arXiv subjects

Itzhak Tamo

Publications and source records attributed to Itzhak Tamo.

At least 19 recordsLinked to original sources

arXiv2026

Quantum Codes with Transversal $CCZ$ Gates and Sublinear $Z$-Stabilizers

We construct asymmetric quantum CSS codes with transversal $CCZ$ gates from algebraic expander codes \cite{KT26}. For every fixed $m\ge 3$, our growing-alphabet codes have length $N$, dimension $Θ(N)$, and distances \[ d_X=Θ(N), \qquad d_Z=Θ(N^{1/m}). \] Moreover, the $Z$-stabilizer space has an explicit generating set of weight $O(N^{1/m})$. We build on the algebraic puncturing framework of Golowich and Guruswami \cite{GG24}, which turns classical codes with the required Schur-product and distance conditions into CSS codes with transversal $CCZ$. However, applying the framework directly to the algebraic expander codes runs into their small dual distance, and therefore produces only sublinear dimension. Our main technical step is a refined puncturing theorem in which the global dual-distance assumption is replaced by a condition only on the selected puncturing set. We also reduce the alphabet to a fixed prime field using a projective-multiplicity version of multiplication-friendly codes. The resulting fixed-prime-field CSS code triples, of length $n$, still have transversal $CCZ$ gates. Their dimension is $Θ(n/(\log n)^4)$, with distances \[ d_X=Ω\!\left(\frac{n}{(\log n)^4}\right), \qquad d_Z=Ω\!\left(\frac{n^{1/m}}{(\log n)^{4/m}}\right), \] and the $Z$-stabilizer generating set remains sublinear.

cs.IT↗

arXiv2026

Breaking the bicycle frame: Coset-based quantum LDPC codes

Generalizing the construction of two-block group algebra (2BGA) codes, we introduce a family of two-block quantum LDPC codes constructed using the action of a group on the cosets of its subgroup. This replaces the regular group actions of the earlier two-block constructions and significantly expands the search space, yielding new quantum LDPC codes outside the 2BGA family. Through a computer search, we identify several new quantum LDPC codes, including weight-6 codes with parameters $[[48,8,6]]$, $[[96,8,10]]$, and $[[224,12,16]]$, as well as weight-8 codes with parameters $[[84,16,8]]$, $[[112,16,10]]$, $[[128,16,12]]$, and $[[168,16,15]]$. Furthermore, we introduce a maximally packed syndrome extraction schedule of depth $w+2$, including initialization and measurement steps, for any code with a maximum stabilizer weight of $w$ from our family. Under a standard circuit-level noise model, our codes, when decoded using BP-OSD, perform competitively with BB codes, achieving thresholds of $\approx0.65\%$ for the weight-6 family and $\approx0.35\%$ for the weight-8 family. Finally, we introduce a group-theoretic framework to generate sequences of graph-based covers of 2BGA codes, recovering and extending recent results on code constructions of this type.

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arXiv2026

Codes with Large Minimum Distance in Product Codes: Explicit Constructions and Bounds

Products of MDS codes are of major practical importance; for a recent example, they are used in Data Availability Sampling (DAS) in blockchain networks such as Celestia and as part of the Ethereum roadmap. This motivates us to consider subcodes of such codes with the goal of obtaining a larger minimum distance. In this paper, we present explicit constructions of subcodes of Reed--Solomon product codes, along with bounds on their minimum distance. In particular, they achieve an optimal or near-optimal dimension--distance tradeoff. For component codes of dimension $r$, our construction requires a field whose size is bounded linearly by the overall product code length, and attains the maximum possible minimum distance for subcode dimensions $r^2-1$, $r^2-2$, and all dimensions at most $2r-1$. Furthermore, we establish a new upper bound on the minimum distance of subcodes of the product of two codes with identical parameters.

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arXiv2026

Algebraic Expander Codes

Expander (Tanner) codes combine sparse graphs with local constraints, enabling linear-time decoding and asymptotically good distance--rate tradeoffs. A standard constraint-counting argument yields the global-rate lower bound $R\ge 2r-1$ for a Tanner code with local rate $r$, which gives no positive-rate guarantee in the low-rate regime $r\le 1/2$. This regime is nonetheless important in applications that require algebraic local constraints (e.g., Reed--Solomon locality and the Schur-product/multiplication property). We introduce \emph{Algebraic Expander Codes}, an explicit algebraic family of Tanner-type codes whose local constraints are Reed--Solomon and whose global rate remains bounded away from $0$ for every fixed $r\in(0,1)$ (in particular, for $r\le 1/2$), while achieving constant relative distance. Our codes are defined by evaluating a structured subspace of polynomials on an orbit of a non-commutative subgroup of $\mathrm{AGL}(1,\mathbb{F})$ generated by translations and scalings. The resulting sparse coset geometry forms a strong spectral expander, proved via additive character-sum estimates, while the rate analysis uses a new notion of polynomial degree and a polytope-volume/dimension-counting argument.

