code words of weight i, i.e. wi

=

B(Q,il, while Ai is the average over all code words of the number of code words at distance i from a given code word, i.e.

I

e

l

- l \ Ai L B(_£1i) CEC we observe that A₀ =

Iw

.

~

DEFINITION 2.3.3. A code cis distance invariant if B(_£,i), .£ E c, only depends on i and not on the particular choice of c E c.

Clearly in a distance invariant code C, with

Q

E c, one has A(z)

=

W (z). The next theorem provides us a recurrence relation of the coefficients wi in W(z).

THEOREM 2.3.4, Let C be a code and a₀,ai'''''ar the coefficients of the Krawtchouk expansion of the characteristic polynomial F

c

{x) • Then the weight enumerator

W(z)

I

satisfies the recurrence relation given by (2.3. 3)

for all k, 0 $ k $ n.

For q

=

2, this relation reduces to

(2. 3.4) 0 S k $ n ,

where wn+i

=

w_i

=

0 for i ~ 1.

6

PROOF. Let Wi . (t) denote the set of words, which can be obtained by ,J

selecting a code word of weight i, changing i zero coordinates into a nonzero, changing j nonzero coordinates into a zero and changing 6 - i - j nonzero coordinates into a different nonzero coordinate. Clear-ly

(2,3. 5)

Since

W~

. (t) is the set of words of weight i + i - j 1 which are at

dis-l.,J

tance 6 to a code word of weight t, it follows that the words of weight k in

c

*

Y

0 form the set u

O$i+j$cS cS

W, , (k-i + j) , l.,J

Formula 2.3.3 now readily follows from theorem 2.1.8 and (2.3.5). Formula (2.3,4) is a direct consequence of (2,3.3), since the factor

we shall now derive a more explicit expression for A(z) and W(z).

DEFINITION 2.3.5. For 0 $ k $ n,

(2. 3.6) X (C)

u

we remark that the last expression in (2.3.6), together with formula (2.1.9) prove that the number Ck are real.

DEFINITION 2.3.6. Let C be a

linear

code, i.e. a linear subspace of V(n,q). Then the

dual

code C~ is defined by

(2. 3. 7) c~ := {u € V(n,q)

I

v

[

(_u,_c) = O]} , CE:C

i.e. ~ is the orthogonal complement of c.

The next theorem will show that the numbers Bk and Ck have a combina-torial meaning if the code is linear.

THEOREM 2.3.7. Let C be a linear code with parameters Bk and Ck as de-fined in definition 2.3.5. Then the weight enumerator wL(z) of c~ sa-tisfies

~(z)

PROOF. We first observe that

(2. 3. 8)

{

lei

,

i f u E

c~

X (C)

-u - 0 , otherwise 1

the case u E c~ being trivial, the other case following from the fact

that C is a linear subspace, that Xu is a linear functional and final-ly that

I

X(a)

=

0.

aE:GF(q)

From (2.3.8) and definition 2.3.5 it is clear that Bk and ck equal the

THEOREM 2.3.8. (The MacWilliams identities). Let C be a code with weight enumerator W (z), distance enumerator A(z) and parameters Bk and ck, as defined in definition 2.3.5. Then

(2. 3.9) (2. 3.10) .L£L n k n-k A(z) = n

L

Bk(l-z) (1+(q-1)z) , q k=O W(z)=.L£L

f

~(1-z)k(1+(q-l)z)n-k.

qn k=O

PROOF. Let

r

be the group of monomial transfonnations of V(n,q), i.e.· each element of

r

is the product of a n x n permutation matrix and a n x n nonsingular diagonal matrix. Let v be a particular element in

n-k

-:-Yk. Denote k! (n -k)! (q -1) by ck. Clearly

By (2.3.6), (2.3.1), (2.1.9) and (2.2.2) one has

X (C) u

I

YET X (y (C)) v 1 n

c~

XV(

e

c~W~Y~)

- ~=0

L

Xy (v) (C) Hf -y(C))

Multiplying both sides of this equality by Pm(n,k) and then summing over k yields, with the aid of (2.2.3),

\c\

This proves (2.3.10). The proof of (2.3.9) is entirely the same since

X (C)X (C)

u u

L

xu

(C

*

(-C))

L

xy

(_v) (C

*

(-C)) y€f n n n Ck-1

x

<

~

z; CR. A Y R, R, >

:!..

R.=O

I

cR.AR.PR.(n,k) R.=O

L

AR.Pk (n,R.) • R.=O

In this paragraph we shall show that codes with external distance r ~ d satisfy certain regularity conditions. Moreover we shall show that the code words of fixed weight in such codes form a design, as-suming that the code contains the origin.

