Quantum low-density parity-check (qLDPC) codes can be implemented by measuring only low-weight checks, making them compatible with noisy quantum hardware and central to the quest to build noise-resilient quantum computers. A fundamental open question is how constraints on check weight limit the achievable parameters of qLDPC codes. Here, we study stabilizer and subsystem codes with constrained check weight, combining analytical arguments with numerical optimization to establish strong upper bounds on their parameters. We show that stabilizer codes with checks of weight at most three cannot have nontrivial distance. We also prove tight tradeoffs between rate and distance for broad families of CSS stabilizer and subsystem codes with checks of weight at most four and two, respectively. Notably, our bounds are applicable to general qLDPC codes, as they rely only on check-weight constraints without assuming geometric locality or special graph connectivity. In the finite-size regime, we derive numerical upper bounds using linear programming techniques and identify explicit code constructions that approach these limits, delineating the landscape of practically relevant qLDPC codes with tens or hundreds of physical qubits.

We study the generalised Chv & aacute;tal-Sankoff constant $\gamma _{k,d}$ , which represents the normalised expected length of the longest common subsequence of d independent uniformly random strings over an alphabet of size k. We derive asymptotically tight bounds for $\gamma _{2,d}$ , establishing that $\gamma _{2,d} = \tfrac 12 + \Theta ({1}/{\sqrt {d}})$ . We also derive asymptotically near-optimal bounds on $\gamma _{k,d}$ for $d\ge \Omega (\log k)$ .

For quantum error-correcting codes to be realizable, it is important that the qubits subject to the code constraints exhibit some form of limited connectivity. The works of Bravyi & Terhal (BT) and Bravyi, Poulin & Terhal (BPT) established that geometric locality constrains code properties -- for instance $[[n,k,d]]$ quantum codes defined by local checks on the $D$-dimensional lattice must obey $k d^{2/(D-1)} \le O(n)$. Baspin and Krishna studied the more general question of how the connectivity graph associated with a quantum code constrains the code parameters. These trade-offs apply to a richer class of codes compared to the BPT and BT bounds, which only capture geometrically-local codes. We extend and improve this work, establishing a tighter dimension-distance trade-off as a function of the size of separators in the connectivity graph. We also obtain a distance bound that covers all stabilizer codes with a particular separation profile, rather than only LDPC codes.

In this paper, we prove that with high probability, random Reed-Solomon codes approach the half-Singleton bound - the optimal rate versus error tradeoff for linear insdel codes - with linear-sized alphabets. More precisely, we prove that, for any epsilon > 0 and positive integers n and k, with high probability, random Reed-Solomon codes of length n and dimension k can correct (1 - epsilon)n - 2k + 1 adversarial insdel errors over alphabets of size n + 2(poly(1/epsilon)) k. This significantly improves upon the alphabet size demonstrated in the work of Con, Shpilka, and Tamo (IEEE TIT, 2023), who showed the existence of Reed-Solomon codes with exponential alphabet size (O) over tilde((n/2k-1)(2)) precisely achieving the half-Singleton bound. Our methods are inspired by recent works on list-decoding Reed-Solomon codes. Brakensiek-Gopi-Makam (STOC 2023) showed that random Reed-Solomon codes are list-decodable up to capacity with exponential-sized alphabets, and Guo-Zhang (FOCS 2023) and Alrabiah-Guruswami-Li (STOC 2024) improved the alphabet-size to linear. We achieve a similar alphabet-size reduction by similarly establishing strong bounds on the probability that certain random rectangular matrices are full rank. To accomplish this in our insdel context, our proof combines the random matrix techniques from list-decoding with structural properties of Longest Common Subsequences.