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arXiv2025

Pure-State Quantum Tomography with Minimal Rank-One POVMs

Quantum state tomography seeks to reconstruct an unknown state from measurement statistics. A finite measurement (POVM) is \emph{pure-state informationally complete} (PSI-Complete) if the outcome probabilities determine any pure state up to a global phase. We study \emph{rank-one} POVMs that are minimally sufficient for this task. We call such a POVM \emph{vital} if it is PSI-Complete but every proper subcollection is not PSI-Complete. We prove sharp upper bounds on the size of vital rank-one POVMs in dimension $n$: the size is at most $\binom{n+1}{2}$ over $\mathbb{R}$ and at most $n^{2}$ over $\mathbb{C}$, and we give constructions that attain these bounds. In the real case, we further exhibit a connection to block designs: whenever $w \mid n(n-1)$, an $(n,w,w-1)$ design produces a vital rank-one POVM with $n + n(n-1)/w$ outcomes. We provide explicit constructions for $w=2,n-1$, and $n$.

quant-ph↗

arXiv2025

Making it to First: The Random Access Problem in DNA Storage

In this paper, we study the Random Access Problem in DNA storage, which addresses the challenge of retrieving a specific information strand from a DNA-based storage system. In this framework, the data is represented by $k$ information strands which represent the data and are encoded into $n$ strands using a linear code. Then, each sequencing read returns one encoded strand which is chosen uniformly at random. The goal under this paradigm is to design codes that minimize the expected number of reads required to recover an arbitrary information strand. We fully solve the case when $k=2$, showing that the best possible code attains a random access expectation of $1+\frac{2}{\sqrt{2}+1}\approx 0.914\cdot 2$ for $q$ large enough. Moreover, we generalize a construction from~\cite{GMZ24}, specifically to $k=3$, for any value of $k$. Our construction uses $B_{k-1}$ sequences over $\mathbb{Z}_{q-1}$, that always exist over large finite fields. We show that for every $k\geq 4$, this generalized construction outperforms all previous constructions in terms of reducing the random access expectation.

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arXiv2025

New Bounds and Constructions for Variable Packet-Error Coding

In this paper, we consider the problem of variable packet-error coding, which emerges in network communication scenarios where a source transmits information to a destination through multiple disjoint paths. The objective is to design codes with dynamic error-correcting capabilities that adapt to varying numbers of errors. Specifically, we first provide several bounds on the rate--distortion trade-off for general variable packet-error coding schemes. Then, we present two explicit constructions of variable packet-error coding schemes. The first construction uses higher-order MDS codes and provides a coding scheme that achieves a better rate--distortion trade-off compared to known results for general parameter regimes. The second construction is based on a variant of the repetition code and yields a coding scheme with an optimal rate--distortion trade-off, with respect to our bound, for certain parameter regimes.

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arXiv2025

Private Information Retrieval on Multigraph-Based Replicated Storage

We consider the private information retrieval (PIR) problem for a multigraph-based replication system, where each set of $r$ files is stored on two of the servers according to an underlying $r$-multigraph. Our goal is to establish upper and lower bounds on the PIR capacity of the $r$-multigraph. Specifically, we first propose a construction for multigraph-based PIR systems that leverages the symmetry of the underlying graph-based PIR scheme, deriving a capacity lower bound for such multigraphs. Then, we establish a general upper bound using linear programming, expressed as a function of the underlying graph parameters. Our bounds are demonstrated to be tight for PIR systems on multipaths for even number of vertices.

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arXiv2025

A point-variety incidence theorem over finite fields, and its applications

Incidence problems between geometric objects is a key area of focus in the field of discrete geometry. Among them, the study of incidence problems over finite fields have received a considerable amount of attention in recent years. In this paper, by characterizing the singular values and singular vectors of the corresponding incidence matrix through group algebras, we prove a bound on the number of incidences between points and varieties of a certain form over finite fields. Our result leads to a new incidence bound for points and flats in finite geometries, which improves previous results for certain parameter regimes. As another application of our point-variety incidence bound, we extend a result on pinned distance problems by Phuong, Thang, and Vinh, and independently by Cilleruelo, Iosevich, Lund, Roche-Newton, and Rudnev, under a weaker condition.

math.CO↗

arXiv2025

New Capacity Bounds for PIR on Graph and Multigraph-Based Replicated Storage

In this paper, we study the problem of private information retrieval (PIR) in both graph-based and multigraph-based replication systems, where each file is stored on exactly two servers, and any pair of servers shares at most $r$ files. We derive upper bounds on the PIR capacity for such systems and construct PIR schemes that approach these bounds. For graph-based systems, we determine the exact PIR capacity for path graphs and improve upon existing results for complete bipartite graphs and complete graphs. For multigraph-based systems, we propose a PIR scheme that leverages the symmetry of the underlying graph-based construction, yielding a capacity lower bound for such multigraphs. Furthermore, we establish several general upper and lower bounds on the PIR capacity of multigraphs, which are tight in certain cases.