DEFINITION 2.4.1. A code is t-regular if for all~ € V(n,q) with

0

p(~ s t, B(~₁k), 0 ~ k ~ n, only depends on p(~) and k. A completely regular code is a r-regular code.

According to definition 2.3.3 distance invariant codes are 0-regular codes and vice versa.

We shall now derive 2 lemmas, which will enable us to prove a theorem that states that certain codes are t-regular.

LEMMA 2.4.2. Let r be the external distance of a code. Then the weight enumerator

n

W(z)

l:

i=O

is uniquely determined by its first r coefficients w

0,w1, ••• ,wr_1•

PROOF. This is a consequence of theorem 2.3.4 which gives a recurrence relation in Wk-r'Wk-r+₁, ••• ,wk+r fork= 0,1, ••• ,n, in which the

coef-ficient of wk+r is nonzero.

0

LEMMA 2.4.3. The parameters Bk' 0 ~ k ~ nand r, the external distance, of a code are invariant under a translation of V(n,q).

PROOF. Clearly the distance enumerator of a code is invariant under

k n-k

translation of V(n,q). Since the polynomials (1-z) (1 + (q- 1) z) ,

0

s

k

s

n, form a base in the set of polynomials of degree

s

n, i t follows by theorem 2.3.8 that the parameters Bk, 0 s k s n, are inva-riant under translation. Since the definition of r only involves these Bk, it follows that also r is invariant under translation.

0

THEOREM 2. 4. 4. Let r be the external distance of a code and d the mi-nimum distance. Then

i) r

s

d

s

2r - 2 ~ C is (d- rl -regular ii) 2r - 1

s

d ~ C is completely regular

PROOF. Let ~ € V(n,q), p (~) w s d-r. Since d-w <:: d- (d-r) <:: r, one has that B (~,k)

=

0 for k S r- 1, k f w and B (~,w) = 1. Since B (~,kl , 0 s k s n·, equals. the coefficient Wk of the weight enumerator of the code {~- ~

I

~ €

c},

the assertion follows directly from

lem-ma 2.4.2.

Similarly in the case 2r-1 S d, one has that Cis (r-1)-regular, since d- (r- 1) <:: r. However for any~ € V(n,q) with p (~)

=

r, one has B (~,0) = B (~, 1) B(~,r-1) = 0, and one can apply the same reasoning as above. Hence in this case C is r-regular i.e. completely regular.

Let us say that a vector~ € V(n,q) is covered by a vector~ € V(n,q) if for every nonzero component xi one has xi yi.

DEFINITION 2.4.5. A q-ary t - (n,k,;l.) dEsign is a collections of vec-tors of weight k in V(n,q) with the property that every vector x € V(n,q) of weight t is covered by exactly

A

vectors ~ € S.

0

REMARK. For q

=

2 this definition coincides with the normal definition of a t - (n,k,;l.) design.

LEMMA 2.4.6. The.collection s o f a q-ary t - (n,k,;l.) design defines al-so a q-ary i - (n,k,;l.i) design for 0 S i S t - 1, where

(2. 4.1)

PROOF. Let x € V(n,q) of weight i. This vector x is covered by

(~=~)

(q-1) t-i vectors

~~of

weight t. Each of these vectors x 1 _is

covered by exactly A vectors ~ €

s.

However for each vector~ € S there are exactly

(k-~)

vectors x 1

such that

~

is covered by

~

1

and x 1

t-l.

is covered by ¥..

REMARK. The numbers Ai' i = 0,1, ••• ,t-1, yield strong divisibility conditions for the existence of a q-ary t - (n,k,A) design.

0

THEOREM 2.4.7. Let C beat-regular code, containing the all zero vec-tors. Let the minimum distance d of C satisfy d ~ 2t. Then for each k,

0 ~ k s n, the collection of code words of weight k form a q-ary t-design.

PROOF. The proof is by induction on k. We first prove the theorem for k = d. Let x € V(n,q), w(x) = t. Since d ~ 2t alsop(~) t. Hence by the regularity one has that B (~, d- t) takes on the same value, let us say c, for all~ with w (x) = t. However by the triangle inequality for any ;;_ € C\ {2_}, d (~, ;;_) = d- t iff w (;;_) d and ;;_ covers ~· Consequently the code words of weight d form a q-ary t - (n,d,A(d)) design, where A (d) = c if d > 2t and A (d) = c- 1 if d = 2t (in the latter case, also

t = d- t ) .