A simple, recently observed generalization of the classical Singleton bound to list-decoding asserts that rate $R$ codes are not list-decodable using list-size $L$ beyond an error fraction $\frac{L}{L+1} (1-R)$ (the Singleton bound being the case of $L=1$, i.e., unique decoding). We prove that in order to approach this bound for any fixed $L >1$, one needs exponential alphabets. Specifically, for every $L>1$ and $R\in(0,1)$, if a rate $R$ code can be list-of-$L$ decoded up to error fraction $\frac{L}{L+1} (1-R -\varepsilon)$, then its alphabet must have size at least $\exp(\Omega_{L,R}(1/\varepsilon))$. This is in sharp contrast to the situation for unique decoding where certain families of rate $R$ algebraic-geometry (AG) codes over an alphabet of size $O(1/\varepsilon^2)$ are unique-decodable up to error fraction $(1-R-\varepsilon)/2$. Our bounds hold even for subconstant $\varepsilon\ge 1/n$, implying that any code exactly achieving the $L$-th generalized Singleton bound requires alphabet size $2^{\Omega_{L,R}(n)}$. Previously this was only known only for $L=2$ under the additional assumptions that the code is both linear and MDS. Our lower bound is tight up to constant factors in the exponent -- with high probability random codes (or, as shown recently, even random linear codes) over $\exp(O_L(1/\varepsilon))$-sized alphabets, can be list-of-$L$ decoded up to error fraction $\frac{L}{L+1} (1-R -\varepsilon)$.

We study the tradeoffs between the locality and parameters of subsystem codes. We prove lower bounds on both the number and lengths of interactions in any D-dimensional embedding of a subsystem code. Specifically, we show that any embedding of a subsystem code with parameters [[ n, k, d]] into R-D must have at least M* interactions of length at least l*, where M* = Omega(max( k, d)), and l* = Omega( max (d/n(D-1/D), (kd(1/D-1)/n) (D-1/D))). We also give tradeoffs between the locality and parameters of commuting projector codes in D-dimensions, generalizing a result of Dai and Li [8]. We provide explicit constructions of embedded codes that show our bounds are optimal in both the interaction count and interaction length. 2012 ACM Subject Classification Hardware -> Quantum error correction and fault tolerance; Theory of computation -> Error-correcting codes

We construct deletion error-correcting codes in the oblivious model, where errors are adversarial but oblivious to the encoder's randomness. Oblivious errors bridge the gap between the adversarial and random error models, and are motivated by applications like DNA storage, where the noise is caused by hard-to-model physical phenomena, but not by an adversary. (1) (Explicit oblivious) We construct t oblivious deletion codes, with redundancy ∼ 2tlog n, matching the existential bound for adversarial deletions. (2) (List decoding implies explicit oblivious) We show that explicit list-decodable codes yield explicit oblivious deletion codes with essentially the same parameters. By a work of Guruswami and Håstad (IEEE TIT, 2021), this gives 2 oblivious deletion codes with redundancy ∼ 3log n, beating the existential redundancy for 2 adversarial deletions. (3) (Randomized oblivious) We give a randomized construction of oblivious codes that, with probability at least 1-2^-n, produces a code correcting t oblivious deletions with redundancy ∼(t+1)log n, beating the existential adversarial redundancy of ∼ 2tlog n. (4) (Randomized adversarial) Studying the oblivious model can inform better constructions of adversarial codes. The same technique produces, with probability at least 1-2^-n, a code correcting t adversarial deletions with redundancy ∼ (2t+1)log n, nearly matching the existential redundancy of ∼ 2tlog n. The common idea behind these results is to reduce the hash size by modding by a prime chosen (randomly) from a small subset, and including a small encoding of the prime in the hash.

We prove several results on linear codes achieving list-recovery capacity. We show that random linear codes achieve list-recovery capacity with constant output list size (independent of the alphabet size and length). That is, over alphabets of size at least l(Omega(1/epsilon)), random linear codes of rate R are (1 - R - epsilon, l, (l/epsilon)(O(l/epsilon)))-list-recoverable for all R is an element of (0, 1) and l. Together with a result of Levi, Mosheiff, and Shagrithaya, this implies that randomly punctured Reed-Solomon codes also achieve list-recovery capacity. We also prove that our output list size is near-optimal among all linear codes: all (1 - R - epsilon, l, L)-list-recoverable linear codes must have L >= l(Omega(R/epsilon)). Our simple upper bound combines the Zyablov-Pinsker argument with recent bounds from Kopparty, Ron-Zewi, Saraf, Wootters, and Tamo on the maximum intersection of a "list-recovery ball" and a low-dimensional subspace with large distance. Our lower bound is inspired by a recent lower bound of Chen and Zhang.

