distributed storage engineers

Abstract: In this paper, we study the minimum subpacketization of $(n,k)$ Reed-Solomon (RS) codes over $\mathbb{F}_{q^\ell}$ admitting cut-set-optimal exact $\mathbb{F}_q$-linear repair of every coordinate from any $d$ helpers. For $3\le k<d\le n-1$, write $s=d+1-k$ and denote this minimum by $\ell_{min}(n,k,d;q)$. Using divisibility and relative-degree constraints on cross-ratio fields of evaluation sets, we establish $\ell_{min}(n,k,d;q)\ge sG_s(d+1)$, where $G_s(m)$ is the minimum product of $m$ pairwise coprime integers, each at least $s$. Let $Δ_s(n,d)$ be the minimum of $lcm(a_1,\ldots,a_n)$ over positive integers $a_1,\ldots,a_n$ satisfying $lcm(a_j:j\in J\cup\{i\})\ge s~lcm(a_j:j\in J)$ for every $d$-subset $J\subseteq[n]$ and every $i\in[n]\setminus J$. We construct RS codes with subpacketization $sΔ_s(n,d)$ and further prove that $\ell_{min}(n,k,d;q)=sΔ_s(n,d)=sG_s(n)$ whenever $d(d+2)\geq 4(n+1)$, for every prime power $q$. In particular, this determines the minimum subpacketization in the full-helper case $d=n-1$, resolving the corresponding exact linear-repair case of Open Problem 20 posed by Ramkumar et al. in their survey on codes for distributed storage. We further give a finite formula for $G_s(n)$ and determine the exact threshold for its closed-form expression in terms of the first $n$ primes. For fixed $k$ and $d$, we prove $\log\ell_{min}(n,k,d;q)= Θ_{k,d}(\log n\log\log n)$ as $n\to\infty$, using divisor-function estimates and cover-free families. An affine-plane construction shows that $Δ_s(n,d)<G_s(n)$ for some parameters, so $sG_s(n)$ is not a universal formula. We conjecture that $\ell_{min}(n,k,d;q)=sΔ_s(n,d)$ for every prime power $q$ and all $3\le k<d\le n-1$.

Abstract: We study random access in linear codes when codeword symbols are sampled independently and uniformly with replacement. A code is recovery balanced if the expected number of samples required to recover a codeword coordinate is the same for every coordinate. We express the recovery expectations in terms of the associated matroid and derive a coordinatewise identity relating these values for a code and its dual. This proves that a linear code is recovery balanced if and only if its dual is recovery balanced, establishing a conjecture of Gruica et al. We further define the recovery balance for arbitrary matroids and show that asymptotically almost every sparse paving matroid is not recovery balanced.

Abstract: This note provides exact information-theoretic outer bounds that completely characterize the storage-repair-bandwidth tradeoff regions for the cases $(n,k,d)=(6,5,5)$ and $(7,6,6)$. The layered code proposed by Tian et al. achieves the tradeoff and is thus optimal. GPT-6 Astra assisted in the discovery of these bounds, based on a new conditioning argument and a structured search over the restricted subsets of entropy linear program constraints. The proofs are given as exact integer-valued certificates, which are difficult for direct human verification, and a Python verifier is included as part of the package, together with the certificates.

Abstract: The torn paper channel breaks a codeword into fragments and returns all fragments without their original order. The noisy torn paper channel additionally introduces bit substitution errors to each fragment. Recovering the message therefore requires accounting for both bit errors and uncertainty about the fragment order. We establish a capacity upper bound for the noisy torn paper channel that matches the achievable rate of Ravi, Vahid, and Shomorony and resolves their conjecture. Our results also determine the capacity of the noisy shuffling channel of Shomorony and Heckel, in which all fragments have the same length, and resolve their open question.

Abstract: In this paper, we focus on decoding optimal Reed-Solomon codes of dimension two for insdel errors up to the half-Singleton bound. We present a decoding algorithm for the general insertion-and-deletion scenario with a time complexity smaller than brute-force decoding. Our algorithm corrects insdel errors up to the half-Singleton bound, which exceeds the scope of existing methods. Additionally, for the special cases of insertion-only and deletion-only insdel error patterns, we present two different algorithms with a time complexity smaller than our general algorithm.