Assume that the theorem is proved for all k < w, i.e. the code words of weight k form a q-ary t - (n,k,A(k)) design for all k < w. By lemma 2.4.6 we may conclude that the number of code words of weight k at a given distance ~ from ~ with w(~) = t is constant. In particular the number of code words of weight s w - 1 and distance w - t from x where w (~)

=

t, is constant, let us say a. Since by our assumption B (~,w-t) is constant, let us say b, for all~ with w(~) = t, we deduce that the number of code words of weight w, at distance w- t from ~' w (~) = t, is b- a. Since a code word ;;_ of weight w is at distance w- t from ~, w(x)

=

t iff c covers~' we have proved that the code words of weight

We remark that the A of the designs in this theorem can be determined from the weight enumerator.

CHAPTER

3

PERFECT ANV [STRONGLY) UNIFORMLY PACKEV COVES

Since perfect and (strongly) uniformly packed codes have already been defined in chapter 1, it is not proper to define these concepts again. The reader, however could regard theorems 3.2 and 3.7 as new algebraic definitions of perfect resp. uniformly packed codes.

LEMMA 3.1. Let a

0,a1, ••• ,ar be the Krawtchouk expansion of the charac-teristic polynomial FC(x) of an e-error-correcting code

c

with exter-nal distance r and minimum distance d. Let r < d. Then

PROOF. If r 2:. e + 1 then d- r - 1 :S 2e + 2- (e + 1) - 1

=

e. If r

=

e, it follows from the remark at the end of paragraph 2.1 that p equals e, i.e. that C is perfect. However this implies that d

=

2e + 1 and conse-quently d-r-1 =e. Hence in all cases d-r-1:;; e. Let!:. E V(n,q)

where w(!:_) = w :5: d-r-1. By the triangle inequality B(!:_,i) = 0 for i :S d-1-w, i #wand B(!:_,w) = 1. Since d-1-w 2: r, corollary 2.1.9 reduces to a = 1.

w D

The next three theorems are on perfect codes. We shall omit the proofs, since perfect codes are not the object of this study and because the proofs are along the same lines as the proofs of the next four theo-rems.

THEOREM 3.2. An e-error-correcting code with external distance r is perfect iff r = e.

THEOREM 3.3. (Lloyd). If a perfect, e-error-correcting code C exists then the polynomial F(x) defined by

(3. 1)

e

F(x) :=

L

Pk(n,x) = Pe(n-1,x-1) k=O

has e integral zeros in [1,n] and

REMARK. Formula (3.2) is identical to formula (1.4).

THEOREM 3.4. A perfect code is completely regular.

THEOREM 3.5. Let C be an a-error-correcting code. Then Cis uniformly

packed with parameters A and ~ iff, as an element in ~(V), C satisfies

(3.3) _C

_{* (}

e-1

t

_Yi $ (1 - -)Y A $ ~ 1 Ye+1)

i=O ~ e ~

e

s

_n

PROOF. Let D := S - C

*

n more than e from

c.

t

Yi' i.e. the set of words at distance

i=O

By (2.1.4) and the triangle inequality

t

B(~1 e+1)~

~EV(n,q)

t

B (~, e + 1) ~Ell

f

B (~, e + 1).:! •

XED

With definition 1.2 this implies that C is uniformly packed with

para-meters A and ~ iff

e

Substitution of D Sn - C

*

i~O Yi now yields the theorem. D

THEOREM 3.6. An a-error-correcting code C is uniformly packed with

pa-rameters A and~ iff its characteristic polynomial Fe satisfies

(3. 4) e-1

L

Pk(n,x) + (1 - -)P (n,x) + - P A 1

1 (n,x)

k=O ~ e ~ e+

PROOF. Let FC(x) be given by (3.4). By theorem 2.1.8, C satisfies (3.3). Hence the if-part follows from theorem 3.5.

So let us now assume that c is uniformly packed with parameters

A and

~· Let F c (x) be its characteristic polynomial. Since p (C)

= e + 1, r

the external distance, is at least e + 1. Let F (x) be the polynomial

F(x) e-1

L

Pk (n, x) + ( 1 - -) P (n, x) + - P ). . 1

1 (n, x)

k=O ~ e ~ e+

we have to show F(x) = FC(x).

Let FC(w) = 0. By (2.1.10), (2.1.11) and (2.1.12), there exists an element u e: V(n,q), w(u) = w ~ 0, such that

x

(C)~ 0. Taking

x

of

- - u u

both sides of formula (3.3) yields with the aid of (2.1.9) and lemma 2.1.4 that F(w) = 0. Consequently FC(x) divides F(x). Since the degree of F(x) = e + 1, and the degree of Fc(x) is at least e + 1, we have F(x) = aFC(x). By (1.12) F(O) Hence a 1. n L _

=

FC(O) •

IC

I

THEOREM 3.7. An e-error-correcting code is uniformly packed iff its external distance r equals e + 1.

D

PROOF. The "only if" part is a direct consequence of the previous theo-rem.