Approximating the graph diameter is a basic task of both theoretical and practical interest. A simple folklore algorithm can output a 2-approximation to the diameter in linear time by running BFS from an arbitrary vertex. It has been open whether a better approximation is possible in near-linear time. A series of papers on fine-grained complexity have led to strong hardness results for diameter in directed graphs, culminating in a recent tradeoff curve independently discovered by [Li, STOC'21] and [Dalirrooyfard and Wein, STOC'21], showing that under the Strong Exponential Time Hypothesis (SETH), for any integer $k\geq 2$ and $\delta > 0$ , a $2-\frac{1}{k}-\delta$ approximation for diameter in directed $m$ -edge graphs requires $mn^{1+1/(k-1)-o(1)}$ time. In particular, the simple linear time 2-approximation algorithm is optimal for directed graphs. In this paper we prove that the same tradeoff lower bound curve is possible for undirected graphs as well, extending results of [Roditty and Vassilevska W., STOC'13], [Li'20] and [Bonnet, ICALP'21] who proved the first few cases of the curve, $k=2,3$ and 4, respectively. Our result shows in particular that the simple linear time 2-approximation algorithm is also optimal for undirected graphs. To obtain our result we develop new tools for fine-grained reductions that could be useful for proving SETH-based hardness for other problems in undirected graphs related to distance computation.

This paper proves optimal tradeoffs between the locality and parameters of quantum error-correcting codes. Quantum codes give a promising avenue towards quantum fault tolerance, but the practical constraint of locality limits their quality. The seminal Bravyi-Poulin-Terhal (BPT) bound says that a [[n,k,d]] quantum stabilizer code with 2D-locality must satisfy kd(2) <= O(n). We answer the natural question: for better code parameters, how much non-locality is needed? In particular, (i) how long must the long-range interactions be, and (ii) how many long-range interactions must there be? We give a complete answer to both questions for all n,k,d: above the BPT bound, any 2D-embedding must have at least Omega(M*) interactions of length Omega(l*), where M*= max(k,d) and l*=max ((d)/(root n), ( kd(2)/(n) )(1/4) ). Conversely, we exhibit quantum codes that show, in strong ways, that our interaction length l* and interaction count M* are asymptotically optimal for all n,k,d. Our results generalize or improve all prior works on this question, including the BPT bound and the results of Baspin and Krishna. One takeaway of our work is that, for any desired distance d and dimension k, the number of long-range interactions is asymptotically minimized by a good qLDPC code of length Theta(max (k,d)). Following Baspin and Krishna, we also apply our results to the codes implemented in the stacked architecture and obtain better bounds. In particular, we rule out any implementation of hypergraph product codes in the stacked architecture.

Reed–Solomon codes are a classic family of error-correcting codes consisting of evaluations of low-degree polynomials over a finite field on some sequence of distinct field elements. They are widely known for their optimal unique-decoding capabilities, but their list-decoding capabilities are not fully understood. Given the prevalence of Reed-Solomon codes, a fundamental question in coding theory is determining if Reed–Solomon codes can optimally achieve list-decoding capacity. A recent breakthrough by Brakensiek, Gopi, and Makam, established that Reed–Solomon codes are combinatorially list-decodable all the way to capacity. However, their results hold for randomly-punctured Reed–Solomon codes over an exponentially large field size 2O(n), where n is the block length of the code. A natural question is whether Reed–Solomon codes can still achieve capacity over smaller fields. Recently, Guo and Zhang showed that Reed–Solomon codes are list-decodable to capacity with field size O(n2). We show that Reed–Solomon codes are list-decodable to capacity with linear field size O(n), which is optimal up to the constant factor. We also give evidence that the ratio between the alphabet size q and code length n cannot be bounded by an absolute constant. Our techniques also show that random linear codes are list-decodable up to (the alphabet-independent) capacity with optimal list-size O(1/ε) and near-optimal alphabet size 2O(1/ε2), where ε is the gap to capacity. As far as we are aware, list-decoding up to capacity with optimal list-size O(1/ε) was not known to be achievable with any linear code over a constant alphabet size (even non-constructively), and it was also not known to be achievable for random linear codes over any alphabet size. Our proofs are based on the ideas of Guo and Zhang, and we additionally exploit symmetries of reduced intersection matrices. With our proof, which maintains a hypergraph perspective of the list-decoding problem, we include an alternate presentation of ideas from Brakensiek, Gopi, and Makam that more directly connects the list-decoding problem to the GM-MDS theorem via a hypergraph orientation theorem.