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arXiv2025

Bounds on Box Codes

Let $n_q(M,d)$ be the minimum length of a $q$-ary code of size $M$ and minimum distance $d$. Bounding $n_q(M,d)$ is a fundamental problem that lies at the heart of coding theory. This work considers a generalization $n^\bx_q(M,d)$ of $n_q(M,d)$ corresponding to codes in which codewords have \emph{protected} and \emph{unprotected} entries; where (analogs of) distance and of length are measured with respect to protected entries only. Such codes, here referred to as \emph{box codes}, have seen prior studies in the context of bipartite graph covering. Upper and lower bounds on $n^\bx_q(M,d)$ are presented.

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arXiv2024

Quantum Locally Recoverable Codes via Good Polynomials

Locally recoverable codes (LRCs) with locality parameter $r$ can recover any erased code symbol by accessing $r$ other code symbols. This local recovery property is of great interest in large-scale distributed classical data storage systems as it leads to efficient repair of failed nodes. A well-known class of optimal (classical) LRCs are subcodes of Reed-Solomon codes constructed using a special type of polynomials called good polynomials. Recently, Golowich and Guruswami initiated the study of quantum LRCs (qLRCs), which could have applications in quantum data storage systems of the future. The authors presented a qLRC construction based on good polynomials arising out of subgroups of the multiplicative group of finite fields. In this paper, we present a qLRC construction method that can employ any good polynomial. We also propose a new approach for designing good polynomials using subgroups of affine general linear groups. Golowich and Guruswami also derived a lower bound on the minimum distance of their qLRC under the restriction that $r+1$ is prime. Using similar techniques in conjunction with the expander mixing lemma, we develop minimum distance lower bounds for our qLRCs without the $r+1$ prime restriction.

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arXiv2024

Efficient Algorithms for Constructing Minimum-Weight Codewords in Some Extended Binary BCH Codes

We present $O(m^3)$ algorithms for specifying the support of minimum-weight words of extended binary BCH codes of length $n=2^m$ and designed distance $d(m,s,i):=2^{m-1-s}-2^{m-1-i-s}$ for some values of $m,i,s$, where $m$ may grow to infinity. The support is specified as the sum of two sets: a set of $2^{2i-1}-2^{i-1}$ elements, and a subspace of dimension $m-2i-s$, specified by a basis. In some detail, for designed distance $6\cdot 2^j$, we have a deterministic algorithm for even $m\geq 4$, and a probabilistic algorithm with success probability $1-O(2^{-m})$ for odd $m>4$. For designed distance $28\cdot 2^j$, we have a probabilistic algorithm with success probability $\geq 1/3-O(2^{-m/2})$ for even $m\geq 6$. Finally, for designed distance $120\cdot 2^j$, we have a deterministic algorithm for $m\geq 8$ divisible by $4$. We also present a construction via Gold functions when $2i|m$. Our construction builds on results of Kasami and Lin (IEEE T-IT, 1972), who proved that for extended binary BCH codes of designed distance $d(m,s,i)$, the minimum distance equals the designed distance. Their proof makes use of a non-constructive result of Berlekamp (Inform. Contrl., 1970), and a constructive ``down-conversion theorem'' that converts some words in BCH codes to lower-weight words in BCH codes of lower designed distance. Our main contribution is in replacing the non-constructive argument of Berlekamp by a low-complexity algorithm. In one aspect, we extends the results of Grigorescu and Kaufman (IEEE T-IT, 2012), who presented explicit minimum-weight words for designed distance $6$ (and hence also for designed distance $6\cdot 2^j$, by a well-known ``up-conversion theorem''), as we cover more cases of the minimum distance. However, the minimum-weight words we construct are not affine generators for designed distance $>6$.