So assume r = e + 1. Let ~ E V(n,q), p (~)

=

e. It follows from the tri-angle inequality and (2.1.15) that ae + ae+

1B (~,e + 1)

=

1. Hence B (~,e + 1) is a constant, being (1-ae)/ae+₁• Similarly for~ e: V(n,q),

B(~,e + 1) is a constant, p (~) > e one 1 . being -a--e+1 rameters ). = has a e+ 1 B (~, e + 1)

=

1 , i.e. According to definition 1.2 -1 (1-ae)/ae+₁ and~ _{= ae+ 1.}

C is uniformly packed with

pa-COROLLARY 3.8. A uniformly packed code is completely regular.

PROOF. Apply theorem 2.4.4 with r e + 1 and d 2e + 1 or d 2e + 2.

D

COROLLARY 3.9. Let C be a uniformly packed code in V(n,q) containing the origin. Then the code words of fixed weight form a q-ary e-design

if d = 2e + 1 and a q-ary (e + 1)-design if d

=

2e + 2.

PROOF. This is an immediate consequence of corollary 3.8 and theorem

COROLLARY 3.10. Let c be a binary, strongly uniformly packed code (i.e. l.l - A = 1) containing the origin. Let d = 2e + 1. Moreover let Cext be the extended aode of c, i.e. the code Obtained from C by annexing an

overall parity check symbol. Then the code words of fixed weight in cext form (e + 1) -design •

PROOF. The proof is analogous to the proof of theorem 2.4.7 and will be omitted. We shall proof the statement of this corollary for the words of minimal weight in Cext. Let x €

v

(n + 1, 2) with w (~) e + 1.

If~= (~', 1), ~· € V(n,2) then, by the uniformly packedness of c, ~·

is covered by exactly A code words of weight 2e + 1 in C. Consequently

(~',

1) is covered by A code words of weight 2e + 2 in cext. Similarly if x = (~' ,0), ~· € V(n,2) then x' is at distance e + 1 to 0 and either at distance e from one code word in C of weight 2e + 1 and at distance e+ 1 from A-1 code words in C of weight 2e + 2 or at distance e + 1

from l.l - 1 code words in C of weight 2e + 2. Since A + 1 = l.l, one has in both cases that x is covered by l.l - 1 = A words of weight 2e + 2 in

cext.

0

The next theorem gives an easy way of checking whether a code is uni-formly packed or not. It will often be used in the next chapter.

THEOREM 3.11. Let C be an e-error-correcting, linear code. Then Cis

. ~

uniformly packed iff the dual code c contains exactly e + 1 nonzero weights.

PROOF. By theorem 2.3.7 and formula (2.1.12) one has that the occurr-ing nonzero weights inC~ are exactly the zeros of FC(x). This theorem

is now a direct consequence of theorem 3.7.

0

The next theorem is a direct consequence of theorem 3.6 and the defi-nition of FC(x). It will yield a very strong necessary condition for the existence of uniformly packed codes.

THEOREM 3.12. Let C be an a-error-correcting, uniformly packed code with parameters A and l.l· Then the polynomial Q(x) defined by

(3. 5) Q(x) : = .!.{ 1 P 1 (n - 1, x - 1) + (\l - A - 1) Pe (n - 1, x - 1) + lJ e+ +AP 1cn-1,x-1)}

e-has (e + 1) distinct integral zeros in the interval [ 1,n] and (3.6)

PROOF. By lemma 2.2.4 and theorem 3.6 Q(x)

=

FC(x). Formula (2.1.12) implies that F C has e + 1 distinct integral zeros in [ 1,n] and that

Fc(O)

=

qn/lcl.

D

Because of the significance of this theorem in chapter 5, we shall now derive a number of properties of the polynomial Q(x), given by (3.5). Especially the fact that the zeros of Q(x) are all integral will prove to be a very powerful tool in (non)existence theorems. To obtain more information on these zeros, we shall now derive some theorems on this subject.

THEOREM 3.13. The polynomial Q(x), as given by (3.5), with

0 :SA < (n-e)(q-1)/(e+1) and 1 :S \l :S n(q-1)/(e+1) has (e+1) dis-tinct positive zeros x

1 < x2 < the zeros of Pe (n- 1,x -1) and of P e+ ₁ (n - 1, e - 1) . Then (3. 7) 0 < _{x1 < a1 < x2} < _a2 (3. 8) X e+1 $ n iff \ l -A - 1 (3.9) lJ - A - 1 ~ 0 ~ x1 ~ < xe+ 1• Let a1 < a2 < •.. < ae be let b

1 < b2 < ••• < be+1 be the zeros

<

...

