  Let $A$ be an $m\times n$ parity-check matrix of a linear binary code of
length $n$, rate $R$, and distance $\delta n$. This means that for every
$0<k<\delta n$, every $m\times k$ submatrix of $A$ has a row of odd weight.
Message-passing decoding algorithms require the stronger unique-neighbor
property. Namely, that every such submatrix has a row of weight $1$. This
concept arises naturally in the context of efficient decoding of LDPC expander
codes as well as in the study of codes for the binary erasure channel, where
$A$ is said to have stopping distance $\delta n$. It is well known that if
$\frac12\le\delta$, then $R=o_n(1)$ whereas for every $\delta<\frac12$ there
exist linear codes of length $n\to\infty$ and distance $\ge\delta n$ with $R$
bounded away from zero. We believe that the unique-neighbor property entails
sharper upper bounds on the rate. Concretely, we conjecture that for a proper
choice of $f(m) = o(m)$ and some positive constant $\epsilon_0$, every
$m\times(m+f(m))$ binary matrix has an $m\times m'$ submatrix with
$0<m'\le(\frac12-\epsilon_0)m$ where no row has weight $1$. In other words,
that every linear code of non-vanishing rate has a normalized stopping distance
of at most $\frac12-\epsilon_0$. We make several contributions to the study of
this conjecture. Concretely, we (1) prove the conjecture for sufficiently dense
matrices (2) find tight upper bounds for the stopping distance of matrices in
standard form, and prove the conjecture in this special case (3) show that the
conjecture can hold only if $f(t)\ge\log_2(t)$ (4) find tight upper bounds for
both distance and stopping distance of matrices where $n-m$ is small, and
provide minimal $(m,n)$ where the upper bound on the stopping distance is
strictly smaller than that of the distance of such matrices.
