The Griesmer bound is a classical technique (developed in 1960) for estimating the minimum length n required for a binary linear code with a given dimension k and minimum distance d. In this article, a unified derivation of the Griesmer bound and two new variations on it are presented. The first variation deals with linear codes which contain the all-ones vector; such codes are quite common and are useful in practice because of their 'transparent' properties. The second variation deals with codes that are constrained to contain a word of weight greater than or equal to M. In both cases these constraints (the all-ones word or a word of high weight) can increase the minimum length of a code with given k and d

Mceliece, R. J.

Solomon, G.

English

NASA Technical Reports Server

N94-29618
TDA Progress Report 42-105 May 15, 1991
Modifications of the Griesmer Bound
R. J. McEliece1 and G. Solomon2
Communications Systems Research Section
The Griesmer bound is a classical technique (developed in 1960) for estimating
the minimum length n required for a binary linear code with a given dimension
k and minimum distance d. In this article, a unified derivation of the Griesmer
bound and two new variations on it are presented. The first variation deals with
linear codes which contain the all-ones vector; such codes are quite common and are
useful in practice because of their "transparent" properties. The second variation
deals with codes that are constrained to contain a word of weight > M. In both
cases these constraints (the all-ones word or a word of high weight) can increase the
minimum length of a code with given k and d.
I. Introduction and Review of the Classical
Griesmer Bound
The key notion for the Griesmer bound is what Solomon
and Stiffler [8] called puncturing. If x = (x i , . . . , x n ) is
a binary vector of length n and if / C {!,...,n}, the
/-puncturing of x is the vector obtained by deleting the
components of x indexed by 7. Thus, for example, the
{1,4} puncturing of (10101) is (Oil). Puncturing is thus
just a special kind of linear transformation, i.e., a projec-
tion onto certain coordinate positions, but here the tradi-
tional terminology will be retained. All of the results in
this article, old and new, are based on the following simple
combinatorial lemma.
Lemma. Let a = ( a \ , . . . ,an) be a fixed binary vector
of length n. If b — (61, . . . , 6n) is another binary vector of
1
 Consultant, California Institute of Technology, Engineering De-
partment.
2
 Consultant.
length n, let 6' be the vector obtained by puncturing 6 at
the positions where a,- = 1. Then
wt(6') = (i)
(2)
Proof: Without loss of generality, take
n—w w
a = ooooooo iTTmlT
6 = 0001111 11100000
a + 6 = 0001111 00011111
where w = wt(a). Then, if x = ( a r 1 , z 2 i - - • > ^ n ) is
vector of length n, then x' = ( x i , . . . , xn_«,). Similarly,
define the complementary puncturing of x—at the com-
ponents where a,- = 0 by x" = (xn-w+i,. . . , xn), so that
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wt(z) = wt(x') + wt(x") for any vector x. Applying this
rule to the second and third line of Eq. (2) yields, noting
that wt[(a + 6)'] = wt(fc') and wt[(a + 6)"] = w- wt(6"),
wt(6') + wt(6") = wt(6)
wt(6') + [w- wt(6")] = wt(a + 6)
is an [n - d,k - l,d'] code with d' > [d/2~\, and so its
length must be > n(k - 1, fd/2]). Hence, n(k,d) - d >
Corollary 1 (Griesmer, 1960).
n(M)>d+r<V2l + |"d/221 + "- + for k > 1
Adding these two equations, 2wt(6') = wt(6) + wt(a + b)
— wt(a), which is the same as Eq. (1). G
In the rest of the article, the MacWilliams-Sloane ([6],