We prove that there exists an absolute constant $\delta >0$ such that any binary code $C\subset \{0,1\}^{N} \vphantom {_{\int }}$ tolerating $(1/2-\delta)N$ adversarial deletions must satisfy $|C|\le 2^{ \mathop {\mathrm {poly}} \log N}$ and thus have rate asymptotically approaching 0. This is the first constant fraction improvement over the trivial bound that codes tolerating $N/2$ adversarial deletions must have rate going to 0 asymptotically. Equivalently, we show that there exists absolute constants $A$ and $\delta >0$ such that any set $C\subset \{0,1\}^{N}$ of $2^{\log ^{A} N}$ binary strings must contain two strings $c$ and $c'$ whose longest common subsequence has length at least $(1/2+\delta)N$ . As an immediate corollary, we show that $q$ -ary codes tolerating a fraction $1-(1+2\delta)/q$ of adversarial deletions must also have rate approaching 0. Our techniques include string regularity arguments and a structural lemma that classifies binary strings by their oscillation patterns. Leveraging these tools, we find in any large code two strings with similar oscillation patterns, which is exploited to find a long common subsequence.

The Longest Common Subsequence (LCS) is a fundamental string similarity measure, and computing the LCS of two strings is a classic algorithms question. A textbook dynamic programming algorithm gives an exact algorithm in quadratic time, and this is essentially best possible under plausible fine-grained complexity assumptions, so a natural problem is to find faster approximation algorithms. When the inputs are two binary strings, there is a simple 1/2-approximation in linear time: compute the longest common all-0s or all-1s subsequence. It has been open whether a better approximation is possible even in truly subquadratic time. Rubinstein and Song showed that the answer is yes under the assumption that the two input strings have equal lengths. We settle the question, generalizing their result to unequal length strings, proving that, for any ε>0, there exists δ>0 and a (1/2+δ)-approximation algorithm for binary LCS that runs in n 1+ε time. As a consequence of our result and a result of Akmal and Vassilevska-Williams, for any ε>0, there exists a (1/ q +δ)-approximation for LCS over q -ary strings in n 1+ε time. Our techniques build on the recent work of Guruswami, He, and Li who proved new bounds for error-correcting codes tolerating deletion errors. They prove a combinatorial “structure lemma” for strings which classifies them according to their oscillation patterns. We prove and use an algorithmic generalization of this structure lemma, which may be of independent interest.

Computing the diameter of a graph, i.e. the largest distance, is a fundamental problem that is central in fine-grained complexity. In undirected graphs, the Strong Exponential Time Hypothesis (SETH) yields a lower bound on the time vs. approximation trade-off that is quite close to the upper bounds. In \emph{directed} graphs, however, where only some of the upper bounds apply, much larger gaps remain. Since $d(u,v)$ may not be the same as $d(v,u)$, there are multiple ways to define the problem, the two most natural being the \emph{(one-way) diameter} ($\max_{(u,v)} d(u,v)$) and the \emph{roundtrip diameter} ($\max_{u,v} d(u,v)+d(v,u)$). In this paper we make progress on the outstanding open question for each of them. -- We design the first algorithm for diameter in sparse directed graphs to achieve $n^{1.5-\varepsilon}$ time with an approximation factor better than $2$. The new upper bound trade-off makes the directed case appear more similar to the undirected case. Notably, this is the first algorithm for diameter in sparse graphs that benefits from fast matrix multiplication. -- We design new hardness reductions separating roundtrip diameter from directed and undirected diameter. In particular, a $1.5$-approximation in subquadratic time would refute the All-Nodes $k$-Cycle hypothesis, and any $(2-\varepsilon)$-approximation would imply a breakthrough algorithm for approximate $\ell_{\infty}$-Closest-Pair. Notably, these are the first conditional lower bounds for diameter that are not based on SETH.