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arXiv2024

$\varepsilon$-MSR Codes for Any Set of Helper Nodes

Minimum storage regenerating (MSR) codes are a class of maximum distance separable (MDS) array codes capable of repairing any single failed node by downloading the minimum amount of information from each of the helper nodes. However, MSR codes require large sub-packetization levels, which hinders their usefulness in practical settings. This led to the development of another class of MDS array codes called $\varepsilon$-MSR codes, for which the repair information downloaded from each helper node is at most a factor of $(1+\varepsilon)$ from the minimum amount for some $\varepsilon > 0$. The advantage of $\varepsilon$-MSR codes over MSR codes is their small sub-packetization levels. In previous constructions of epsilon-MSR codes, however, several specific nodes are required to participate in the repair of a failed node, which limits the performance of the code in cases where these nodes are not available. In this work, we present a construction of $\varepsilon$-MSR codes without this restriction. For a code with $n$ nodes, out of which $k$ store uncoded information, and for any number $d$ of helper nodes ($k\le d<n$), the repair of a failed node can be done by contacting any set of $d$ surviving nodes. Our construction utilizes group algebra techniques, and requires linear field size. We also generalize the construction to MDS array codes capable of repairing $h$ failed nodes using $d$ helper nodes with a slightly sub-optimal download from each helper node, for all $h \le r$ and $k \le d \le n-h$ simultaneously.

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arXiv2024

Combinatorial alphabet-dependent bounds for insdel codes

Error-correcting codes resilient to synchronization errors such as insertions and deletions are known as insdel codes. Due to their important applications in DNA storage and computational biology, insdel codes have recently become a focal point of research in coding theory. In this paper, we present several new combinatorial upper and lower bounds on the maximum size of $q$-ary insdel codes. Our main upper bound is a sphere-packing bound obtained by solving a linear programming (LP) problem. It improves upon previous results for cases when the distance $d$ or the alphabet size $q$ is large. Our first lower bound is derived from a connection between insdel codes and matchings in special hypergraphs. This lower bound, together with our upper bound, shows that for fixed block length $n$ and edit distance $d$, when $q$ is sufficiently large, the maximum size of insdel codes is $ \frac{q^{n-\frac{d}{2}+1}}{{n\choose \frac{d}{2}-1}}(1 \pm o(1))$. The second lower bound refines Alon et al.'s recent logarithmic improvement on Levenshtein's GV-type bound and extends its applicability to large $q$ and $d$.

math.CO↗

arXiv2024

Non-Binary Covering Codes for Low-Access Computations

Given a real dataset and a computation family, we wish to encode and store the dataset in a distributed system so that any computation from the family can be performed by accessing a small number of nodes. In this work, we focus on the families of linear computations where the coefficients are restricted to a finite set of real values. For two-valued computations, a recent work presented a scheme that gives good feasible points on the access-redundancy tradeoff. This scheme is based on binary covering codes having a certain closure property. In a follow-up work, this scheme was extended to all finite coefficient sets, using a new additive-combinatorics notion called coefficient complexity. In the present paper, we explore non-binary covering codes and develop schemes that outperform the state-of-the-art for some coefficient sets. We provide a more general coefficient complexity definition and show its applicability to the access-redundancy tradeoff.

cs.IT↗

arXiv2024

Optimal Two-Dimensional Reed--Solomon Codes Correcting Insertions and Deletions

Constructing Reed--Solomon (RS) codes that can correct insertions and deletions (insdel errors) has been considered in numerous recent works. For the special case of two-dimensional RS-codes, it is known [CST23] that an $[n,2]_q$ RS-code that can correct from $n-3$ insdel errors satisfies that $q=Ω(n^3)$. On the other hand, there are several known constructions of $[n,2]_q$ RS-codes that can correct from $n-3$ insdel errors, where the smallest field size is $q=O(n^4)$. In this short paper, we construct $[n,2]_q$ Reed--Solomon codes that can correct $n-3$ insdel errors with $q=O(n^3)$, thereby resolving the minimum field size needed for such codes.

cs.IT↗

arXiv2024

Repairing Reed-Solomon Codes over Prime Fields via Exponential Sums

This paper presents two repair schemes for low-rate Reed-Solomon (RS) codes over prime fields that can repair any node by downloading a constant number of bits from each surviving node. The total bandwidth resulting from these schemes is greater than that incurred during trivial repair; however, this is particularly relevant in the context of leakage-resilient secret sharing. In that framework, our results provide attacks showing that $k$-out-of-$n$ Shamir's Secret Sharing over prime fields for small $k$ is not leakage-resilient, even when the parties leak only a constant number of bits. To the best of our knowledge, these are the first such attacks. Our results are derived from a novel connection between exponential sums and the repair of RS codes. Specifically, we establish that non-trivial bounds on certain exponential sums imply the existence of explicit nonlinear repair schemes for RS codes over prime fields.

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