Free Expander Walks in Balanced Code Constructions

• Free expander walks are non-stationary Markovian processes that traverse diverse expander graphs to achieve superior mixing and pseudorandomness.
• The method leverages all-signings near-Ramanujan expander families, ensuring minimal adversarial bias alignment and robust spectral contraction.
• This technique yields nearly optimal ε-balanced binary codes with rate Ω(ε^(2+o(1))) by efficiently contracting bias over repeated walks.

A free expander walk is a non-stationary Markovian traversal on a sequence of distinct expander graphs, designed so that each step uses a different expander from a predetermined family. Unlike classical expander walks that repeatedly apply the same graph, free expander walks leverage spectral properties of multiple expanders to achieve superior mixing, bias contraction, and pseudorandomness amplification. The core technical innovation is the use of "all-signings near-Ramanujan" families, which ensure that adversarial bias-vectors cannot persistently align with the spectral structure of every expander in the sequence. Free expander walks yield nearly optimal constructions for $\varepsilon$-balanced codes, matching the Gilbert-Varshamov bound in the low-rate, high-distance regime and providing a conceptually and technically streamlined alternative to wide-replacement product constructions (Hsieh et al., 18 Jan 2026).

1. Classical Versus Free Expander Walks

Classical expander walks operate on a single $d$-regular graph $G$ with normalized adjacency matrix $A_G = (1/d)\cdot \operatorname{Adj}(G)$. The spectral gap $1 - \lambda(G)$ governs mixing; after $\ell$ steps the deviation from uniformity contracts as $$\|\pi^\ell - \text{uniform}\|_2 \leq \lambda(G)^\ell \|\pi^0 - \text{uniform}\|_2$$. In contrast, a free expander walk fixes a sequence of $d_i$-regular graphs $G_1, G_2, ..., G_t$ on a common vertex set $V$, each with normalized adjacency $A_i$ and second eigenvalue $\lambda_i$. At step $s$, a move is taken according to $G_{W[s]}$, where $W \in [t]^\ell$ is a walk schedule. After $\ell$ steps, the distribution is $$\pi^\ell = \pi^0 A_{W[1]} A_{W[2]} \cdots A_{W[\ell]}$$. The spectral mixing bound generalizes: $$\|\pi^\ell - \text{uniform}\|_2 \leq \prod_{s=1}^\ell \lambda_{W[s]} \|\pi^0 - \text{uniform}\|_2$$.

Where classical walks have contraction strictly controlled by powers of a single $\lambda$, free walks can dramatically improve contraction when the adjacency matrices are sufficiently "incoherent." Specifically, the product of the individual $\lambda_i$ can be much smaller than $\lambda(G)^\ell$ for comparable walk length.

2. Construction of All-Signings Near-Ramanujan Expander Families

Optimal performance of free expander walks requires sequences of expanders with strong spectral incoherence. This is realized via the all-signings near-Ramanujan family construction of O'Donnell–Wu [OW20]. Formally, for $V$ of size $n$, a collection $\{H_j: j = 1, ..., t\}$ of $d$-regular graphs is called all-signings near-Ramanujan if:

• For each $j$, $\|A_{H_j}\|_{1^{\perp}} \leq 2/\sqrt{d}$.
• For every signing vector $\sigma \in \{\pm1\}^t$, the signed sum $H(\sigma) = \sum_{j=1}^t \sigma_j H_j$ yields a $d t$-regular (multi)graph with adjacency matrix $A_{H(\sigma)}$ satisfying $\|A_{H(\sigma)}\|_{1^{\perp}} \leq 2/\sqrt{d t}$.

This construction guarantees that for any fixed bias-vector, coherence with the expander family is limited, enforcing strong contraction for most sequences and ensuring exponentially small bias in code applications. Explicit constructions for such families are known for all $n$, $t$, $d \geq 2$.

3. Operator Norm Decomposition and Spectral Mixing Bounds

Let $D_x = \operatorname{diag}(x)$ encode the sign-vector $x$ (with coordinates in $\{\pm1\}$). For a word $w = (w_1, ..., w_\kappa) \in [t]^\kappa$, define the linear operator $M_w = A_{w_\kappa} D_x \cdots A_{w_1} D_x$. In coding-theoretic applications, $x$ depends on the underlying codeword.

For any $\ell$, schedule $W$ of length $\ell$, and $x \neq 0$, the bias is bounded as

$$\text{bias}(f_W(x)) = |\langle 1, D_x A_{W[1]} ... D_x A_{W[\ell]} D_x 1 \rangle| \leq \|M_W\|_{\ell_2 \to \ell_2}.$$

The operator norm splits over the constant part $1$ and its orthogonal complement $1^{\perp}$. For appropriately chosen schedule and expander family, $\|M_W\|$ can be bounded by products of individual $\lambda_i$, up to combinatorial factors reflecting adversarial alignments.