Heterotopic pancreas (HP), less accurately known as ectopic, accessory or aberrant pancreas is an isolated, firm, somewhat lobulated, nodule that is anatomically separate from the normal pancreas. It is present in at least 1% of individuals and is most often found in the wall of the stomach, duodenum or a Meckel diverticulum but can be present elsewhere in the small bowel, oesophagus, appendix or other intra-abdominal viscera.1 In the gastrointestinal tract, HP is most often submucosal rather than subserosal and may extend into the muscularis propria to a variable degree.2 In children, HP is usually asymptomatic and detected incidentally but may rarely cause intussusception, gastric outlet obstruction or gastrointestinal bleeding.1 Malignant degeneration in this age group is exceedingly uncommon.3 We report a boy with heterotopic pancreatitis causing gastric outlet obstruction. An 8-year old boy presented to hospital with recurrent coffee ground vomiting. As a neonate, he had undergone repair of oesophageal atresia and a tracheo-oesophageal fistula. An upper gastrointestinal endoscopy revealed circumferential erosive oesophagitis in the mid oesophagus and he was treated with oral Omeprazole. Nine days later he returned with upper abdominal pain and vomiting. Laboratory analysis revealed a plasma lipase of 600 U/L and CRP of 60 mg/L. An abdominal CT scan showed focal thickening of the gastric antrum with an adjacent triangular wedge of tissue suggestive of heterotopic pancreas (Fig. 1). He was initially managed conservatively with intravenous fluids, analgesia and nasogastric drainage. Total parenteral nutrition was started because of persistent large nasogastric losses. A repeat CT scan 1 week later revealed multiple pseudocysts around the porta hepatis and anterior to the liver. The patient had ongoing abdominal pain, persistent large volume gastric losses and continuing raised inflammatory markers and plasma lipase. An MRI scan showed acutely inflamed heterotopic pancreatic tissue anterior to the pancreatic head adjacent to the duodenal bulb. There was an associated circumferential fluid collection around the pylorus with gastric outlet obstruction. The pancreas appeared normal. After 3 weeks of failed conservative management a laparotomy was performed. Multiple large inflammatory pseudocysts around the liver, pylorus and within the lesser sac were drained. The gallbladder was thick walled and distended. A 3 cm long firm pyramidal wedge of heterotopic pancreas was fused to the serosa of the anterior wall of the first part of the duodenum (Fig. 2). A small pancreatic duct entering the heterotopic pancreas from the duodenum was ligated. The duodenal serosa was repaired and an omental patch applied. He had an uneventful postoperative course and was discharged home 10 days later. Histology revealed pancreatic tissue with normal exocrine and endocrine glands, small ducts and mild fibrosis. The patient remains well 6 months later. His parents kindly gave their permission to report his clinical course. Pancreatitis within HP tissue in children is very rare. There are only six previous case reports of this complication: two were from HP within an enteric duplication cyst, two were in HP nodules within the jejunal mesentery and one was in the wall of the jejunum.2, 4-6 The sixth report documented heterotopic pancreatitis and gastric outlet obstruction in a neonate who also had infantile hypertrophic pyloric stenosis; the relevance of the HP is not clear as the antropyloric heterotopic tissue was not resected and the patient had no further sequelae during a 2-year follow up.7 Our patient also had other unusual features related to the HP. Firstly, the heterotopic pancreatitis caused severe gastric outlet obstruction which resolved after simple resection of the lesion. Secondly, the HP was subserosal rather than submucosal. Thirdly, he had a history of repaired oesophageal atresia. However, it is worth emphasizing that up to 40% of children with a history of oesophageal atresia have evidence of asymptomatic gastric HP on endoscopy.1 The pathogenesis of HP is uncertain but the most popular theory is that pancreatic tissue from the lateral buds of the developing dorsal or ventral pancreas becomes attached to an adjacent organ and then detaches from the pancreas. Heterotopic pancreas may contain any of the components of normal pancreatic tissue, namely acini, ducts and islets of Langerhans; all elements were present in the resected specimen in our patient. What triggered the onset of pancreatitis in our patient with HP who had no previous symptoms to suggest similar episodes is unknown. We wish to thank the parents for their permission to report this patient's medical details and Louise Goossens, Senior Medical Photographer, Capital & Coast DHB for her expert assistance with the operative image. Open access publishing facilitated by University of Otago, as part of the Wiley - University of Otago agreement via the Council of Australian University Librarians. William Fleischl: Data curation; writing – original draft. Ray Li: Data curation. Gianluca Valsenti: Data curation; investigation. Tim Foster: Investigation. Mark D. Stringer: Conceptualization; project administration; supervision; writing – review and editing.

