arXiv subjects

Isaac Barouch Essayag

Publications and source records attributed to Isaac Barouch Essayag.

6 recordsLinked to original sources

arXiv2026

Auxiliary Codes and the Generalized Packing-Covering Conjecture

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. We also prove the bound on generalized Hamming weights $d_t(C)\le2R_t(C)+1$ whenever the alphabet size $q$ satisfies $q\ge R_t(C)$. Both results use an auxiliary-code criterion that converts a covering property in the syndrome space into a weight bound. For binary primitive BCH codes, the packing radius is strictly smaller than the covering radius for every fixed error parameter and order, both at least two, once the extension degree is sufficiently large; this follows from existing covering bounds.

cs.IT↗

arXiv2026

From Round-Trip State Echo to Error Recovery: Snapshot-Resolved Quantum-Hardware Diagnostics

Quantum-hardware scores vary across workloads and execution times. We study these variations through repeated communication and deletion-recovery experiments on superconducting and trapped-ion hardware. Our opening test is round-trip state echo (RTSE): prepare one of four tetrahedral qubit states, move it along a route and back by swaps, undo the preparation at the root, and measure the return probability. Repeating identical native communication programs on IQM Emerald changed the differences between tasks on the same routes, while the aggregate RTSE estimates differed by only $0.00125$. A separate two-window experiment found a drop in root return probability from route length 2 to 10 for both RTSE and its remote-inverse do-nothing predecessor; the prespecified directional test did not support better retention by RTSE. Across 64 fixed combinations of placement, input state, and deleted position, the IQM recovery mean fell from $0.738$ to $0.624$. On IonQ's five submitted virtual wires, with advertised all-to-all connectivity, recovery means of $0.911$ and $0.923$ exceeded the prespecified two-thirds reference in both windows. The corresponding differences from a control using the decoder's adjoint were $0.446$ and $0.443$. These measurements characterize complete compiled workloads at dated executions. Recovery is tested after subsystem discard in the circuit model, and each hardware interface is analyzed separately.

quant-ph↗

arXiv2026

The Exact Second Generalized Covering Radius of Binary Primitive Triple-Error-Correcting BCH Codes

Let $C_m=\operatorname{BCH}(3,m)$ 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 $m\geq 5$. Equivalently, every two-dimensional syndrome subspace is contained in the binary span of at most eight parity-check columns, and eight columns are necessary in the worst case. This matches the known lower bound and settles the parameter for the entire binary primitive triple-error-correcting BCH family.

cs.IT↗

arXiv2026

Multiset Deletion Codes: Cyclic Constructions, Bounds, and Exact Results

We study deletion-correcting codes in the space of length-$n$ multisets over a $q$-ary alphabet. We present an explicit cyclic Sidon-type construction for arbitrary alphabet size $q$ and deletion radius $t$, defined by a single congruence modulo $t(t+1)^{q-2}+1$. The construction has redundancy at most $\log_q(t(t+1)^{q-2}+1)$ and admits linear-time online decoding for fixed $q$ and $t$ after finite preprocessing. We prove that its syndrome classes are asymptotically balanced and compare several general upper bounds. For a single deletion, we show that the natural sum-modulo construction is asymptotically optimal for every fixed $q$. We also obtain exact results for $q=3$ and $q=4$, including uniqueness results for optimal codes in the relevant parameter ranges, and formulate conjectures for prime alphabets.

cs.IT↗

arXiv2026

Polynomial Constructions and Deletion-Ball Geometry for Multiset Deletion Codes

We study error-correcting codes in the space $\mathcal{S}_{n,q}$ of length-$n$ multisets over a $q$-ary alphabet under the deletion metric, motivated by permutation channels in which ordering is completely lost and errors act only on symbol multiplicities. We develop two complementary directions. First, we present polynomial Sidon-type constructions over finite fields, in both projective and affine forms, yielding multiset $t$-deletion-correcting codes in the regime $t<q$ with redundancy $t+O(1)$, independent of the blocklength $n$. Second, we develop a geometric analysis of deletion balls in $\mathcal{S}_{n,q}$. Using difference-vector representations together with a diagonal reduction of the relevant generating functions, we derive exact generating-function expressions for individual deletion-ball sizes, exact formulas for the number of ordered pairs of multisets at a fixed distance $m$, and consequently for the average ball size. We prove that radius-$r$ deletion balls are minimized at extreme multisets and maximized at the most balanced multisets, giving a formal global characterization of extremal centers in $\mathcal{S}_{n,q}$. We further relate the maximal-ball value to the ideal difference set $S_{q-1}(r,r)$ through boundary truncation, obtaining explicit closed forms for $q=2$ and $q=3$. These geometric results lead to volume-based bounds on code size, including sphere-packing upper bounds, a boundary-aware analysis of code--anticode arguments, and Gilbert--Varshamov-type lower bounds governed by exact average ball sizes. For fixed $q$ and $t$, the resulting average-ball lower bound matches the interior-difference-set scale asymptotically.

cs.IT↗

arXiv2025

Optimal Functional $2^{s-1}$-Batch Codes: Exploring New Sufficient Conditions

A functional $k$-batch code of dimension $s$ consists of $n$ servers storing linear combinations of $s$ linearly independent information bits. These codes are designed to recover any multiset of $k$ requests, each being a linear combination of the information bits, by $k$ disjoint subsets of servers. A recent conjecture suggests that for any set of $k = 2^{s-1}$ requests, the optimal solution requires $2^s-1$ servers. This paper shows that the problem of functional $k$-batch codes is equivalent to several other problems. Using these equivalences, we derive sufficient conditions that improve understanding of the problem and enhance the ability to find the optimal solution.

cs.IT↗