When averaging over sequences, a key lemma asserts: for any fixed $\delta > 0$, a random $w \in [t]^\kappa$ satisfies $\|M_w\| \leq C ( \max_j \lambda_j)^{\kappa-1 }$ except with probability $\delta$, where $C$ grows only exponentially in $\kappa$ and polynomially in $1/\delta$. This ensures that most schedules are contractive, yielding efficient bias amplification.

4. Lemmas and Theorems Governing Bias Contraction

Primary results formalize that almost all sequences are contractive:

• Lemma 5.1: For $\kappa \geq 1$, large $t$, and $\lambda \leq 1/2$, a random $w \in [t]^\kappa$ satisfies

$\|M_w\| \leq 2^{\kappa + 1} \lambda^{\kappa - 1}$

with failure probability at most $\kappa^2 / t^{1/4}$.

• Lemma 5.2: For contractive $w$,
  • (a) $\|M_w 1\|_2 \leq (2\lambda)^\kappa$,
  • (b) $$\|M_w v\|_2 \leq 2^\kappa \lambda^{\kappa-1} \|v\|_2$$ for all $v \perp 1$,
  • Together, $\|M_w\| \leq 2^{\kappa+1}\lambda^{\kappa-1}$.
• Main Theorem 5.3: If $C_0 \subseteq \{\pm1\}^{n_0}$ is a base code of bias $\varepsilon_0 \asymp 1/\sqrt{d}$ and rate $r_0 = \operatorname{poly}(1/\varepsilon_0)$, concatenating via a free expander walk of total length $\ell = R\kappa$ (for $R = t^\kappa$, $$\kappa \approx \frac{\log\log 1/\varepsilon}{\log\log\log 1/\varepsilon}$$) yields a binary linear code $C \subseteq \{\pm1\}^n$ of rate $\Omega(\varepsilon^{2 + o(1)})$ and bias at most $\varepsilon$. The bias of every nontrivial codeword $x$ is contracted by $(2^{\kappa+1}\lambda^{\kappa-1})^R$, which rapidly becomes subpolynomial in $1/\varepsilon$.

5. Worked Example: Explicit Parameters

Consider $d = 3$, $t = 5$, $\kappa = 2$:

• Let $V = \{1, \dots, n_0\}$ and choose 5 distinct 3-regular graphs $H_1, ..., H_5$ on $V$ with $\lambda(H_j) \leq 2/\sqrt{3} < 1/2$.
• With $\kappa = 2$, schedule $W^*$ comprises all $5^2 = 25$ words of length 2 over $\{1, ..., 5\}$, repeated $R = 3$ times (for $\ell = 75$).
• For base codeword $x$ with bias $\varepsilon_0 \leq 1/2$, define $M_{(i,j)} = A_j D_x A_i D_x$. By Lemma 5.1, almost all $M_{(i,j)}$ satisfy $\|M_{(i,j)}\| \leq 8 (1/2) = 4$.
• Each block (length 2) reduces bias by at least factor 4, so over $R = 3$ repetitions, total bias contracts to at most $4^{75}$.
• Lifting back yields binary codes of relative distance near $1/2$ and rate near $\varepsilon^2$. For practical parameters, choose $d \gg 3$ so $\lambda \ll 1$, and take $\kappa = O(\log\log 1/\varepsilon)$ and $R = O(\log 1/\varepsilon)/\kappa$, ensuring code rate $\Omega(\varepsilon^{2+o(1)})$ and exponentially small bias.

6. Integration into $\varepsilon$-Balanced Code Construction

The construction proceeds systematically:

1. Base code: Select explicit linear code $C_0 \subseteq \{\pm1\}^{n_0}$ with bias $\leq \varepsilon_0 = 2/\sqrt{d}$, rate $r_0 = \operatorname{poly}(1/\varepsilon_0)$.
2. Expander family: Use O'Donnell–Wu's all-signings near-Ramanujan family $\{H_1, ..., H_t\}$ of $d$-regular graphs on $n_0$ vertices.
3. Schedule: Specify $\kappa, R$; enumerate all words in $[t]^\kappa$, repeated $R$ times to form $W^*$.
4. Lift: Define mapping $f_{W^*}(x)$, labeling each coordinate by the walk product of $x$ along the sequence. Resulting code $C = f_{W^*}(C_0)$ in $\{\pm1\}^{n_0 d^\ell}$.
5. Bias analysis: Equations and lemmas guarantee bias of each nonzero $x \in C_0$ is reduced to at most $\varepsilon$.
6. Rate: Mapping is linear, so $C$ has rate $r_0/(d^\ell)$. Selection of $d, \kappa, R$ yields $\ell = \varepsilon^{-o(1)}$, so rate is $\Omega(\varepsilon^{2 + o(1)})$.

This framework demonstrates how free expander walks give nearly optimal $\varepsilon$-balanced codes by strategically exploiting spectral incoherence across a sequence of expanders rather than via construction of highly complex single expanders (Hsieh et al., 18 Jan 2026).