Functional Batch Conjecture

• Functional Batch Conjecture is the assertion that a binary functional batch code of dimension s with 2^(s-1) requests attains the optimal length of 2^s-1 using the simplex code.
• It employs a coding-theoretic framework where n servers store linear combinations of s bits, ensuring any multiset of requests is recoverable via disjoint recovery sets.
• Recent advances verify the conjecture for small s and reformulate it algebraically and combinatorially, though full optimality for all dependent requests remains open.

In coding theory, the Functional Batch Conjecture is the assertion that the optimal length of a binary functional batch code of dimension $s$ with $2^{s-1}$ requests is $2^s-1$; equivalently, the binary simplex code of length $2^s-1$ should be a $2^{s-1}$-functional batch code (Yohananov et al., 2021, Kilic et al., 4 Aug 2025). In a functional $k$-batch code of dimension $s$, $n$ servers store linear combinations of $s$ linearly independent information bits, and any multiset of $k$ requested linear combinations must be recoverable by $2^{s-1}$0 pairwise disjoint subsets of servers (Yohananov et al., 2021). The conjecture occupies a central position because it asks whether the shortest code already known to be optimal for functional PIR at the extremal point $2^{s-1}$1 is also optimal for the strictly stronger functional batch requirement (Yohananov et al., 2021, Kong et al., 2023).

1. Formal statement and coding-theoretic setting

A binary functional $2^{s-1}$2-batch code of dimension $2^{s-1}$3, denoted $2^{s-1}$4, consists of $2^{s-1}$5 servers storing nontrivial linear combinations of $2^{s-1}$6 information bits over $2^{s-1}$7. A request is any vector $2^{s-1}$8, interpreted as the linear form $2^{s-1}$9, and a multiset of $2^s-1$0 such vectors must be served by $2^s-1$1 pairwise disjoint recovery sets whose column sums equal the requested vectors (Yohananov et al., 2021). The optimal length is

$2^s-1$2

Within the same framework, functional PIR is the special case in which all $2^s-1$3 requests are identical, while ordinary batch codes restrict requests to unit vectors (Yohananov et al., 2021).

The conjecture, stated in Zhang–Etzion–Yaakobi and restated as Conjecture 1 in later work, is

$2^s-1$4

for all $2^s-1$5 (Yohananov et al., 2021). Since $2^s-1$6 is the natural lower-bound target in the binary setting, the conjecture asserts that no redundancy beyond the simplex length is needed even when the $2^s-1$7 requests are arbitrary linear combinations rather than repeated copies of one request or multisets of coordinates (Kilic et al., 4 Aug 2025).

A later generalization introduces functional $2^s-1$8-batch codes, where a batch of total size $2^s-1$9 is allowed to contain at most $2^s-1$0 distinct request vectors. This interpolates between functional PIR ($2^s-1$1) and functional batch ($2^s-1$2), and it provides a finer language for partial progress toward the extremal case $2^s-1$3 (Kong et al., 2023).

2. Simplex code and equivalent formulations

The conjecture is centered on the binary simplex code. The $2^s-1$4 simplex code has generator matrix whose columns are all nonzero vectors of $2^s-1$5, so its redundancy is $2^s-1$6 (Kong et al., 2023). In the original binary notation, this is the $2^s-1$7 code appearing in the conjecture (Yohananov et al., 2021).

Several formulations are used in the literature.

Formulation	Statement	Source
Length form	$2^s-1$8	(Yohananov et al., 2021)
Simplex form	The binary simplex code is a $2^s-1$9-functional batch code	(Kilic et al., 4 Aug 2025)
Generalized redundancy form	$2^{s-1}$0 for all $2^{s-1}$1	(Kong et al., 2023)
Pair-partition form	If $2^{s-1}$2, $2^{s-1}$3, $2^{s-1}$4, and $2^{s-1}$5, then $2^{s-1}$6 can be partitioned into pairs $2^{s-1}$7 with $2^{s-1}$8	(Yohananov et al., 19 Jan 2025)
All-symbol form for simplex	The simplex code is a $2^{s-1}$9-all-symbol batch code	(Boruchovsky et al., 7 Jan 2026)

The simplex formulation is particularly natural because every nonzero vector of $k$0 already appears as a column. In consequence, for the simplex code, the functional batch property and the all-symbol batch property coincide: serving all linear combinations is the same as serving all stored symbols (Boruchovsky et al., 7 Jan 2026). This also explains why the conjecture is stronger than standard batch: the standard $k$1-batch property for simplex concerns only unit vectors, whereas the functional version quantifies over all nonzero vectors in $k$2 (Boruchovsky et al., 7 Jan 2026).

