We construct systematic $q$-ary codes correcting two deletions with three redundant symbols over polynomial-size alphabets. For every information length $n\ge2$, our first construction has length $n+3$ over an alphabet of size $q=2^s<2^{32}n^7$ for some integer $s$, with polynomial-time encoding and decoding. A cardina...
The $2\times3$ magic rectangle is a nonlocal game in which two players match entries subject to incompatible parity constraints. We prove that its quantum value is $(1+\sqrt{2/3})/2$ and classify all optimal strategies in arbitrary finite local dimensions, allowing general measurements. Every optimal strategy contains...
Isaac Barouch Essayag, Aryeh Lev Zabokritskiy· 0 citations
We study leakage from Shamir secret sharing when each device uses its share to produce one qubit from an initial quantum resource independent of the secret and sharing randomness. We prove exponentially small trace distance between the leakage states of any two secrets, when reconstruction requires a sufficiently large...
Isaac Barouch Essayag, Aryeh Lev Zabokritskiy· 0 citations
We determine the least Greenberger-Horne-Zeilinger (GHZ) extractability compatible with a Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell score for every number of parties greater than two. Extractability is the largest squared overlap with a GHZ state obtainable by separate local quantum channels. Between the biseparabl...
Let $s\geq 2$ and let $v_1,\ldots,v_{2^{s-1}}\in F_2^s\setminus\{0\}$ have sum zero. The prescribed-difference matching problem asks whether $F_2^s$ can be partitioned into pairs whose differences, counted with multiplicity, are exactly these vectors. Fix a hyperplane $H\leq F_2^s$. An $h$-hole instance is one in which...
Quantum-hardware scores vary across workloads and execution times through repeated communication and deletion-recovery experiments on superconducting and trapped-ion hardware, and these measurements characterize complete compiled workloads at dated executions.
Isaac Barouch Essayag, Aryeh Lev Zabokritskiy· 0 citations
We study linear codes whose coordinates are the ordinary edges and self-loops of complete undirected graphs; a node erasure removes all coordinates incident with a failed vertex. The construction results are binary. For triple-node erasures, we extend the published cyclic construction by allowing a suitable cyclic chec...
Let $C_m:=\mathrm{BCH}(3,m)$ be the binary primitive triple-error-correcting BCH code of length $2^m-1$. We determine its second generalized covering radius exactly: $R_2(C_m)=8$ for every $m\geq5$. Equivalently, every two-dimensional syndrome subspace is contained in the binary span of at most eight parity-check colum...
Isaac Barouch Essayag, Aryeh Lev Zabokritskiy· 6 citations
We study the generalized covering radii of binary primitive BCH codes, which measure how many parity-check columns suffice to span several prescribed syndromes. For the four-error-correcting family of length $2^m-1$, the second radius is exactly $11$ for $m\geq55$, and it is either $11$ or $12$ for $m\geq16$. For every...
Zeev Vladimir Belinsky, Aryeh Lev Zabokritskiy· 5 citations
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 a...
The generalized packing--covering conjecture asks whether, at every order, the packing radius of a linear code is at most its covering radius. We prove the conjecture for every linear code of redundancy at most fourteen over every finite field, extending the previously established range of redundancies at most seven. W...
Isaac Barouch Essayag, Aryeh Lev Zabokritskiy· 2 citations· ⚡1
Quadratic ball discrepancy defines an energy on codes in finite Hamming spaces. At perfect-code parameters, its exact minimizers are the perfect codes. We fix the alphabet size, length, and code cardinality and compare all codes with these parameters. We prove tiling-defect stability: excess discrepancy above the perfe...
V. Grishin, Aryeh Lev Zabokritskiy· 0 citations
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