Section 1.1) terminology of an [n,fc,d] code is used to de-
scribe a binary linear code with length n, dimension k, and
minimum distance d.
Theorem 1. Let C be an [n, k, d] code, and let a be a
. codeword of weight d. Let C' be the code obtained from
C by puncturing each codeword at the coordinates where
ai = 1. Then C' is an [n-d, k-1, d'] code with d' > |"<f/2] •
Proof: The code C" is by definition of length n — d,
since there are d punctured coordinates. To compute the
dimension of C", use the fact that the puncturing map-
ping P from C to C" is a linear transformation, so that
rank(P) + nullity(P) = dim((7) = k ([4], Theorem 3.1.3).
To find nullity (P), examine the set of codewords x 6 C
such that x' = 0. If x' is such a codeword, then the 1's of
x must be confined to the d coordinates where a is nonzero,
so that either x = 0 or wt(a + z) < d. But since x + a
is a codeword and d is the minimum weight of C, it fol-
lows that x + a = 0, i.e., x = a. Thus, there are just
two words in C that, when punctured, yield 0—0 and a,
and so nullity(/7) = 1, so that rank(P), i.e., the dimen-
sion of C', is one less than the dimension of C, i.e., k — 1.
Finally, if b is an arbitrary codeword of C not equal to
0 or a, wt(6 + a) > d = wt(a), and so by the Lemma,
wt(&') > fwt(6)/21 > \d/2}. Thus, every nonzero word in
C' has weight > fd/2]. O
Let n(k,d) be the minimum length of a binary code
with Hamming distance > d and dimension k. The original
Griesmer bound can now be stated and proven ([3] or [6],
p. 546).
Theorem 2 (Griesmer, 1960). If A: > 2, then
Proof: This follows from Theorem 2, combined with
the self-evident result that n(l, d) = d for all d > 1, with
mathematical induction, and that |"fx]/2] = fa;/2] (see
[1] or [5], solution to exercise 1.2.4, p. 476). D
II. The Griesmer Bound for Codes
Containing the All-Ones Word
In many applications, it is necessary to consider codes
that contain the all-ones vector, e.g., "transparent codes"
for synchronizing phase-shift-keyed-modulated data ([2],
Section 6.6.1), or for synthesizing good finite state-codes
[7]. It is therefore useful and interesting to study the pos-
sible loss in performance induced by requiring a code to
contain the all-ones vector. Thus let N(k,d) denote the
minimum length of a binary code with Hamming distance
> d and dimension k that contains the all-ones vector.
Theorem 3. If k > 1, then (cf. Theorem 2).
Proof: Let C be an [n, k, d] binary linear code contain-
ing the all-ones vector with n = N(k,d). Then the punc-
tured code C' described in Theorem 2 is an [n — d, k — l,d']
code that contains the all-ones vector (since puncturing
an all-ones vector leaves another all-ones vector) with
d' > I'd/2"! , and so its length must be > 'N(k -l,d- 1).
Hence, N(k,d) - d> N(k - l,\d/2]). O
Theorem 4. Both N(l,d) = d and N(2,d) = 2d for
all d > 1.
Proof: For the k = 1 result, take as the generator
matrix
G = ( l 1 1)
Proof: Let C be an [n, k, d] binary linear code with
n = n(k,d). Then the code C' described in Theorem 1
For the k = 2 result, note that an [n, 2,d] code with the
all-ones vector has a 2 x n generator matrix of the form
42
1 1 ••• 1 1'
G =
,0 0 1 1
Denote by no the number of columns of G of the form (J)
and by HI the number of columns of the form (*). Then,
since the code has minimum weight d, it must follow that
HI > d and no > d. Hence, n = no + n\ > 2d. On the
other hand, by taking no = d and n\ = d, one obtains a
[2d,2,d] code containing the all-ones vector. O
Theorem 5. If k > 3, then
N(k, d ) > d (3)
Proof: This follows by mathematical induction on k,
using Theorem 3 as the boundary value and Theorem 4
