Linear Monotone Erasure Codes

• Linear monotone erasure codes are defined by a full-rank matrix and a node labeling scheme that ensures file reconstruction when the rank condition is met.
• They generalize MDS threshold codes by supporting non-threshold recoverability and heterogeneous fragment allocation under arbitrary trust assumptions.
• Recursive and LP-based constructions balance storage overhead with recovery guarantees, extending the concept to erasure channels via monotone set analysis.

Linear monotone erasure codes occupy a junction of coding theory, access-structure design, and monotone-threshold analysis. In the formal model introduced for arbitrary trust assumptions, an $[m,k]$-linear monotone erasure code over $\mathbb{F}_q$ is specified by a full-rank matrix $G\in \mathbb{F}_q^{k\times m}$ together with a labeling of columns to nodes, so that a file $f\in \mathbb{F}_q^k$ is encoded as $fG$ and a node set reconstructs exactly when the columns assigned to that set have rank $k$ (Bammert et al., 21 May 2026). A complementary line of work on the binary erasure channel uses linearity to represent decoding failure as an upward-closed family of erasure patterns, turning EXIT functions and block/bit failure events into monotone Boolean quantities amenable to sharp-threshold analysis (Kudekar et al., 2015). The subject therefore includes both a concrete non-threshold coding primitive and a structural viewpoint in which erasure decoding of linear codes is analyzed through monotonicity.

1. Formal definition and access-structure model

Let $\mathcal{P}=\{p_1,\dots,p_n\}$ be a set of nodes. An access structure $\mathcal{A}$ on $\mathcal{P}$ is a collection of subsets of $\mathcal{P}$ such that no set is contained in another, and each set $\mathbb{F}_q$0 is an access set. Monotonicity is semantic rather than explicit in the representation: if an access set can reconstruct, then every superset can also reconstruct, so only minimal sufficient sets need be listed (Bammert et al., 21 May 2026).

A monotone erasure code for $\mathbb{F}_q$1 consists of an encoder and decoder with fragment space $\mathbb{F}_q$2, where $\mathbb{F}_q$3 is the fragment domain of node $\mathbb{F}_q$4. Completeness requires that for every $\mathbb{F}_q$5, the fragments $\mathbb{F}_q$6 reconstruct the file. The model explicitly allows $\mathbb{F}_q$7, so a node may store nothing. Unlike secret sharing, the requirement is reconstructability rather than secrecy: a non-access set may still happen to reconstruct (Bammert et al., 21 May 2026).

In the linear specialization, the file is a vector

$\mathbb{F}_q$8

the total stored length is $\mathbb{F}_q$9, and the code is defined by a full-rank matrix

$G\in \mathbb{F}_q^{k\times m}$0

and a labeling function

$G\in \mathbb{F}_q^{k\times m}$1

If $G\in \mathbb{F}_q^{k\times m}$2 is the submatrix of columns assigned to node $G\in \mathbb{F}_q^{k\times m}$3, then encoding is

$G\in \mathbb{F}_q^{k\times m}$4

For a node set $G\in \mathbb{F}_q^{k\times m}$5, let $G\in \mathbb{F}_q^{k\times m}$6 be the matrix formed by all columns assigned to nodes in $G\in \mathbb{F}_q^{k\times m}$7. The central characterization is that $G\in \mathbb{F}_q^{k\times m}$8 is sufficient if and only if

$G\in \mathbb{F}_q^{k\times m}$9

This rank criterion is the exact non-threshold generalization of the reconstruction rule for MDS threshold codes (Bammert et al., 21 May 2026).

Classical threshold erasure codes arise as the special case

$f\in \mathbb{F}_q^k$0

with one column per node. In this sense, linear monotone erasure codes strictly generalize Reed–Solomon and other MDS threshold codes. Storage overhead is measured by

$f\in \mathbb{F}_q^k$1

and in the linear setting reduces to

$f\in \mathbb{F}_q^k$2

Thus the formal theory treats non-threshold recoverability and heterogeneous fragment allocation as first-class objects rather than as perturbations of the $f\in \mathbb{F}_q^k$3-out-of-$f\in \mathbb{F}_q^k$4 model (Bammert et al., 21 May 2026).

2. Construction paradigms and storage overhead

A basic recursive construction starts from a monotone Boolean formula or access tree. AND nodes split data among children, while OR nodes replicate data to each child. In the non-linear version, if a node is an AND with $f\in \mathbb{F}_q^k$5 children, the assigned string is divided into $f\in \mathbb{F}_q^k$6 chunks; if a node is an OR, the same string is passed to every child. This gives a direct correctness proof by structural recursion, but it can incur large overhead and is not inherently linear over a finite field (Bammert et al., 21 May 2026).