< a < X I e e+1 $ (n-e) (n- e- 1) + Ae (e + 1) (e + 1) (n- e) ~ _b1 I (3 .10) \.1 - A - 1 $ 0 X e+1 $ b e+1

PROOF. The statement on the existence of distinct positive zeros xi and formula (3.7) follow from the next three observations:

i) sign Q(O) e-1 sign{

L

i=O n i A n e (.) (q - 1) + ( 1 - -) ( ) (q - 1) + ~ lJ e +

.!.(

_{lJ e+1} n )( _ 1,e+1_q } ₊ 1 _'

since

n e+ 1 n e n e n - e

( )(q-1) -A( )(q-1) = ( )(q-1) ( - - ( q - 1 ) -A)> 0

e+1 e e e + 1 '

ii) by the definition of ak, the formulas (2.2.5) and (2.2.10) and the inequality on A,

sign Q(ak) = sign{Pe+₁ (n -1,ak -1) + APe_

1 (n -1,~ -1)} = sign{(A_(n-e)(q- 1 ))P (n-1 a -1)} = (-1)k, 1 :S k s e,

e + 1 e-1 ' K

iii) sign Q(x) = (-1) e+1 ,

since the coefficient of xe+1 in Q(x) has sign (-1)e+1•

Since

e+1

Q(n) = (-1) { (n-1) _ (J.l _A_ _{1 )} (n-1) + A (n-1)}

J.l e+1 e e-1

by (2.2.7), one has ae < xe+

1 :Sniff sign Q(n) = (-1)e+1

i.e. i f f

thus proving (3.8). Since Pe_

1(n-1,x-1), Pe(n-1,x-1) and Pe+1(n-1,x-1) are all posi-tive on [O,b

1J, (3.9) follows immediately from the definition of Q(x).

e-1

Similarly, these three polynomials have alternating sign ((-1) e e+1

(-1) , (-1) ) on (b

1,n]. So (3.10) becomes obvious. 0

e+

THEOREM 3.14. Let x₁ < x₂ < ••• < xe+₁ be the zeros of

where Q(x) =~P 1cn-l,x-1) + (J.l-A-l)P (n-1,x- 1) + J.l e+ e 0 s A < (n- e) (q-1) e + 1 +AP 1cn-l,x-1)}, e-1 s ll s n (q-1) e + 1

Then (3. 11) (3. 12) (3. 13) where (3. 14) where e+1

L

X. i=1 ~

(e+ 1){ _{(q- 1)n} +ll -A_ (q_;2)e}

q 2 e+1 II _xi i=1 e-1 IJ (e + 1) ! ( \ n i ).. n e L (. ) (q - 1) + (1 - -) ( ) (q - 1) + qe+1 i=O ~ 1J e + _!, ( n ) ( _ ₁₎ e+ 1 } 1J e+1 q ' e+1 II (xi-1) i=1 (q _ ₁) e-1 e+ 1 (n-1)(n-2) ••• (n-e+1)p2(n) q e+1 II i=1 2

(n-e)(n-e-1)(q-1) +(IJ-A-1)(n-e)(e+1)(q-1)+

(X. - 2) ~ + A (e + 1) e , ( _₁)e-2 q e+l (n-2) (n-3) .•• (n-e+l)p 3(n) q 3 := (q-1) (n-1) (n -e)(n-e-1)-2 (q-1) q(n-e)(n-e-1)(e+1) + (IJ-A-1){(q-1)2 (n-1)(n-e)(e+1)-(q-1)q(n-e) (e+ 1)e} +

A{ (q -1) (n - 1) (e + 1)e -q(e + 1)e(e - 1)}

PROOF. By (2.2.11) ce+1 (Q(x) l c e+ 1 1 -;;-P e+ 1 (n - 1 , x - 1) Hence e+ 1 e+ 1 (3. 15) Q(x) q II (x. - x) \l(e+1)! i=l ~ ( -q ) e+1 \l (e + 1)!

A different expression can be obtained by (2.2.4),

(3.16) Q(x)

e-1 A 1

I

P (n,x) + (1 --)P (n,x) + -P

1 (n,x)

Using this expression, one can easily verify with the aid of lemma 2.2.8 Since e+1

L

xi i=1 -Ce(Q(x))/Ce+₁ (Q(x))

formula (3.11) now follows readily.

Formula (3;12) results from a substitution of x

=

0 in (3.15) and (3.16), using (2.2.6).

Formulas (3.13) and (3.14) follow similarly after substitution of x = 1 resp. x

=

2 in (3.15) and the definition of Q(x). Here one can fruit-fully make use of lemma 2.2.6.

For a specific result on binary, e-error-correcting uniformly packed codes in chapter 5 we shall need one more theorem.