There is also a stronger “size-at-most-two” version. A reformulation due to Hollmann, Khathuria, Riet, and Skachek asks whether every sequence of $k$3 nonzero vectors in $k$4 can be served by subsets of size at most two in the simplex generator matrix; this is stronger than being $k$5-functional batch and is equivalent to a special additive matching problem in $k$6 (Hollmann et al., 2021).

3. Verified cases and constructive progress

The conjecture is known to hold for small dimensions. Earlier work verified $k$7 for $k$8, and this is recorded explicitly in the later Hadamard-code construction paper (Yohananov et al., 2021). The same paper proves several progressively stronger constructions at or near the conjectured length $k$9. In particular, it constructs

$s$0

then improves this to

$s$1

and also gives a family

$s$2

for all $s$3 (Yohananov et al., 2021). At the neighboring extremal point $s$4, the same work proves the optimal equality

$s$5

via a double-Hadamard construction (Yohananov et al., 2021).

A different line of progress isolates large subclasses of request lists that the simplex code can already serve at the conjectured threshold. The simplex code is known to be a $s$6-batch code in the standard, non-functional sense (Boruchovsky et al., 7 Jan 2026). More recently, it was shown that the same code can serve any list

$s$7

of total size $s$8 provided the distinct requested vectors $s$9 are linearly independent (Boruchovsky et al., 7 Jan 2026). This is substantially stronger than standard batch and narrows the unresolved portion of the conjecture to linearly dependent request patterns.

An intermediate result of a different flavor shows that the simplex code is a $n$0-odd batch code: every sequence of $n$1 odd-weight vectors can be served, and more generally every sequence of $n$2 vectors from the complement of a $n$3-dimensional subspace can be served with subsets of size at most two (Hollmann et al., 2021). This places the full conjecture between the classical batch property and a hyperplane-complement functional property.

4. Generalized functional $n$4-batch codes and exact progress for small numbers of distinct requests

The generalized theory of functional $n$5-batch codes reframes the conjecture by separating the total number of requests from the number of distinct request vectors (Kong et al., 2023). In that notation, a binary functional $n$6-batch code of dimension $n$7 must serve any multiset of $n$8 requests using at most $n$9 distinct vectors, and its minimal redundancy is denoted $s$0 (Kong et al., 2023).

Within this framework, the original simplex conjecture becomes

$s$1

which says that at the PIR extremal point $s$2, the optimal redundancy is independent of how many distinct requests appear in the batch (Kong et al., 2023). This reformulation is useful because it isolates the combinatorial source of difficulty: the conjecture is already solved when the number of distinct requests is very small.

The strongest exact result in this direction is that the $s$3 simplex code is a functional $s$4-batch code for all $s$5. Since the redundancy $s$6 is already optimal for $s$7, it follows that

$s$8

(Kong et al., 2023). Thus the generalized conjecture is completely proved for batches containing at most four distinct requested linear combinations.

The same paper also proves an asymptotic version for sparse distinct-request regimes. For $s$9, the simplex code is a functional $k$0-batch code with

$k$1

and therefore if $k$2, then simplex achieves optimal redundancy while supporting

$k$3

requests (Kong et al., 2023). This gives strong asymptotic evidence that the full extremal statement should remain true well beyond the exactly solved cases $k$4.

5. Algebraic and additive-combinatorial reformulations

One of the most substantial recent developments is the reduction of the conjecture to explicit algebraic and matching problems. In the polynomial approach, set $k$5 and $k$6. For nonzero $k$7 with $k$8, the conjecture becomes the existence of a partition of $k$9 into disjoint pairs $2^{s-1}$00 such that $2^{s-1}$01 for every $2^{s-1}$02 (Yohananov et al., 19 Jan 2025). This pair-partition statement is equivalent to the coding-theoretic conjecture.

