Applied
Mathematics
Letters
Applied Mathematics Letters 13 (2000) 109-113
PERGAMON
www.elsevier.nl/locate/aml
Projective Codes W h i c h
Satisfy the Chain Condition
S. ENCHEVA
Stord/Haugesund College
B~rnsonsg. 45, 5528 Haugesund, Norway
G. D. COHEN
Department of hff., ENST and CNRS
46 rue Barrault, 75634 Paris, France
(Received November 1998; revised and accepted March 1999)
A b s t r a c t - - B i n a r y linear codes with length at most one above the Griesmer bound were proven
to satisfy the chain condition by Helleseth et al. [11. Binary linear projective codes with length
two above the Griesmer bound which satisfy the chain condition are found. Necessary conditions
for binary linear codes for which the two-way chain condition holds are derived. @ 2000 Elsevier
Science Ltd. All rights reserved.
Keywords--Linear
codes, Chain condition, Two-way chain condition.
1.
INTRODUCTION
Let C be an In, k] b i n a r y linear code.
It is called projective if a n y two of its c o o r d i n a t e s are linearly i n d e p e n d e n t , i.e., if t h e d u a l
code C ± has m i n i m u m d i s t a n c e d ± > 3.
F o r ~my s u b c o d e D of C, t h e s u p p o r t of D, s u p p (D), is t h e set of p o s i t i o n s w h e r e n o t all t h e
c o d e w o r d s of D are zero, a n d it is d e n o t e d by x ( D ) . T h e s u p p o r t of a b i n a r y v e c t o r is t h e set of
its n o n z e r o c o o r d i n a t e s . T h e m i n i m u m s u p p o r t weight, d,., of a code C is t h e size of t h e s m a l l e s t
s u p p o r t of a n y r - d i m e n s i o n a l s u b c o d e of C. In p a r t i c u l a r , dl
d. T h e weight hierarchy of C
is {dl, d 2 , . . • , dk}.
T h e c o n c e p t s of chain condition (CC) and two-way chain condition ( T C C ) were i n t r o d u c e d
by W e i a n d Yang [2] a n d F o r n e y [3], respectively; our definitions are slightly different. T h e T C C
implies t h a t t h e code possesses an efficient c o o r d i n a t e ordering, a highly d e s i r a b l e f e a t u r e for
trellis d e c o d i n g (see [3]).
DEFINITION 1. A n [n, k] code C satisfies the clmin condition K i t is equivalent to a code C such
that there exists a chain o f s u b c o d e s of C~, D1 c D2 c . . - C Dk = C, where tbr t < r < k. we
have d i m ( D r ) = r, a n d x ( D ~ ) = { 1 , 2 , . . . , d r } .
DEHNITION 2. A n [n, k] code C satisfies the two-way chai~ condition if it is equivalent to a
('ode 6" with the following property: there exist two chains of subcodes of C, the left chain D~~ C
0893-9659/00/$ - see front matter @ 2000 Elsevier Science Ltd. All rights reserved.
PII: S0893-9659 (99)00219-0
Typeset by .@¢~S-TF/K
110
S. E N C H E V A AND G . D . C O H E N
D L C ' - - C D L = C, and the right chain D R C D R C - - . C DkR = C, where, for 1 < r < k, we
have d i m ( D L) = dim(DR) = r, x ( D L) = {1, 2 . . . . . d~}, and x ( D ~ ) = { n - d ~ + l , n - d ~ + 2 , . . . , n}.
In Definition 2, the chain of subcodes D L is actually the chain Di from Definition 1• The chain
of subcodes D R is similar to the chain Di but the order of coordinates is counted from the right
to the left.
THEOREM 1. (See [1].) A code C of length n = g(k, d) + 1 satisfies the chain condition.
THEOREM 2. (Duality) (See [4].) Let C be an [n, k] code and C ± be its dual code.
{dr:l <r<k}=
{1,2,...,n} \ {n+ X-drl:X <rKn-k}.
