Spectral Distribution of Random Matrices From Binary Linear Block Codes

Let
C
be a binary linear block code of length
n
, dimension
k
and minimum Hamming distance
d
over GF(2)
n
. Let
d
^⊥
denote the minimum Hamming distance of the dual code of
C
over GF(2)
n
. Let ε:GF(2)
n
→{-1,1}
n
be the component-wise mapping ε(
vi
):=(-1)
vi
, for
v
=(
v
₁
,
v
₂
,...,
vn
) ∈ GF(2)
n
. Finally, for
p
<;
n
, let \mmbΦ
C
be a p × n random matrix whose rows are obtained by mapping a uniformly drawn set of size
p
of the codewords of
C
under ε. It is shown that for
d
^⊥
large enough and
y
:=
p
/
n
∈ (0,1) fixed, as
n
→∞ the empirical spectral distribution of the Gram matrix of [1/(√
n
)]\mmbΦ
C
resembles that of a random i.i.d. Rademacher matrix (i.e., the Marchenko-Pastur distribution). Moreover, an explicit asymptotic uniform bound on the distance of the empirical spectral distribution of the Gram matrix of [1/(√
n
)]\mmbΦ
C
to the Marchenko-Pastur distribution as a function of
y
and
d
^⊥
is presented.