The main linear recursive construction operates on access trees whose internal vertices are threshold gates. For a node with threshold $f\in \mathbb{F}_q^k$7 and $f\in \mathbb{F}_q^k$8 child subtrees $f\in \mathbb{F}_q^k$9, the algorithm first recursively computes $fG$0 for each child. It then sets

$fG$1

lifts each child matrix via

$fG$2

chooses a $fG$3 Vandermonde matrix $fG$4, and forms

$fG$5

The resulting subtree parameters are

$fG$6

With

$fG$7

the construction is complete and has polynomial complexity in the size of the access tree (Bammert et al., 21 May 2026).

The same framework yields an explicit overhead formula. If $fG$8 is the set of vertices whose children are leaves, $fG$9 is the path from the root to $k$0, $k$1 is the threshold label of $k$2, and $k$3 is its number of children, then

$k$4

the matrix has

$k$5

rows and

$k$6

columns, so

$k$7

This makes the tradeoff between access-tree structure and storage overhead explicit (Bammert et al., 21 May 2026).

An overhead-optimal construction instead reduces the problem to fractional fragment allocation over an $k$8-MDS base code. If $k$9 and $\mathcal{P}=\{p_1,\dots,p_n\}$0 is the incidence matrix of access sets, the optimization problem is

$\mathcal{P}=\{p_1,\dots,p_n\}$1

With $\mathcal{P}=\{p_1,\dots,p_n\}$2 equal to the least common multiple of the denominators of an optimal rational solution and $\mathcal{P}=\{p_1,\dots,p_n\}$3, assigning $\mathcal{P}=\{p_1,\dots,p_n\}$4 columns of an $\mathcal{P}=\{p_1,\dots,p_n\}$5-MDS code to node $\mathcal{P}=\{p_1,\dots,p_n\}$6 yields

$\mathcal{P}=\{p_1,\dots,p_n\}$7

The construction is presented as optimal in storage overhead, and if

$\mathcal{P}=\{p_1,\dots,p_n\}$8

then

$\mathcal{P}=\{p_1,\dots,p_n\}$9

asymptotically for large $\mathcal{A}$0 (Bammert et al., 21 May 2026).

For partitioned access structures, where every atom appears at most once in the access tree, the paper gives a dynamic-programming-style algorithm FA($\mathcal{A}$1) with running time

$\mathcal{A}$2

At a threshold-$\mathcal{A}$3 parent with child costs $\mathcal{A}$4, sorted increasingly, the algorithm chooses

$\mathcal{A}$5

or $\mathcal{A}$6 otherwise, and computes optimal $\mathcal{A}$7 for the resulting linear monotone erasure code. This gives an efficient optimal construction for an important hierarchical class of non-threshold trust models (Bammert et al., 21 May 2026).

3. Monotone-set representation on erasure channels

On the binary erasure channel, linearity induces a different but closely related monotone structure. For a binary linear code $\mathcal{A}$8 of length $\mathcal{A}$9 and a fixed bit $\mathcal{P}$0, the bit-MAP failure event can be written as a set of erasure patterns on the other $\mathcal{P}$1 coordinates. In subset notation,

$\mathcal{P}$2

Equivalently, in vector notation,

$\mathcal{P}$3

The event is monotone upward: if an erasure pattern already prevents indirect recovery of $\mathcal{P}$4, adding more erasures cannot restore recoverability (Kudekar et al., 2015).

This representation is exact because the code is linear and the BEC is symmetric. If the all-zero codeword is assumed transmitted, then under erasure pattern $\mathcal{P}$5 the compatible codewords form a linear subcode. If that subcode contains a codeword with $\mathcal{P}$6-th bit equal to $\mathcal{P}$7, then exactly half the compatible codewords have $\mathcal{P}$8-th bit $\mathcal{P}$9 and half have $\mathcal{P}$0, so bit-MAP decoding fails; otherwise bit $\mathcal{P}$1 is uniquely determined. Hence failure depends only on whether the erasures cover the support of a codeword with a $\mathcal{P}$2 in position $\mathcal{P}$3, and not on the unerased values themselves (Kudekar et al., 2015).

The EXIT function is therefore the product-measure mass of a monotone set. For the scalar $\mathcal{P}$4,

$\mathcal{P}$5

and for the full decoder one has

$\mathcal{P}$6

At the average level,

$\mathcal{P}$7

The BEC area theorem thus fixes the integral of the average EXIT function at the code rate, while monotonicity ensures that larger erasure probability can only increase bit uncertainty (Kudekar et al., 2016).