Then
From now on, C is a binary linear projective code.
The rest of this letter is organized as follows. In Section 2, we study conditions under which
codes of length 2 above the Griesmer bound satisfy the chain condition. In Section 3, we give
necessary and sufficient conditions for codes to satisfy the two-way chain condition.
2. P R O J E C T I V E
CODES
AND
THE
CHAIN
CONDITION
For the description of generator matrices of projective codes, we need some further notations.
A column in this description represents a sequence of columns in the generator matrix and the
number of columns in the sequence is written above the column. Thus, a generator matrix of C
can be written in the form
a
0
0
G8
0
0
...
...
0
0
0
0
*..•*
*...*
1
*
0
...
0
0
1
...
0
0
,...,
,...*
*
*
...
1
0
*
*
...
*
1.
(1)
k-I
For given k and d, there exists an [n, k, d] code where n(k, d) >_ g(k, d) := ~ i = o
the least integer not smaller than x; this is known as the Griesmer bound.
d
I•l,
[xl being
LEMMA 1. Let C be an [n = g(k, d) + 2, k, d < 2 k-2] code such that d~ = 3, d& = 5. Then C
satisfies the CC.
PROOF. From Theorem 2 and conditions d~ = 3, d2J- = 5 it follows t h a t the weight hierarchy of
Cisdi=g(i,d)
for 1 < i < k - 3 ,
dk_2=g(k-2, d)+l, anddi=g(i,d)+2,
fori=k-l,k.
Since d _< 2 k-2, dk-2 = g(k - 2, d) + 1 and d~ = 5, then dk-3 = g(k - 3, d). By Theorem 2,
the highest di's are dk = n, dk-1 = n - 1,dk-2 = n -- 3, dk-3 = n -- 5. Since d~ = 3, a generator
matrix of C can be written in the form (1) with s = k - 2, a = 2, where Gk-2 generates an
[n -- 3 = g(k - 2, d) + 1, k - 2, d] code Ck-2. The code Ck-2 generated by the first k - 2 rows has
length n - 3 = g ( k - 2 , d)+1, and according to Theorem 1 satisfies the CC. From dk_ 3 = g ( k - 3 , d),
it follows t h a t the weight hierarchy of Ck-2 coincides with the first k - 2 elements of the weight
hierarchy of C, completing the proof•
|
DEFINITION 3. Let c be a nonzero codeword of C and G a generator matrix for C, with c as first
row. The code generated by the restriction of G to the columns in which c has zero coordinates
is called the residual code of C with respect to c and denoted by Res (C, c).
THEOREM 3. A necessary and sufficient condition for an [n = g( k, d) +2, k, d] code C with weight
hierarchy {dl = d, d2 . . . . , dk} to s a t i J y the chain condition is that there exists an I n - d , k - 1, d~ =
d2 - d ] code C ' = Res (C, c) with weight hierarchy d~, d~ = d3 - d . . . . , d'k_ 1 = dk - d which satisfies
the CC.
Projective Codes
11 t
PROOF. Let C be a code as described in t h e t h e o r e m a n d satisfies the CC. W.l.o.g. its g e n e r a t o r
m a t r i x G c a n be w r i t t e n in the form (1) in such a way t h a t its rows display the chain of subcodes
of Definition 1. Let c be the first row in G. Now, we should prove t h a t C ' = Res (C, c) satisfies
the CC.
Since C is a (:ode with length 2 above the Griesmer b o u n d , there are integers 0 < m _< p < k
such t h a t dl = d, d2 = g(2, d ) , . . . , d m = g ( m , d ) , d m + , = g ( m + 1,d) + 1 , . . . , dr, = 9(p,d) +
[,@+1 = g(P + 1,d) + 2 , . . . , d k .
