Bounds on Unique-Neighbor Codes
Nati Linial, Edan Orzech
TL;DR
This work investigates the unique-neighbor property for binary parity-check matrices and its impact on the rate–distance tradeoffs for linear codes. By formalizing $u(A)$ (minimum size of a nonempty $1$-free set) and $ε(A)$ (minimum nonempty even set), the authors study asymptotic bounds $R_U(δ)$ and $R_L(δ)$, conjecturing a strict bound gap with $δ_U< frac{1}{2}$ while $δ_L= frac{1}{2}$. They prove several results: high row weights enforce stronger bounds; standard-form matrices satisfy $u_I(n,n+k)\le rac{n}{H_k}+k$ with tight behavior for fixed $k$; and constructive families like $U_k$ illuminate when $u$ and $ε$ coincide or differ. They also show sub-logarithmic slack can invalidate the conjecture and characterize exact values of $u$ and $ε$ for bounded row/column regimes, advancing understanding of stopping sets, LDPC decoding, and unique-neighbor expanders. Overall, the paper provides both theoretical bounds and explicit constructions that sharpen the limitations on unique-neighbor codes and guide future explorations of the rate–distance landscape under stronger neighbor constraints.
Abstract
Recall that a binary linear code of length $n$ is a linear subspace $\mathcal{C} = \{x\in\mathbb{F}_2^n\mid Ax=0\}$. Here the parity check matrix $A$ is a binary $m\times n$ matrix of rank $m$. We say that $\mathcal{C}$ has rate $R=1-\frac mn$. Its distance, denoted $δn$ is the smallest Hamming weight of a non-zero vector in $\mathcal{C}$. The rate vs.\ distance problem for binary linear codes is a fundamental open problem in coding theory, and a fascinating question in discrete mathematics. It concerns the function $R_L(δ)$, the largest possible rate $R$ for given $0\leδ\le1$ and arbitrarily large length $n$. Here we investigate a variation of this fundamental question that we describe next. Clearly, $\mathcal{C}$ has distance $δn$, if and only if for every $0<n'<δn$, every $m\times n'$ submatrix of $A$ has a row of odd weight. Motivated by several problems from coding theory, we say that $A$ has the unique-neighbor property with parameter $δn$, if every such submatrix has a row of weight $1$. Let $R_U(δ)$ be the largest possible asymptotic rate of linear codes with a parity check matrix that has this stronger property. Clearly, $R_U(\cdot),R_L(\cdot)$ are non-increasing functions, and $R_U(δ)\le R_L(δ)$ for all $δ$. Also, $R_U(0)=R_L(0)=1$, and $R_U(1)=R_L(1)=0$, so let $0\leδ_U \leδ_L\le1$ be the smallest values of $δ$ at which $R_U$ resp.\ $R_L$ vanish. It is well known that $δ_L=\frac12$ and we conjecture that $δ_U$ is strictly smaller than $\frac12$, i.e., the rate of linear codes with the unique-neighbor property is more strictly bounded. While the conjecture remains open, we prove here several results supporting it. The reader is not assumed to have any specific background in coding theory, but we occasionally point out some relevant facts from that area.