The same monotone viewpoint extends beyond single-bit ambiguity. For a linear code $\mathcal{P}$8, if

$\mathcal{P}$9

then block decoding succeeds if and only if $\mathbb{F}_q$00. More generally,

$\mathbb{F}_q$01

is a monotone Boolean function for every $\mathbb{F}_q$02, because erasing more coordinates can only increase the dimension of the hidden subcode. This formulation exposes block failure as a nested family of monotone events parameterized by the dimension of the ambiguity subcode (Pfister et al., 10 Jan 2025).

4. Sharp thresholds, generalized Hamming weights, and capacity

When the underlying code family has strong permutation symmetry, monotonicity becomes a sharp-threshold statement. If the permutation group is doubly transitive, then all bit EXIT functions are identical, and the partial derivatives with respect to the other coordinates are equal. In the Boolean-function language, the associated failure sets have equal influences. The sharp-threshold inequality used in this setting is

$\mathbb{F}_q$03

for monotone $\mathbb{F}_q$04 with equal influences. Combined with the EXIT area theorem, this forces the decoding transition to occur at the Shannon point $\mathbb{F}_q$05 for doubly transitive code families (Kudekar et al., 2016).

This symmetry-based mechanism yields a generic capacity theorem on the BEC. If $\mathbb{F}_q$06 is a sequence of binary linear codes with $\mathbb{F}_q$07, $\mathbb{F}_q$08, and doubly transitive permutation group for each $\mathbb{F}_q$09, then $\mathbb{F}_q$10 is capacity achieving on the BEC under bit-MAP decoding. Reed–Muller codes are the canonical example because $\mathbb{F}_q$11 is doubly transitive; the same framework also applies to affine-invariant codes, extended primitive narrow-sense BCH codes, and yields the existence of a capacity-achieving sequence of binary cyclic codes on the BEC (Kumar et al., 2015).

The distinction between bit-MAP and block-MAP remains fundamental. The original sharp-threshold proof is inherently bitwise because the EXIT area theorem controls bit uncertainty rather than block ambiguity. Block-MAP capacity can follow either from sufficiently large minimum distance together with a sharp EXIT derivative lower bound, or from stronger symmetry yielding a more powerful derivative inequality. Reed–Muller codes require the latter route: the argument exploits additional symmetry of the bit-failure set to obtain a derivative lower bound with an extra $\mathbb{F}_q$12 factor, from which block-MAP capacity follows (Kudekar et al., 2016).

A later bit-to-block framework makes the monotone structure more explicit by tracking not only whether $\mathbb{F}_q$13 is nontrivial but also its dimension. The key control parameter is the generalized Hamming weight

$\mathbb{F}_q$14

The derived inequality

$\mathbb{F}_q$15

shows that low-dimensional subcodes with small support are the obstruction to transferring a bit threshold into a block threshold. Under suitable lower bounds on $\mathbb{F}_q$16, the block-MAP threshold lies within $\mathbb{F}_q$17 of the bit-MAP threshold; this framework provides a new proof that Reed–Muller codes achieve capacity on the erasure channel with respect to block error probability (Pfister et al., 10 Jan 2025).

5. Ordered and parity-check formulations

A distinct monotone erasure model arises over extension fields through hierarchical erasures. For a code

$\mathbb{F}_q$18

and an ordered basis $\mathbb{F}_q$19, a symbol is expanded as

$\mathbb{F}_q$20

An $\mathbb{F}_q$21-hierarchical erasure removes left-justified prefixes of these coordinate expansions: there exists $\mathbb{F}_q$22 with $\mathbb{F}_q$23 such that, in symbol $\mathbb{F}_q$24, coordinates $\mathbb{F}_q$25 are erased. The corresponding indistinguishable subspace is

$\mathbb{F}_q$26

and the fundamental criterion is

$\mathbb{F}_q$27

This converts ordered monotone erasure correction into a forbidden-intersection problem in $\mathbb{F}_q$28 (Raviv et al., 2018).