THEOREM 3.15. Let x

1 < x2 < ••• < xe+1 be the zeros of Q(x), as de-fined in theorem 3.14. Let q

=

2. Then

(3. 17) 2 e (e + ₄ 1) { ( ). ) 2 _{2 2} e - 1} n+\J- - \ J - 3 -(3. 18)

0

n(3\l+3A+2e-2) +61!+2(e-2)(1J-A)}.

PROOF. We omit the proof, which is a straightforward calculation like in the previous theorem and in theorem 2.2.9. We shall need this theo-rem only for e

=

3. For this case one can easily check the formulas

(3.17) and (3.18) by the explicit expressions of P

2(n,x), P3(n,x) and P

CHAPTER 4

CONSTRUCTIONS OF UNIFORMLY PACKEV COVES

In the tables 4.1.1 and 4. 2. 1 the reader can find all the parameters, for which uniformly packed codes have been found. These tables appear-ed before in a paper by Goethals and van Tilborg (1975), except for the series of codes with number 1 in table 4.1.1.

We are aware that these tables are almost certainly not complete. It is for instance still unknown, whether there exist uniformly packed codes with the parameters

a) n

=

25, q

=

2, e

=

1, A 6, )1

=

9,

lei

=

3.218,

b) n

=

70, q

=

2, e

=

2, A 16, )1 10,

lei

=

258

,

c) n

=

(22m+1 + 1)/3, q

=

4, e

=

2, )1

=

A + 1

=

(22m- 1) /3.

In the following two paragraphs the reader should realize that the pa-rameters A and )1 of a linear, uniformly packed code C in V(n,q) are

uniquely determined by the occurring weights inC~ by means of (2.1.12).

Before we start the constructions, we derive some very nice properties of linear, single error correcting, uniformly packed codes.

Let C beak-dimensional linear code in V(n,q). A k x n matrix over

GF(q) whose rows form a base for C is called a generator matrix of

c.

A (n- k) x n matrix over GF (q) whose rows form a base for C~ (the

or-thogonal complement of C) is called a parity check matrix of

c.

A pari-ty check matrix of Cis clearly a generator matrix of C~.

THEOREM 4.1.1. Let C be a k-dimensional linear code in V(n,q), with parity check matrix H. Then C is single error correcting iff the co-lumns of H are pairwise linearly independent, i.e. iff the columns of

H are distinct point in PG (n- k- 1 ,q), where PG (m,q) denotes the m-di-mensional projective geometry over GF(q).

PROOF. If two columns of H are linearly dependent then C does contain a code word of weight 2 (or 1) and vice versa. The reader should rea-lize that in a linear code the minimum distance equals the smallest

m_1

DEFINITION 4.1. 2. Hm (q) is a m x

T,

matrix of which the col=s as

vectors are all the points of PG (m-1 ,q) (for instance all vectors in

V(m,q)\{Q) of which the first nonzero coordinate equals 1). Hm(q) is

called a Hamming matrix. The linear code C (q) with H (q) as parity

m m

check is called the Hamming code.

By theorem 4.1.1 Cm(q) is single error correcting. Since

where n (qm- 1) I (q- 1) , we have by ( 1. 4) that Hamming codes are

per-feet.

Let E..= (c₁,c₂, .•• ,em) E PG(m-1,q). The set of points (x

1,x2, ••• ,xm)

E PG(m-1,q) satisfying c

1x1 + c2x2 + ... + cmxm

=

0 forms a hyperplane,

containing (qm-1 - 1) I (q- 1) points. This proves that the vreight of

T E.. Hm (q) equals n -qm-1 _ ₁ q-1 m-1 q

In other words: Hm(q) is the generator matrix of a one-weight code,

which result also follows from theorem 3.2.

DEFINITION 4.1.3. Let C be a linear, single error correcting code in

V(n,q) of dimension k. Let H be its parity check matrix. Then the

complementary code Ccompl of

c

is a linear code with parity check

ma-trix H', such that the matr~x (HjH') is a Hamming matrix.

Clearly by theorem 4.1.1 Ccompl is also a single error correcting,

linear code. A, for our goal, more important property is expressed in

the following theorem.

THEOREM 4.1.4. Let C be a linear, single error correcting, uniformly

packed code. Then Ccompl is also a linear, single error correcting,

uniformly packed code.

compl .

PROOF. One knows already that C ~s linear and single error

cor-compl

of the rows of H can take two possible. weights, let us say w

1 and w2• Since the nonzero linear combinations of the rows Hm(q) have weight

m-1

q , it follows that the nonzero linear combinations of the rows of

m-1 m-1 compl

H' must have weight q -w

1 or q -w2• By theorem 3.11 C is

uniformly packed.