as the induction step, again with the help of the result
\\x]/2] = ["z/2~| cited above. O
Examples. Let k = 3 and d = 3. Then by the
Corollary 1 and Theorem 5, n ( 3 , 3 ) > 3 + 2 + l = 6 and
W(3,3) > 3 + 2 + 2 = 7. In both cases the bound is sharp,
since there is a [6,3,3] code, namely, a punctured [7,3,4]
simplex code with generator matrix
/110100\
011010
V001101/
and a [7,3,3] code with the all-ones word, namely,
G = 1000011
\ 0100101 /
Since Af(5,9.) > 9 + 5 + 3 + 2-2 = 21, there is no [20,5,9]
code with the all-ones word. There is, however, a [21,5,9]
code with the all-ones word, obtained from the [16,5,8]
biorthogonal code by repeating the information bits.
Theorem 6.
N(k',2) =
k+l if A; is odd
k + 2 if k is even
Proof: Since there is plainly no [fc , fc ,2] code, with or
without the all-ones word, it follows that N(k,2) > k + 1
for all k. The only [k + l ,fc, 2] code has the parity-check
matrix
This code contains the all-ones vector if and only if k
is odd, which proves that N(k,"2) = k + 1 if k is odd,
and N(k,2) > k + 2 if k is even. If k is even, there is a
[k+2, k, 2] code containing the all-ones word, with a parity-
check matrix (illustrated for k = 6)
11000000, /
so that N(k, 2) = k + 2 when k is even, as asserted.
Corollary 2.
>
k + 3 if k is even
k + 4 if k is odd
Proof: From Theorem 3, N(k,3) > 3 + N(k - 1,2).
The result now follows from Theorem 6. O
III. The Griesmer Bound for Codes
Containing a Word of Bounded Weight
As another variation on the Griesmer bound, let
N(k,d,M) denote the length of the shortest [n,k,d] bi-
nary linear code that contains a word of weight > M .
Theorem 7.
N(k,d,M) >
Proof: Let C be an [n, k, d] code containing a word of
weight > M . As in the proof of Theorem 2, consider the
code C', which is an [n - d, k - 1, d'} code with d' > [rf/2] .
Now let 6 be a word of weight > M in C. Then, by the
Lemma, wt(fc') > [wt(6)/2"| > [M/2]. Thus, C' is an
[n - d, k — 1, d'} code with d' > \d/1~\ containing a word of
weight > [M/2], i.e., n - d > N(k - 1, fd/2], [M/2]).
43
Theorem 8. If M > d and k > 2,
N (k, d ,M)>d
Proof: This follows from Theorem 3 and the boundary
value N(l,d,M) = max(M,d). O
Example. According to Theorem 5, n(3,4) > 7, and
there is a [7,3,4] code, i.e., the simplex code. However,
this code has words only of weight 4. If one looks for a
[7,3,4] code with a word of weight 5 or more, an appeal to
Theorem 8 shows that #(3,4,5) > 4+ [4/21 + [5/41 = 8.
There is an [8,3,4] code with a word of weight 6, namely,
the code with generator matrix
G = 00001111
\11001010/
but it is unknown whether there is an [8,3,4] code with a
word of weight 5.
References
[1] L. D. Baumert and R. J. McEliece, "A Note on the Griesmer Bound," IEEE
Trans. Inform. Theory, vol. IT-19, pp. 134-135, January 1973.
[2] G. C. Clark, Jr. and J. B. Cain, Error-Correction Coding for Digital Communi-
cations, New York: Plenum Press, 1981.
[3] J. H. Griesmer, "A Bound for Error-Correcting Codes," IBM J. Res. Develop.,
vol. 4, pp. 532-542, 1960.
[4] K. Hoffman and R. Kunze, Linear Algebra, Englewood Cliffs, New Jersey:
Prentice-Hall, 1961.
[5] D. E. Knuth, The Art of Computer Algorithms, vol. 1, Reading, Massachusetts:
Addison-Wesley, p. 476, 1973.
[6] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes,
Amsterdam: North-Holland, 1977.
[7] F. Pollara, R. J. McEliece, and K. Abdel-Ghaffar, "Finite-State Codes," IEEE
Trans. Inform. Theory, vol. IT-34, pp. 1083-1088, September 1988.
[8] G. Solomon and J. J. Stiffler, "Algebraically Punctured Cyclic Codes," Informa-
tion and Control, vol. 8, pp. 170-179, 1965.
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