--- title: 'Monotone Erasure Codes: Thresholds & Applications' url: https://www.emergentmind.com/topics/monotone-erasure-codes type: topic --- # Monotone Erasure Codes: Thresholds & Applications In the cited literature, “monotone erasure codes” refers to settings in which recoverability or non-recoverability is monotone in an erasure pattern. On the binary erasure channel (BEC), the key object is the monotone set of erasure patterns that prevent maximum a posteriori (MAP) recovery of a bit; combined with transitivity, double transitivity, EXIT identities, and sharp-threshold theorems, this yields capacity results for deterministic linear codes such as Reed–Muller, affine-invariant, and BCH families [1601.04689]. In distributed systems, the term also denotes erasure codes for arbitrary monotone access structures, including linear monotone erasure codes (LMECs) whose reconstruction condition is expressed as a rank constraint on node-assigned generator columns [2605.22426]. Related variants appear in hierarchical erasures over extension fields and in ordered deletion–erasure channels, where monotonicity is imposed by left-justified or ordered corruption patterns rather than by subsets of erased coordinates [1802.02265] [1806.07848]. ## 1. Monotonicity on the binary erasure channel The BEC has input alphabet $X=\{0,1\}$ and output alphabet $Y=\{0,1,*\}$; each transmitted bit is erased independently with probability $\varepsilon$ and otherwise received without error, and its capacity is $C(\varepsilon)=1-\varepsilon$ [1601.04689]. For a binary linear code $C\subseteq\{0,1\}^n$ and a uniform prior on codewords, MAP decoding on the BEC reduces to unique recoverability from the linear equations induced by the unerased positions. Block-MAP seeks a unique codeword, whereas bit-MAP for coordinate $i$ seeks a unique value of $X_i$ from the extrinsic observation $Y_{\sim i}$. The monotone structure is encoded by the failure set $$ \Omega_i \triangleq \Big\{ A \subseteq [N]\setminus\{i\} \mid \exists\, B\subseteq A,\; B\cup\{i\}\in C \Big\}. $$ Equivalently, in the formulation using erasure indicators $\omega\in\{0,1\}^{n-1}$, $$ \Omega_i=\{\omega\in\{0,1\}^{n-1}: \exists\, c\in C \text{ with } c_i=1 \text{ and } c_{\sim i}\preceq \omega\}. $$ If an erasure pattern prevents recovery of bit $i$, any superset of erasures also prevents recovery: adding erasures cannot help the decoder. Thus $\Omega_i$ is an increasing set, and its indicator is a monotone Boolean function of the erasure pattern. With the product measure $\mu_p$ on $\{0,1\}^{n-1}$, the bit-MAP EXIT function satisfies $$ h_i(p)=H(X_i\mid Y_{\sim i})=\sum_{A\in\Omega_i} p^{|A|}(1-p)^{N-1-|A|}=\mu_p(\Omega_i'). $$ On the BEC, this is exactly the probability that bit $i$ is not recoverable from the extrinsic observation. The bit erasure probabilities obey $$ P_{b,i}(p)=p\,h_i(p), \qquad P_b(p)=p\,h(p), $$ because bit $i$ can remain unknown only when $Y_i$ is itself erased and the remaining observations fail to determine it. ## 2. Symmetry, EXIT functions, and sharp thresholds The threshold mechanism depends not only on monotonicity but also on code symmetry. For a linear code of length $N$, the vector EXIT function is $$ h_i(\vec p)\triangleq H\{X_i\mid Y_{\sim i}(\vec p_{\sim i})\}, $$ with scalar specialization $h_i(p)$ at $\vec p=(p,\dots,p)$ and average EXIT $$ h(p)\triangleq \frac{1}{N}\sum_{i=1}^N h_i(p). $$ If the permutation group of the code is transitive, then all bits have identical EXIT functions and $$ h(p)=h_i(p). $$ If the permutation group is doubly transitive, then for distinct $i,j,k$, $$ \frac{\partial h_i(\vec p)}{\partial p_j}\Big|_{\vec p=(p,\dots,p)} = \frac{\partial h_i(\vec p)}{\partial p_k}\Big|_{\vec p=(p,\dots,p)}, $$ so the influences of the coordinates are equal at symmetric points [1601.04689]. The relevant analytic tool is the Margulis–Russo lemma: $$ \frac{d}{dp}\mu_p(\Omega)=I^{(p)}(\Omega), $$ where $I^{(p)}(\Omega)$ is the total influence. For $\Omega_i'$, this yields $$ \frac{d h_i(p)}{d p} = \sum_{j\in[N]\setminus\{i\}} \frac{\partial h_i(\vec p)}{\partial p_j}\Big|_{\vec p=(p,\dots,p)}. $$ Under equal influences, the symmetric sharp-threshold theorem gives a universal constant $C\ge 1$ such that $$ \frac{d}{dp}\mu_p(\Omega)\ge C(\log M)\mu_p(\Omega)(1-\mu_p(\Omega)), $$ and therefore, for any $0<\delta\le 1/2$, $$ p_{1-\delta}-p_\delta \le \frac{2}{C}\, \frac{\log\!