0

Let C be a linear, single error correcting, uniformly packed code of length n and dimension n - m. Let the two occurring weights in c.l be w

1 < w2• With C.l we associate the graph f(C.l) on qm points as follows.

The vertices of f(C.l) are identified with the elements of C.l and two vertices are adjacent iff the corresponding vectors have distance w₁•

Clearly the automorphism group of f(C.l) contains a subgroup, isomor-phic to the elementary abelian group of order qm.

We quote the following two results of Delsarte (1972), The reader who is not familiar with the terminology used here, is referred to Cameron and van Lint (1975).

THEOREM 4.1.5. Let C be a linear, single error correcting, uniformly packed code in V(n,q) of dimension n -m, Let w

1 < w2 be the nonzero

weights in C.l. Then the associated graph f(C.l) on qm points is strong-ly regular, with eigenvalues given by

i m-1 w

1 +w2- (1 + (-1) )q , i 1,2,

where N n (q-1).

THEOREM 4.1.6. Let r be a strongly regular graph on pm vertices, p prime, whose adjacency matrix has integral eigenvalues p

0,p1,p2,

IP

01 < pm- 1, P1 > 1, P2 < O. Assume that the automorphism group of r

contains a regular subgroup isomorphic to the elementary abelian group of order pm. Then there exists a linear, single error correcting, uni -formly packed code C in V(n,p) of dimension n- m of which the associa-ted graph f(C.l) is isomorphic tor, while nand the two occurring non-zero weights w

1 and w2 in c.l are determined by

(4.1. 2) (p1 + (-1) )p i m-1 I i 1,2 •

We shall need two more definitions before we describe the constructions of the codes in table 4.1.

DEFINITION. Let

c

be a code. By puncturing

c

on position i, one dele-tes the i-th coordinate of

c.

By shortening C on position i, one only considers the code words with a fixed value at the i-th coordinate and then deletes this coordinate.

Clearly the punctured code of C has the same number of code words as

c,

while the shortened code of C has a minimum distance not less than the minimum distance of C.

No. 1. Let 1 :;: k :;: m- 1. We renumber the columns of Hm (q) in such a

way that the first (qk - 1) I (q- 1) are the points of PG (k - 1 ,q) . Since each hyperplane c

1x1 + c2x2 + ••• + cmxm = 0 either contains PG(k-1,q) or intersects i t in (qk-1_1)l(q-1) points, i t follows that cTH (q)

k k-1 - m

has either (q -1)l(q-1) or (q -1)/(q-1) zeros on its first (qk - 1) 1 (q - 1) coordinates. If one punctures the code generated by this H (q) on its first (qk- 1) I (q- 1) one obtains a m-dimensional

m m-1 m-1 m-1 k-1

2-weight linear code, the weights being q -0 = q and q -q By theorem 3.11 this code is the dual of a uniformly packed code. For q = 2 and k = m- 1 one obtains the extended Hamming code. For q

=

2 and k

ly perfect.

one obtains the shortened Hamming code, which is

near-No. 2. Let w be a primitive element of GF (qm) and let 2 :;: r :;: qm- 1.

Then

w

*

is clearly the generator matrix of a 2-dimensional, 2-weight code C in V(r,qm), the weights being r-1 and r. We define the trace

(4.1. 3)

2 Tr(a) := a + aq + aq

m-1 + ••• + aq

Since (Tr(a))q

=

Tr(aq)

=

Tr(a), we have indeed that Tr(a) E GF(q). Moreover since (a+ Slq = aS+ Sq for a,S E GF(qm) and aq = a for

a E GF(q), clearly Tr is a linear mapping. Finally there exists an ele-ment a E GF(qm) with Tr(a) # 0, because the equation

m-1 +. • .+ xq

₌

0 has at most qm-1 solutions in GF(qm).

From these properties it follows that Tr assumes every value in GF(q) m-1 exactly q times. Let be defined by crm- 1 + V(~1 ,q) q -qm-1 q-1 .!_(a) := (Tr(a) ,Tr(aw), .•• ,Tr(aw ) )

It follows from the properties of Tr that .!_(a) has weight 0 for a 0 m-1 and q otherwise (use: m

.s...:l

q-1 w qm-1 i + 1 E GF(q), so TR(awi) F 0 # Tr(aw q- ) F 0) •

We remark that {.!_(a)

I

a E GF(qm)} is the dual of the Hamming code.