Several constructions follow. A UDM-based parity-check construction yields $\mathbb{F}_q$29-correcting $\mathbb{F}_q$30-linear codes of dimension at least $\mathbb{F}_q$31. For $\mathbb{F}_q$32, $\mathbb{F}_q$33, and even $\mathbb{F}_q$34, the code

$\mathbb{F}_q$35

with $\mathbb{F}_q$36 a root of an irreducible quadratic over $\mathbb{F}_q$37 is $\mathbb{F}_q$38-correcting. The paper also gives high-rate constructions for balanced and power-pattern families when $\mathbb{F}_q$39, and proves that the Gabidulin code

$\mathbb{F}_q$40

is $\mathbb{F}_q$41-correcting for $\mathbb{F}_q$42 and $\mathbb{F}_q$43. Hierarchical erasures therefore form a concrete subclass of monotone erasures in which coordinate order inside each extension-field symbol is part of the channel model (Raviv et al., 2018).

In the parity-check setting, monotonicity appears through $\mathbb{F}_q$44-separating matrices. Given a parity-check matrix $\mathbb{F}_q$45 of an $\mathbb{F}_q$46 code $\mathbb{F}_q$47 and an erasure set $\mathbb{F}_q$48, let $\mathbb{F}_q$49 be the matrix obtained by discarding every row touching $\mathbb{F}_q$50 and then deleting the columns indexed by $\mathbb{F}_q$51. The matrix $\mathbb{F}_q$52 is $\mathbb{F}_q$53-separating if $\mathbb{F}_q$54 is a parity-check matrix for the punctured code, and $\mathbb{F}_q$55-separating if this holds for every $\mathbb{F}_q$56. The defining rank condition is

$\mathbb{F}_q$57

and the $\mathbb{F}_q$58-separating redundancy $\mathbb{F}_q$59 is the minimum number of rows in such a matrix. If $\mathbb{F}_q$60, every $\mathbb{F}_q$61-separating matrix contains, for each erasure set $\mathbb{F}_q$62 with $\mathbb{F}_q$63, a row that has exactly one nonzero entry among coordinates in $\mathbb{F}_q$64, so no stopping set of size at most $\mathbb{F}_q$65 can occur. The theory provides improved lower bounds based on generalized coverings and improved probabilistic and design-theoretic upper bounds on $\mathbb{F}_q$66, making explicit the parity-check overhead required for uniform erasure-handling guarantees over the downward-closed family $\mathbb{F}_q$67 (Tsunoda et al., 2016).

6. Applications, scope, and limitations

The main systems application of the formal access-structure model is generalized asynchronous verifiable information dispersal. Given a Byzantine quorum system $\mathbb{F}_q$68, let $\mathbb{F}_q$69 be the set of kernels. The coding step is to use a monotone erasure code whose access structure is the kernel system $\mathbb{F}_q$70. The resulting protocol, GAVID, is proved to be a general asynchronous verifiable information dispersal scheme for $\mathbb{F}_q$71. If the code has parameters $\mathbb{F}_q$72 and overhead $\mathbb{F}_q$73, then disperse communication complexity is

$\mathbb{F}_q$74

storage complexity is

$\mathbb{F}_q$75

retrieve message complexity is $\mathbb{F}_q$76, and retrieve communication complexity is

$\mathbb{F}_q$77

This places linear monotone erasure codes inside reliable broadcast and consensus under non-threshold trust assumptions (Bammert et al., 21 May 2026).

Several misconceptions recur. Linear monotone erasure codes in the access-structure sense are not secret-sharing schemes: the requirement is that every access set can reconstruct, not that every non-access set learns nothing. In the BEC sharp-threshold literature, “capacity achieving” refers to MAP decoding, often specifically bit-MAP decoding, and not to practical low-complexity algorithms. The monotone-set arguments are also channel-specific: they exploit the fact that, on the BEC, recoverability is order-monotone in the erased set. The same sources explicitly note that extending the method to channels such as the BSC is nontrivial because “it is currently unclear if the GEXIT function can be encoded in terms of a monotone function” (Kudekar et al., 2015).

The principal limitations are correspondingly structural. For general access structures, the recursive linear construction is polynomial-time in the size of the access tree but is not always overhead-optimal, whereas the LP-based optimal construction may be expensive when the access structure has exponentially many minimal access sets. In the BEC threshold theory, the strongest asymptotic results depend on high symmetry, especially double transitivity, and the block-MAP upgrade requires either strong minimum-distance growth or stronger derivative estimates. In hierarchical-erasure models, correction depends on a fixed ordered basis and therefore describes a specific prefix-erasure geometry rather than arbitrary partial-symbol erasures. These limitations do not diminish the unifying theme: linear monotone erasure codes are governed by rank conditions, support growth, and downward-closed families of erasure or availability patterns, whether the objective is generalized trust-aware storage, sharp-threshold analysis on erasure channels, or ordered recovery under structured erasure semantics.