On Bounded Weight Codes

The maximum size of a binary code is studied as a function of its length
n
, minimum distance
d
, and minimum codeword weight \ssi
w
. This function
B
(
n
,
d
,
w
) is first characterized in terms of its exponential growth rate in the limit
n
→∞ for fixed δ =
d
/
n
and ω =
w
/
n
. The exponential growth rate of
B
(
n
,
d
,
w
) is shown to be equal to the exponential growth rate of
A
(
n
,
d
) for 0 ≤ ω ≤ 1/2, and equal to the exponential growth rate of
A
(
n
,
d
,
w
) for 1/2 <; ω ≤ 1. Second, analytic and numerical upper bounds on
B
(
n
,
d
,
w
) are derived using the semidefinite programming (SDP) method. These bounds yield a nonasymptotic improvement of the second Johnson bound and are tight for certain values of the parameters.