\big(\frac{1-\delta}{\delta}\big)}{\log M}, \qquad p_t\triangleq h^{-1}(t)=\inf\{p\in[0,1]\mid h(p)\ge t\}. $$ For code sequences with doubly transitive permutation groups, $M=N-1$, so the EXIT transition width shrinks like $1/\log N$. The area theorem fixes the transition point: $$ \int_0^1 h(p)\,dp=\frac{K}{N}=R. $$ As the EXIT function becomes step-like, the area constraint forces the jump to occur at $$ \varepsilon^\ast=1-R. $$ In the formulation of Proposition 7, bit-MAP capacity, pointwise convergence of $h^{(n)}(p)$ to a step function, and vanishing threshold width are equivalent. The 2015 arXiv formulation already emphasized the same chain: monotone EXIT functions, 2-transitivity, and the Area Theorem pin the threshold of Reed–Muller codes at channel capacity under bit-MAP decoding [1505.05831]. ## 3. Capacity-achieving code families on erasure channels For Reed–Muller codes, $$ \mathrm{RM}(m,r):\quad n=2^m,\qquad k=\sum_{i=0}^r \binom{m}{i},\qquad d_{\min}=2^{m-r}, $$ with rate $$ \frac{k}{n} = 2^{-m}\sum_{i=0}^r \binom{m}{i}. $$ Choosing $$ r \approx \frac{m}{2}+\frac{\sqrt m}{2}Q^{-1}(1-R), \qquad Q(t)=\frac{1}{\sqrt{2\pi}}\int_t^\infty e^{-\tau^2/2}\,d\tau, $$ gives asymptotic rate $R$ [1601.04689]. Reed–Muller codes are affine-invariant; their automorphism group contains $\mathrm{AGL}(m,2)$, which acts doubly transitively on coordinates. By Theorem 3, any sequence with $N=2^m\to\infty$ and rate $R\in(0,1)$ achieves BEC capacity under bit-MAP, with threshold at $\varepsilon^\ast=1-R$. For block-MAP, the argument requires an additional conversion from vanishing bit erasure to vanishing block erasure. General inequalities are $$ P_b\le P_B\le N P_b, \qquad P_B\le \frac{N}{d_{\min}}P_b. $$ Theorem 4 uses the route $\log d_{\min}^{(n)}/\log N_n\to 1$, while Theorem 5 uses a stronger derivative lower bound on an interval $(a_n,b_n)$ with $a_n\to 0$, $b_n\to 1$, and $w_n\to\infty$. For Reed–Muller codes, Theorem 4 does not directly apply because the minimum distance is sublinear; instead, stronger symmetry on $\Omega_i'$ yields $$ \frac{d h_N(p)}{d p} \ge C\log(\log N)\log(N)\, h_N(p)(1-h_N(p)) $$ for $p\in(a_n,b_n)$, which implies block-MAP capacity via Theorem 5 [1601.04689]. The same symmetry-based framework extends to affine-invariant codes, hence to extended primitive narrow-sense BCH codes. The extended code $\mathrm{eBCH}(v,n)$ of length $N=2^n$ is affine-invariant and has minimum distance at least $v+1$; for the sequences considered in the paper, $$ d_{\min}^{(n)}\ge 1+\frac{N_n(1-R)}{n}, \qquad \lim_{n\to\infty}\frac{\log d_{\min}^{(n)}}{\log N_n}=1. $$ Therefore affine-invariance gives bit-MAP capacity, while the distance growth gives block-MAP capacity. Since BCH codes are cyclic, this yields sequences of primitive narrow-sense BCH codes that achieve capacity under both bit-MAP and block-MAP on the BEC, resolving affirmatively the existence of capacity-achieving sequences of binary cyclic codes [1601.04689]. A broader family is given by the dual Berman codes $C_n(r,m)$ and Berman codes $B_n(r,m)$ of length $N=n^m$. Their parameters are $$ \dim(C_n(r,m))=\sum_{w=0}^{r}\binom{m}{w}(n-1)^w, \qquad d(C_n(r,m))=n^{m-r}, $$ and $$ \dim(B_n(r,m))=\sum_{w=r+1}^{m}\binom{m}{w}(n-1)^w, \qquad d(B_n(r,m))=2^{r+1}. $$ When $n=2$, $C_2(r,m)$ is exactly $\mathrm{RM}(r,m)$. For $n\ge 3$, these codes are transitive but not doubly transitive; nevertheless, by using the Kumar–Calderbank–Pfister result on transitive codes whose stabilizer orbits grow to infinity, the paper proves BEC capacity under bit-MAP for sequences with target rate $R^\ast\in(0,1)$ [2202.09981]. For odd $n$, the same work identifies a larger abelian family of ideals in $\mathbb{F}_2[G^m]$ that also achieves BEC capacity under bit-MAP. ## 4. Linear monotone erasure codes for arbitrary access structures In the distributed-systems formulation, an access structure is a monotone family $\Gamma\subseteq 2^{[n]}$ of authorized node sets: if $S\in\Gamma$ and $S\subseteq T$, then $T\in\Gamma$. A compact specification is a monotone