Now, we prove t h a t m ¢ p. Suppose t h a t there exists a
(::ode C w i t h p a r a m e t e r s as described in the t h e o r e m a n d an integer s such t h a t 0 < s < k,
Bj = 0, j = 1 , 2 , . . . , k - s - 2 a n d B k ± , - 1 > 0, with f u r t h e r m o r e d.i = g(i,d) for 1 < i < s,
d., = g(i, d) + 2 for s + 1 < i < k. I n other words, suppose t h a t I = p = s. A s s u m i n g w.l.o.g, t h a t
a codeword of C ± has for s u p p o r t the last k - s positions. T h e n a g e n e r a t o r m a t r i x of C can be
w r i t t e n in the form (1) where G~ generate a s u b c o d e of length n - (k - s) a n d d i m e n s i o n s a n d
,~ = 3.
T h e last k - s - 2 c o l u m n s in (1) are linear i n d e p e n d e n t . T h e second c o l u n m after G~, (h..e), is a
tinear c o m b i n a t i o n of all k - s - 2 c o l u m n s on its right since Bk . . . . 1 > 0. T h e first c o l u n m after
C.s differs from h2 by the p r o j e c t i v i t y a s s u m p t i o n . Therefore, it can be a linear c o m b i n a t i o n of
at, most k - s - 3 c o l u m n s chosen a m o n g the last k - s - 2. This c o n t r a d i c t s the a s s u m p t i o n t h a t
By = 0 , j = l , 2 , . . . , k - s - 2 .
Hence, m ¢ p .
/
I
/
/
Let us calculate dl_
1 a n d dl: dr_
1 . dl
g(I
. . d.
1, d l ) , d l = dl+l - d = 9(l,d'l) + i. T h e
!
!
same relations are valid between dp, dp_ 1 a n d dp+l, dp. Therefore, C ' has weight hierarchy
d~ = d2 - d,d~ = da - d , . . . , d g _ l
= dk - d a n d satisfies the CC.
Conversely, let C be a n [n = 9(k, d) + 2, k, d] code with weight hierarchy {all -- d, (t2 . . . . , d k }
~nd such t h a t there exists a n I n - d, k - 1, d~ = d2 - d] code C' = Res (C, c) with weight hierarchy
{d~, d.~ = d3 - d , . . . , d ~ . ~ = dk - d} satisfying the CC. From the definition of a residual code, it
[i)ll()ws t h a t C satisfies the CC.
II
PROPOSITION 1. I [ C is a n [n = g(k, d) + 2, k, d <_ 2 k-l+1] code with d~ = I > 3 containing an
[lz - l, k - l + 1, d] subcode, then C satisfies the CC.
PROOF. Since C is projective, its generator m a t r i x can be w r i t t e n in the form (1) where G.,
s p a n s an [n - l, k - l + 1, d] code C k - l + l .
F r o m T h e o r e m 1, it follows t h a t C k - l + l code satisfies the CC because n - l = 9(~:, d) + 2 - l =
!i(/¢-l+1,d)+1.
3.
CODES
AND
THE
TWO-WAY
CHAIN
CONDITION
Delsarte [5] has shown t h a t the weights of a two-weight code C are of ttie form
wl = u2 t,
w2 = (u + 1)2 t
(2)
tor s u i t a b l e integers u a n d t, u >_ 1, t >_ 0.
According to K10ve [6], if a b i n a r y linear code C satisfies the T C C , t h e n it c o n t a i n s two
(:odewords with m i n i m m n weight a n d disjoint supports. Thus, w2 = 2wl. Using e q u a t i o n (2), we
get (** + 1)2 t = 2u2 t, i.e., u = 1, proving the following lemma.
LEMMA 2. /if C is a two-weight code which satisfies the TCC, then its weights are of the form
~t~l =')t~, w2 = 2 t+l. t_> 0.
PROPOSITION 2. I f C is a n In, k, 2 t] two-weight code with t = k TCC, then n >_ g( k, 2 t ) + 2 j.
1 + j, j >_ O, which satisfies the
PROOF. B y G r i e s m e r b o u n d 9 ( k , 2 t) = ~i=:0
k-1 [(2t/2i)~ = 2t+1 - 2j, if t = k - 1 + j .
know t h a t n >_ w2 = 2 t + l .
