Bounds on Unique-Neighbor Codes

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.