--- title: Functional Batch Conjecture url: https://www.emergentmind.com/topics/functional-batch-conjecture type: topic --- # Functional Batch Conjecture 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 [2101.06722][2508.02586]. 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 \(k\) pairwise disjoint subsets of servers [2101.06722]. 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 \(k=2^{s-1}\) is also optimal for the strictly stronger functional batch requirement [2101.06722][2309.06806]. ## 1. Formal statement and coding-theoretic setting A binary functional \(k\)-batch code of dimension \(s\), denoted \(FB(n,s,k)\), consists of \(n\) servers storing nontrivial linear combinations of \(s\) information bits over \(\mathbb{F}_2\). A request is any vector \(v\in\mathbb{F}_2^s\), interpreted as the linear form \(\langle v,x\rangle\), and a multiset of \(k\) such vectors must be served by \(k\) pairwise disjoint recovery sets whose column sums equal the requested vectors [2101.06722]. The optimal length is \[ FB(s,k):=\min\{n:\text{ an }FB(n,s,k)\text{ code exists}\}. \] Within the same framework, functional PIR is the special case in which all \(k\) requests are identical, while ordinary batch codes restrict requests to unit vectors [2101.06722]. The conjecture, stated in Zhang–Etzion–Yaakobi and restated as Conjecture 1 in later work, is \[ FB(s,2^{s-1}) = 2^s-1 \] for all \(s\) [2101.06722]. Since \(FB(s,2^{s-1})\ge 2^s-1\) 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 \(2^{s-1}\) requests are arbitrary linear combinations rather than repeated copies of one request or multisets of coordinates [2508.02586]. A later generalization introduces functional \((s,t)\)-batch codes, where a batch of total size \(t\) is allowed to contain at most \(s\) distinct request vectors. This interpolates between functional PIR (\(s=1\)) and functional batch (\(s=t\)), and it provides a finer language for partial progress toward the extremal case \(t=2^{n-1}\) [2309.06806]. ## 2. Simplex code and equivalent formulations The conjecture is centered on the binary simplex code. The \([2^n-1,n]\) simplex code has generator matrix whose columns are all nonzero vectors of \(\mathbb{F}_2^n\), so its redundancy is \(2^n-n-1\) [2309.06806]. In the original binary notation, this is the \([2^s-1,s]\) code appearing in the conjecture [2101.06722]. Several formulations are used in the literature. | Formulation | Statement | Source | |---|---|---| | Length form | \(FB(s,2^{s-1})=2^s-1\) | [2101.06722] | | Simplex form | The binary simplex code is a \(2^{s-1}\)-functional batch code | [2508.02586] | | Generalized redundancy form | \(r_F(n;s,2^{n-1})=2^n-n-1\) for all \(1\le s\le 2^{n-1}\) | [2309.06806] | | Pair-partition form | If \(q=2^s\), \(k=2^{s-1}\), \(v_i\in\mathbb{F}_q^\times\), and \(\sum_i v_i=0\), then \(\mathbb{F}_q\) can be partitioned into pairs \(\{x_i,y_i\}\) with \(x_i+y_i=v_i\) | [2501.11122] | | All-symbol form for simplex | The simplex code is a \(2^{k-1}\)-all-symbol batch code | [2601.04041] | The simplex formulation is particularly natural because every nonzero vector of \(\mathbb{F}_2^k\) 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 [2601.04041]. This also explains why the conjecture is stronger than standard batch: the standard \(2^{k-1}\)-batch property for simplex concerns only unit vectors, whereas the functional version quantifies over all nonzero vectors in \(\mathbb{F}_2^k\) [2601.04041]. There is also a stronger “size-at-most-two” version. A reformulation due to Hollmann, Khathuria, Riet, and Skachek asks whether every sequence of \(2^{k-1}\) nonzero vectors in \(\mathbb{F}_2^k\) can be served by subsets of size at most two in the simplex generator matrix; this is stronger than being \(2^{k-1}\)-functional batch and is equivalent to a special additive matching problem in \(\mathbb{F}_2^k\) [2110.07421]. ## 3. Verified cases and constructive progress The conjecture is known to hold for small dimensions. Earlier work verified \(FB(s,2^{s-1})=2^s-1\) for \(s\le 5\), and this is recorded explicitly in the later Hadamard-code construction paper [2101.06722]. The same paper proves several progressively stronger constructions at or near the conjectured length \(2^s-1\). In particular, it constructs \[ FB\!