B u t we
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112
S. ENCHEVAAND G. D. COHEN
LEMMA 3. I f C is a self-complementary two-weight code, then it satisfies the T C C .
PROOF. Let C[n, k, 2 t] be such a code, with generator m a t r i x G. One has Wl = 2 t, w2 = 2 t+l = n,
since the code is self-complementary.
W.l.o.g. assume t h a t the first row of G has s u p p o r t in the first 2 t positions; call it c~. Since
n = 2 t+l, its c o m p l e m e n t c~ has its s u p p o r t in the last 2 t positions.
Let c2 be any codeword not in {c[,c[, 1}. Since wt(c~ + c2) = 2 t, we get d2 = 3.2 t-1 and
d2 - d l -- 2 t-1. Consider the union of the s u p p o r t s of c[ and c2 ws(c~l,C2) where wt(c[) =
wt(c2) = wt(c[ + c2) -- 2 t. Therefore, ws(c~l, c2) = 3.2 t-1. Continuing in the s a m e way, we see
t h a t ws(c~, c2, c3) = w~(c[, c2, c3) and so on. Hence, C satisfies the T C C .
|
PROPOSITION 3. M a c D o n M d codes [2k - 2 u+l, k, 2 k-1 - 2 ~] satisfy the T C C only for u = k - 2.
PROOF. M a c D o n a l d codes [2k - 2 u+l, k, 2 k-1 - 2 u] are two-weight codes with Wl = 2 ~-1 - 2 ~
and w2 = 2 k-1. B y L e m m a s 2 and 3, we conclude t h a t these codes satisfy the T C C only for
u=k-2.
I
LEMMA 4. A necessary condition for a code C to satisfy the two-way chain condition is that
Bl >_ 2, where d~l = l >_ 3.
PROOF. B y Definition 2 (now we need only the left chain of subcodes of C) and the fact t h a t C
is projective, its generator m a t r i x can be written written in the form (1) with s = k - l + 1, a = 2,
where G k - t + l spans the [n - l = g(k - l + 1, d) + 1, k - l + 1, d] code C k - t + l . In other words,
Bt > 1. From the existence of the right chain in Definition 2, we conclude t h a t Bl _> 2.
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PROPOSITION 4. Subcodes of the e x t e n d e d [24, 12, 8] GoIay code with p a r a m e t e r s [24-p, 1 2 - p , 8],
1 <_ p <_ 6 do not satisfy the TCC.
PROOF. All codes described in Proposition 4 are e n u m e r a t e d in [7].
B y [7], there are exactly two [18, 6, 8] codes. T h e first one has weight distribution
A0=l,
As=45,
A12=18.
T h e r e is no codeword of weight 16, hence by L e m m a 5, the code does not satisfy the T C C . T h e
second one has B2 = 1, thus does not satisfy the T C C by L e m m a 6. Let C be a [19, 7, 8] code.
If C satisfies the T C C , there must exist three codewords as follows:
1111111100000090000
0000111111110000000
0000000000011111111.
To see this, note t h a t [6] for the T C C to hold, we need two codewords with m i n i m u m weight and
disjoint supports; for the left chain, we need a two-dimensional subcode with weight hierarchy
8,12. T h e y lead to a codeword of weight 14, which does not belong to the code since by [7], the
unique [19, 7, 8] code has weight distribution
A0 = 1,
As = 78,
A12 = 48,
A16 = 1.
Therefore, C does not not satisfy the T C C . Similar considerations complete the proof for the
other five codes of lengths 20,21,22,23, and 24, respectively.
|
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39, 1709-1713, (1993).
3. G.D. Forney, Jr., Dimension/length profiles and trellis complexity of linear block codes, IEEE Trans. Inform.
Theory 40, 1741-1752, (1994).
Projective Codes
113
4. V.K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37, 1412-1418,
(1991).
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(1972).
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