*

From the code C defined above we now define the code C in

by

m 1

V(r~

₁

,q)

q-*

Since C is a 2-weight code with weights r - 1 and r, it follows from the properties of Tr, mentioned above that also C is a 2-weight code,

m-1 m-1

the weights being (r-1)q and rq • Since (f(O),f(O), ••• ,f(O)) is

I

*I

-

m 2- 2m

-the only all zero vector in

c

and since

c

=

(q )

=

q , it also

From the construction one can easily prove that any two coordinates in care linearly independent. Hence by theorem 3.11 Cis the dual code of a uniformly packed code. Its parameters A and ~ can be determined from (3.4) and (2.1.12).

We remark that for q

=

2 and r

=

2m-1 + 1, one has

~

-A code is strongly uniformly packed.

1, i.e. the

No. 3. This is a sequence of uniformly packed codes C, which exist for

~1 ~1 m

r

=

q -1 or (q-1)q , m ~ 2 and also when r

=

(q -1llt or (t- 1) (qm -1Jit, where m 2i, for some integer i and t

F 1 is any

di-visor of q + 1. Note that the complementary code of such a code also be-longs to this family. For a description of these codes we refer to Delsarte (1971). The graphs f(C~) associated with care called negati-ve Latin square graphs by Mesner (1967). We shall now describe the

con-m-1

struction of the codes with r

=

q - 1, m ~ 2. Consider H

2m (q), its columns representing the points of PG(2m- 1 ,q), q = pa

Let S be a non degenerate elliptic guadric in PG (2m- 1 ,q). The equation

2 2

of scan be written as x

1x2 + x3x4 + ••. + x2m_3x2m_2+x2m_1 +Sx2m = 0, for p

F 2, where -S is a nonsquare element of GF(q), or

X ₁ x + x x _{2 3 4} + _{· • ·} + x _{2m-1 2m} x + _A(x22m-l + _x22ml

=

0 , for p

=

2, where

A(x~m-

1

+ x;m) + x

2m_1x2m is irreducible over GF(q). In both cases S contains (qm- 1 - 1) (qm + 1) I (q- 1) points, which can easily be proved by an induction argument. Moreover the intersection of S with a hyp~rplane H has cardinality

2m-2 v 1 := (q -1JI(q-1) or 2m-2 m m-1 v 2 : = (q - q + q - 1l I (q- 1 J

according asH is a tangent space to S or not (cf. Bose (1966)). Hence by shortening the Hamming code to the positions, corresponding

to points in PG (2m- 1 ,q) \S, one obtains a linear, single error

correc-m-1 m

ting code C of length n

=

(q -1) (q + 1JI(q -1), of which the dual code has as its nonzero weights

Cis uniformly packed by theorem 3.11. The parameters A and~ can easi-ly be computed from (2.1.12) and (3.4). For q

= 2, r

=

2m-1 -1, one finds, as in No. 2, that~ -A

=

1, i.e. the code is strongly uniformly packed. As an example of the construction above, we consider m = 3,

2 2 q

=

2. LetS be defined by x

1x2+x3x4+x5x6+x5+x6

=

0. Since q

=

2, one can rewrite this equation as x

1x2 + x3x4 + (x5 + 1) (x6 + 1) = 1, whi.ch, after the transformation of the base of V(6,2) given by yi := xi, i = 1,2,3,4; yi :=xi+ 1, i = 5,6, can be reduced to y

1y2 + y3y4 + + y5y6

=

1.

Let a

1y1 + a2y2 + ••• + a6y6 = 0 be the equation of a hyperplane H. W.l.o.g. a

1 = 1. The intersection of S with His the set of solutions to both equations. Substitution of y

1 = a2y2 + .•• + a6y6 in y1y2 + +y

3y4 +y5y6 = 1, yields, after the base transformation z2 := y2, z_{2R. := y 2R. + a2R._ 1y2 , z2R.-l := y2R._1 + a2R.y2 ,} R. 2: 2, the equation

It is easily checked that this equation has 12 or 16 solutions, de-pending on whether a

2 + a3a4 + a5a6 equals 0 or 1. Since

ls

i

= 36 and a hyperplane contains 32 points, one obtains by shortening the Hamming

code of length 63 on the positions corresponding to S, a linear code of length 27, of which the dual has as its nonzero weights 12 and 16. By theorem 3.11 this code is uniformly packed, its parameters being A + 1 = 1.1 = 6.

No. 4 and 5 (cf. Delsarte 1971). The codes of No. 5 are the complemen-tary codes of those of No. 4 (see theorem 4.1.4). Let w be a primitive

element of GF (2m). Let H be the 3 x (2m+ 2) matrix over GF (2m) defined by

H

We claim that H is the parity check matrix of the codes of No. 4. By theorem 3.11 i t suffices to show that the nonzero linear combinations of the rows of H take 2 possible weights.

Let (a,B,y) € V(3,2m)\{Q} and a(x) = a+

Bx

+ yx2 • Then