The same paper proves that the pair-partition formulation is equivalent to the nonvanishing of an explicit polynomial

$2^{s-1}$03

in the quotient ring

$2^{s-1}$04

(Yohananov et al., 19 Jan 2025). A necessary condition is

$2^{s-1}$05

and this is shown to be exactly the obstruction visible at the level of the polynomial formulation (Yohananov et al., 19 Jan 2025). The paper then derives structural constraints on the coefficient polynomials of $2^{s-1}$06, proves new sufficient conditions for nonvanishing, and uses them to recover the conjecture for $2^{s-1}$07 and $2^{s-1}$08 within a uniform algebraic framework (Yohananov et al., 19 Jan 2025).

A complementary reformulation comes from additive combinatorics. For the simplex generator matrix $2^{s-1}$09, serving a request $2^{s-1}$10 with subsets of size at most two is equivalent to finding distinct vectors $2^{s-1}$11 such that the vectors $2^{s-1}$12 are also distinct and disjoint from the $2^{s-1}$13’s; this is called a special service in a finite abelian group (Hollmann et al., 2021). The paper formulates a generalized special-service conjecture for finite abelian groups, proves it for cyclic groups of odd prime order by a Combinatorial Nullstellensatz argument, and shows that a specific characteristic-two version would imply the strong size-at-most-two form of the Functional Batch Conjecture for simplex codes (Hollmann et al., 2021). This reformulation is significant because it isolates the unresolved difficulty as a characteristic-two matching problem rather than a purely coding-theoretic anomaly.

6. Variants, extensions, and open problems

The all-symbol framework developed in 2026 places the conjecture into a broader hierarchy. A $2^{s-1}$14-all-symbol PIR code requires $2^{s-1}$15 disjoint recovery sets for every stored symbol, and a $2^{s-1}$16-all-symbol batch code requires disjoint recovery sets for every multiset of $2^{s-1}$17 stored symbols (Boruchovsky et al., 7 Jan 2026). For the simplex code, all-symbol batch is equivalent to functional batch because every nonzero vector of $2^{s-1}$18 already appears as a column (Boruchovsky et al., 7 Jan 2026). The same paper shows that the simplex code is exactly a $2^{s-1}$19-all-symbol PIR code, because the upper bound

$2^{s-1}$20

with $2^{s-1}$21 and $2^{s-1}$22 gives $2^{s-1}$23, and the bound is attained by pairing columns $2^{s-1}$24 for each requested symbol $2^{s-1}$25 (Boruchovsky et al., 7 Jan 2026). This confirms that the conjectured threshold $2^{s-1}$26 is the maximal plausible one even before batch interactions are considered.

A stricter variant imposes small recovery sets. In that setting, it has been conjectured that the simplex code is a $2^{s-1}$27 functional batch code, meaning locality $2^{s-1}$28 for all $2^{s-1}$29 functional requests (Oksner et al., 18 Jan 2026). The general lower-bound theory for locality-constrained functional batch codes shows that, for $2^{s-1}$30 codes,

$2^{s-1}$31

and for fixed locality $2^{s-1}$32,

$2^{s-1}$33

(Oksner et al., 18 Jan 2026). These results do not resolve the simplex locality conjecture, but they show that bounded recovery size fundamentally changes the asymptotic geometry of the problem.

Over non-binary fields, the standard binary conjecture does not have a literal analogue, but recent work argues that the “correct” list sizes should scale like $2^{s-1}$34 rather than $2^{s-1}$35. In that setting, the exact functional PIR identity

$2^{s-1}$36

is known, and the asymptotic equality

$2^{s-1}$37

has been established (Kilic et al., 4 Aug 2025). The binary Functional Batch Conjecture itself, however, remains open in full generality (Kilic et al., 4 Aug 2025).

The current state of the subject is therefore sharply stratified. Exact optimality is known for $2^{s-1}$38 in the original dimension notation, for at most four distinct request vectors in the generalized $2^{s-1}$39-formalism, for all odd-weight or hyperplane-complement request families, and for all linearly independent request lists of total size $2^{s-1}$40 in the simplex code (Yohananov et al., 2021, Kong et al., 2023, Hollmann et al., 2021, Boruchovsky et al., 7 Jan 2026). The unresolved core is the full dependent-request regime at the extremal threshold $2^{s-1}$41, where the simplex code is already optimal for all-symbol PIR but has not yet been proved to sustain arbitrary functional batch lists.