\left(2^s-1,s,\left\lfloor \tfrac{2}{3}\cdot 2^{s-1}\right\rfloor\right), \] then improves this to \[ FB\!\left(2^s-1,s,\left\lfloor \tfrac{5}{6}\cdot 2^{s-1}\right\rfloor - s\right), \] and also gives a family \[ FB\!\left(2^s+\left\lceil(3\alpha-2)\cdot 2^{s-2}\right\rceil -1,\ s,\ \left\lfloor \alpha\cdot 2^{s-1}\right\rfloor\right) \] for all \(2/3\le \alpha \le 1\) [2101.06722]. At the neighboring extremal point \(k=2^s\), the same work proves the optimal equality \[ FB(s,2^s)=2^{s+1}-2 \] via a double-Hadamard construction [2101.06722]. 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 \(2^{k-1}\)-batch code in the standard, non-functional sense [2601.04041]. More recently, it was shown that the same code can serve any list \[ \{v_1^{t_1},\dots,v_\ell^{t_\ell}\} \] of total size \(2^{k-1}\) provided the distinct requested vectors \(v_1,\dots,v_\ell\) are linearly independent [2601.04041]. 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 \(2^{k-1}\)-odd batch code: every sequence of \(2^{k-1}\) odd-weight vectors can be served, and more generally every sequence of \(2^{k-1}\) vectors from the complement of a \((k-1)\)-dimensional subspace can be served with subsets of size at most two [2110.07421]. This places the full conjecture between the classical batch property and a hyperplane-complement functional property. ## 4. Generalized functional \((s,t)\)-batch codes and exact progress for small numbers of distinct requests The generalized theory of functional \((s,t)\)-batch codes reframes the conjecture by separating the total number of requests from the number of distinct request vectors [2309.06806]. In that notation, a binary functional \((s,t)\)-batch code of dimension \(n\) must serve any multiset of \(t\) requests using at most \(s\) distinct vectors, and its minimal redundancy is denoted \(r_F(n;s,t)\) [2309.06806]. Within this framework, the original simplex conjecture becomes \[ r_F(n;s,2^{n-1}) = 2^n-n-1 \qquad\text{for all }1\le s\le 2^{n-1}, \] which says that at the PIR extremal point \(t=2^{n-1}\), the optimal redundancy is independent of how many distinct requests appear in the batch [2309.06806]. 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 \([2^n-1,n]\) simplex code is a functional \((s,2^{n-1})\)-batch code for all \(s\le 4\). Since the redundancy \(2^n-n-1\) is already optimal for \(s=1\), it follows that \[ r_F(n;s,2^{n-1}) = 2^n-n-1 \qquad\text{for }1\le s\le 4 \] [2309.06806]. 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 \(1\le s\le n-\ln n\), the simplex code is a functional \((s,t)\)-batch code with \[ t = 2^{n-1} - \left\lceil\frac{s}{2}\right\rceil 2^{s-1}, \] and therefore if \(s2^s=o(2^n)\), then simplex achieves optimal redundancy while supporting \[ t = 2^{n-1}(1-o(1)) \] requests [2309.06806]. This gives strong asymptotic evidence that the full extremal statement should remain true well beyond the exactly solved cases \(s\le 4\). ## 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 \(q=2^s\) and \(k=2^{s-1}\). For nonzero \(v_1,\dots,v_k\in\mathbb{F}_q\) with \(\sum_i v_i=0\), the conjecture becomes the existence of a partition of \(\mathbb{F}_q\) into disjoint pairs \(\{x_i,y_i\}\) such that \(x_i+y_i=v_i\) for every \(i\) [2501.11122]. 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 \[ f_v(x_1,\dots,x_k) = v_1\cdots v_k \prod_{1\le i