Boolean formula $\phi(v_1,\dots,v_n)$ using only AND, OR, and threshold operators $\Theta_k^m$. A monotone erasure code for $\Gamma$ consists of - $\mathrm{Encode}: F\to G_1\times\cdots\times G_n$, - $\mathrm{Decode}: G_1\times\cdots\times G_n\to F\cup\{\bot\}$, with the correctness condition that for every authorized $S\in\Gamma$, decoding from the fragments in $S$ recovers the file [2605.22426]. There is no privacy requirement; privacy is the domain of secret sharing rather than of MECs. A basic construction follows the access tree. At an OR node, the whole input is assigned to each child. At a $\Theta_t^r$ node, the input is split into $r$ subfragments so that any $t$ suffice, for example by an $[r,t]$ MDS code, and the recursion continues. At a leaf labeled by node $i$, the assigned chunk is appended to node $i$’s fragment. This construction is complete for the access structure, though it does not necessarily minimize overhead. The linear version fixes a finite field $\mathbb{F}_q$, a full-rank generator matrix $G\in\mathbb{F}_q^{k\times m}$, and an assignment $\varphi:[m]\to[n]$ from columns to nodes. For node $i$, the fragment is $$ g_i=x^\top G_{p_i}\in\mathbb{F}_q^{m_i}, $$ where $G_{p_i}$ is the submatrix of columns assigned to $i$ and $m_i$ is the number of those columns. The total coded length is $m=\sum_i m_i$, the overhead is $$ \beta=\frac{m-k}{k}, $$ and the rate is $R=k/m$. The reconstruction theorem is purely algebraic: a subset $S\subseteq[n]$ can reconstruct if and only if $$ \operatorname{rank}(G_S)=k. $$ The paper gives three principal LMEC constructions. First, for a general access tree, it composes small Vandermonde blocks and Kronecker products. If $r_{\max}$ is the maximum number of children of any nontrivial threshold node, choosing a prime power $q\ge \max\{2,r_{\max}\}$ suffices. If $V$ is the set of internal vertices whose children are leaves, and $$ \psi_v=\frac{y_v}{\prod_{u\in R_v} x_u}, $$ then the construction has $$ k=\operatorname{lcm}(\text{denominators of }\psi_v), \qquad m=\Big(\sum_{v\in V}\psi_v\Big)k, $$ so $$ \beta=\sum_{v\in V}\psi_v-1. $$ Second, when the minimal authorized sets $A_1,\dots,A_\omega$ are listed explicitly, optimal LMECs are obtained from an $[m,k]$ MDS base code by solving $$ \min \sum_{i=1}^{n} y_i \quad \text{subject to } \Gamma y^\top \ge 1_\omega^\top,\quad y\ge 0. $$ From an optimal rational solution, one sets $k$ to the least common multiple of the denominators, chooses $m_i=y_i k$, and labels $m_i$ columns of the base MDS code by node $i$. The overhead becomes $$ \beta=\sum_{i=1}^{n} y_i-1. $$ Third, for partitioned access structures, a dynamic program $\mathrm{FA}(T,L)$ computes optimal parameters bottom-up. In the threshold special case $\Gamma=\{S\subseteq[n]: |S|=w\}$, the optimal LMEC is the standard $[n,k]$ MDS code with $k=w$ and one column per node; any $w$ nodes reconstruct by solving a $k\times k$ linear system [2605.22426]. ## 5. Hierarchical and ordered erasure models Hierarchical erasures arise for linear codes over extension fields $\mathbb{F}_{q^\alpha}$ when each symbol is revealed progressively in a fixed basis order. Writing $$ x=\sum_{j=1}^{\alpha} x_j\beta_j, \qquad \phi(x)=(x_1,\dots,x_\alpha)\in\mathbb{F}_q^\alpha, $$ a hierarchical erasure pattern is a tuple $t=(t_1,\dots,t_n)$ with $0\le t_i\le \alpha$ and $\sum_i t_i\le m$, where the first $t_i$ coordinates of symbol $i$ are erased. The induced sets $$ E^{(j)}=\{i\in[n]: t_i\ge j\} $$ satisfy $$ E^{(1)}\subseteq E^{(2)}\subseteq \cdots \subseteq E^{(\alpha)}, $$ so the erasures are left-justified and monotone in the hierarchy index. The central correctability criterion is $$ C\cap X_t=\{0\}, $$ where $X_t$ is the $\,\mathbb{F}_q$-subspace of components hidden by the pattern. Lemma 1 states that a code is $T$-correcting if and only if this condition holds for every $t\in T$ [1802.02265]. This framework yields several constructions. Universally Decodable Matrices (UDMs) give $m$-correcting codes of dimension $n-m$ through trace-based parity checks. Balanced and power erasure patterns are handled by recursive-basis parity constructions when $\alpha=2^\beta$. Gabidulin codes provide a rank-metric route: for $\alpha\ge n$ and $0\le r