In this paper, we present CopyCat2 (CC2), a novel model capable of: a) synthesizing speech with different speaker identities, b) generating speech with expressive and contextually appropriate prosody, and c) transferring prosody at fine-grained level between any pair of seen speakers. We do this by activating distinct parts of the network for different tasks. We train our model using a novel approach to two-stage training. In Stage I, the model learns speaker-independent word-level prosody representations from speech which it uses for many-to-many fine-grained prosody transfer. In Stage II, we learn to predict these prosody representations using the contextual information available in text, thereby, enabling multi-speaker TTS with contextually appropriate prosody. We compare CC2 to two strong baselines, one in TTS with contextually appropriate prosody, and one in fine-grained prosody transfer. CC2 reduces the gap in naturalness between our baseline and copy-synthesised speech by $22.79\%$. In fine-grained prosody transfer evaluations, it obtains a relative improvement of $33.15\%$ in target speaker similarity.

A family of error-correcting codes is list-decodable from error fraction $p$ if, for every code in the family, the number of codewords in any Hamming ball of fractional radius $p$ is less than some integer $L$ . It is said to be list-recoverable for input list size $\ell $ if for every sufficiently large subset of at least $L$ codewords, there is a coordinate where the codewords take more than $\ell $ values. In this work, we study the list size of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below $q$ is the alphabet size, and $ \varepsilon > 0$ is the gap to capacity). (1) A random linear code of rate $1 - \log _{q}(\ell) - \varepsilon $ requires list size $L \ge \ell ^{\Omega (1/ \varepsilon)}$ for list-recovery from input list size $\ell $ . (2) A random linear code of rate $1 - h_{q}(p) - \varepsilon $ requires list size $L \ge \left \lfloor{ {h_{q}(p)/ \varepsilon +0.99}}\right \rfloor $ for list-decoding from error fraction $p$ . (3) A random binary linear code of rate $1 - h_{2}(p) - \varepsilon $ is list-decodable from average error fraction $p$ with list size with $L \leq \left \lfloor{ {h_{2}(p)/ \varepsilon }}\right \rfloor + 2$ . Our lower bounds follow by exhibiting an explicit subset of codewords so that this subset—or some symbol-wise permutation of it—lies in a random linear code with high probability. Our upper bound follows by strengthening a result of (Li, Wootters, 2018).

Given a probability distribution D over the nonnegative integers, a D-repeat channel acts on an input symbol by repeating it a number of times distributed as D. For example, the binary deletion channel (D=Bernoulli) and the Poisson repeat channel (D=Poisson) are special cases. We say a D-repeat channel is square-integrable if D has finite first and second moments. In this paper, we construct explicit codes for all square-integrable D-repeat channels with rate arbitrarily close to the capacity, that are encodable and decodable in linear and quasi-linear time, respectively. We also consider possible extensions to the repeat channel model, and illustrate how our construction can be extended to an even broader class of channels capturing insertions, deletions, and substitutions.Our work offers an alternative, simplified, and more general construction to the recent work of Rubinstein [3], who attains similar results to ours in the cases of the deletion channel and the Poisson repeat channel. It also slightly improves the runtime and decoding failure probability of the polar codes constructions of Tal et al. [1] and of Pfister and Tal [2] for the deletion channel and certain insertion/deletion/substitution channels. Our techniques follow closely the approaches of Guruswami and Li [4] and Con and Shpilka [5]; what sets apart our work is that to obtain our result, we show that a capacity-achieving code for the channels in question can be assumed to have an "approximate balance" in the frequency of zeros and ones of all sufficiently long substrings of all codewords. This allows us to attain near-capacity-achieving codes in a general setting. We consider this "approximate balance" result to be of independent interest, as it can be cast in much greater generality than just repeat channels.A full version of this paper is available at https://arxiv.org/abs/2201.12746.

This paper shows that there exist Reed--Solomon (RS) codes, over large finite fields, that are combinatorially list-decodable well beyond the Johnson radius, in fact almost achieving list-decoding capacity. In particular, we show that for any $\epsilon\in (0,1]$ there exist RS codes with rate $\Omega(\frac{\epsilon}{\log(1/\epsilon)+1})$ that are list-decodable from radius of $1-\epsilon$. We generalize this result to obtain a similar result on list-recoverability of RS codes. Along the way we use our techniques to give a new proof of a result of Blackburn on optimal linear perfect hash matrices, and strengthen it to obtain a construction of strongly perfect hash matrices. To derive the results in this paper we show a surprising connection of the above problems to graph theory, and in particular to the tree packing theorem of Nash-Williams and Tutte. En route to our results on RS codes, we prove a generalization of the tree packing theorem to hypergraphs (and we conjecture that an even stronger generalization holds). We hope that this generalization to hypergraphs will be of independent interest.