Abstract: In distributed storage systems, redundancy enables a single data object to be reconstructed using several disjoint groups of servers. The resulting service rate region (SRR) captures all combinations of object request rates that the system can support simultaneously without overloading any individual server. We analyze the SRR of binary codes obtained via the Plotkin-type construction. We demonstrate that each recovery set for an information object of the Plotkin-type code $\mathcal{C}_p$ induces a corresponding recovery set for the same data object characterized by the underlying code $\mathcal{C}$. Furthermore, by characterizing the recovery sets structure of $\mathcal{C}_p$ in terms of the recovery set structure of $\mathcal{C}$, we identify the additional recovery sets introduced by this construction. Using the associated recovery hypergraphs, we establish bounds on the SRRs of $\mathcal{C}_p$ and iterated code $\mathcal{C}_{p^m}$ in terms of the SRR of $\mathcal{C}$. We consider the family of first-order binary Reed-Muller codes and show how the recovery structure, and consequently, the service rates of R$(1,m)$ for arbitrary $m$ can be recursively derived from a generator matrix of R$(1,2)$ through our results for successive Plotkin-type constructions.

Abstract: A storage code on a graph assigns a symbol to each vertex so that the symbol can be recovered from its neighbors. We prove that the binary full-parity storage codes on triangle-free coset graphs of primitive BCH codes have rate tending to one for every fixed error-correction capability of at least two. The conclusion also holds for an increasing error budget under an explicit growth condition. The proof bounds the binary rank of convolution matrices associated with polynomial maps. Low coordinate degree forces cancellation in the polynomial representing the image, and a polynomial rank bound converts this cancellation into a quantitative estimate for the storage rate. For the double-error-correcting family over $\mathbb{F}_{2^m}$, we prove that the storage deficiency is $Θ(((1+\sqrt{5})/4)^m)$. This refinement follows from an exact Fibonacci formula for a polynomial coefficient rank and a matching exponential lower bound after finite-field evaluation.

Abstract: This paper investigates a fundamental question in stripeless erasure coding based on symmetric balanced incomplete block designs (SBIBDs): how many replicas per object are precisely required to guarantee recovery from any set of at most $p$ node failures? We refine the known sufficient recovery guarantee for the generalized SBIBD $(v,k,λ)$ construction. One replica is necessary and sufficient for $p=1$, and $q=λ(p-1)+2$ replicas ($q\le k$) guarantee recovery for $p\ge2$. Recovery takes one round for $p=2$ and at most two rounds in general. To study the tightness of this count, we define the universal replica threshold $q^\ast(A,p)$ for a fixed zero-diagonal SBIBD representative~$A$. We determine $q^\ast(A,p)$ for $p=1$ and $p=2$. For $p\ge3$, we identify pairwise separated sets and private target permutations as the structures determining whether the sufficient count is tight or can be further reduced. These conditions give exact thresholds for all but the case where $λ>1$ and $A$ contains no pairwise separated set of size~$p$. We improve the bounds for this remaining case and leave its exact threshold open. Finally, we give sufficient conditions for pairwise separated sets and show how affinity relabeling can realize private target permutations.

Abstract: The connection between coding theory and combinatorial $t$-designs is an important research topic at the intersection of coding theory and combinatorics. Let $q=p^m$, where $p$ is an odd prime and $m\geq 2$. In this paper, we investigate a class of linear codes $\mathcal{C}$ over $\mathbb{F}_{q^2}$ and their connection with combinatorial $3$-designs. By analyzing the relevant structural properties of $\mathcal{C}$ and $\mathcal{C}^{\perp}$, we show that the supports of the codewords of every nonzero weight in $\mathcal{C}$ and {the supports of the codewords of weight $4$ in $\mathcal{C}^{\perp}$} form $3$-designs. We further investigate a class of nonlinear codes $\mathcal{C}_2$ associated with $\mathcal{C}$, and prove that the supports of the codewords of every nonzero Hamming weight in $\mathcal{C}_2$ also form $3$-designs. These results identify further classes of linear and nonlinear codes whose codewords support combinatorial $3$-designs. In particular, the nonlinear case provides additional examples of codes supporting $3$-designs, a topic for which relatively few results are currently available. As applications, we construct from $\mathcal{C}^{\perp}$ an entanglement-assisted quantum error-correcting code with parameters $[[q+1,q-3,4;4]]_q$. We also prove that $\mathcal{C}$ is an all-symbol locally repairable code with locality $3$. Furthermore, we show that the code $\mathcal{C}$ {meets the Singleton-type bound for locally repairable codes} and hence is optimal in some cases.