Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.02512

Two-Sided Product Expanding Codes via Rademacher Matrices

Eshan Chattopadhyay·Noam Ringach·Nicholas Spooner
View original source ↗Read source document ↗Export metadata ↓

Abstract

We give a proof of the existence of two-sided product expanding codes which, unlike the earlier result of Kalachev and Panteleev (FOCS, 2025), does not rely on explicit constructions of asymptotically optimal locally testable codes ($c^3$-LTCs). For every fixed number of component codes and dimensions whose rates are bounded away from zero and one, we show that independent random linear codes over a sufficiently large prime field have constant two-sided product expansion with probability tending to one. The tradeoff is that our proof requires the characteristic to grow with the block length, while [KP25] takes extensions of $\mathbb{F}_2$. To replace the use of $c^3$-LTCs, we develop several new techniques that we view as interesting in their own right. Instead of working directly over finite fields, we work over the reals and use random Rademacher matrices for the generator matrices of the component codes. From here, we we show that the extendability of $\varepsilon$-closed sets can be reduced to controlling the operator norm of sparse restrictions of carefully chosen Gram matrices. Applying the trace power method to bound this norm reduces to bounding the number of possible labelings of certain closed walks on bipartite graphs, which we bound using the sparsity of the operators and bounds on the number of equivalence classes of weak Wigner words from Anderson and Zeitouni (Probab. Theory Relat. Fields., 2006), which were originally applied to band matrices. Due to our techniques not relying on explicit $c^3$-LTCs, we believe that they form a promising starting point towards showing the existence of product expanding tensor codes where the component codes have non-trivial automorphism groups and coboundary expanding codes that could be used as the local codes of non-cubical complexes, such as simplicial complexes.

Explore related subjects

cs.ITcs.CCcs.DMmath.IT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Eshan Chattopadhyay, Noam Ringach, Nicholas Spooner. 2026-10-01. Two-Sided Product Expanding Codes via Rademacher Matrices. https://arxiv.org/abs/2610.02512

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Higher-order Common Information

Shannon's mutual information quantifies dependence between two random variables. We introduce \emph{higher-order common information} (HCI), a measure of statistical relevance across $n$ random variables. HCI is defined through a sequential information-bottleneck construction in which an auxiliary representation is generated locally from a single reference source and successively optimized for relevance to the remaining sources. The terminal representation is evaluated by its minimum mutual information with any individual source. Thus, an HCI value of $r$ guarantees a locally generated representation having at least $r$ bits of statistical information about every source. We derive closed-form expressions for jointly Gaussian sources and for a binary common-source model for arbitrary finite $n$. For finite-alphabet sources, we further show that the HCI is lower-bounded by the Gács--Körner common information, while it is upper-bounded by the smallest pairwise mutual information. Finally, we propose an HCI-inspired numerical approximation and illustrate its application to multivariate EEG data.

cs.IT↗
arXiv2026

The kernel-block rank profiles of the $\mathbb{Z}_2\mathbb{Z}_4$-linear and the $\mathbb{Z}_{2^s}$-linear Hadamard codes, and a complete classification of the $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes

The kernel of a binary code containing the zero word partitions the binary coordinates into blocks, two coordinates lying in the same block when every kernel word takes the same value in both, and the \emph{kernel-block rank profile} is the multiset of the dimensions of its linear span punctured on those blocks. For the family $H^{t_1,t_2,t_3}$ of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes, this invariant is known explicitly and gives a complete classification of the family. In this paper, we compute it for the $\mathbb{Z}_2\mathbb{Z}_4$-linear and $\mathbb{Z}_{2^s}$-linear Hadamard families with which those codes are compared, and we prove that it is constant for every nonlinear $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard code and for every nonlinear $\mathbb{Z}_{2^s}$-linear Hadamard code $\bar H^{a_1,\dots,a_s}$ with $s\geq2$. The second statement is obtained without any rank formula, by exhibiting coordinate permutations that preserve the code and act transitively on its kernel blocks, which makes the argument uniform in $s$. We also prove a descent theorem: if $\bar H^{a_1,\dots,a_s}$ is nonlinear and $a_1\geq2$, then the code punctured on one kernel block is the $\mathbb{Z}_{2^{s-1}}$-linear Hadamard code $\bar H^{a_1,\dots,a_{s-1}}$. Consequently, the constant local rank equals $rank(\bar H^{a_1,\dots,a_{s-1}})$ and is at least $t-κ+2$, where $2^t$ is the length and $κ$ the kernel dimension. These results separate every nonlinear $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code from every $\mathbb{Z}_4$-linear, $\mathbb{Z}_2\mathbb{Z}_4$-linear and $\mathbb{Z}_{2^s}$-linear Hadamard code of the same length, except for the single infinite family $H^{1,1,t-4}$ and $\bar H^{2,0,t-5}$ with $t\geq5$. The members of this infinite family agree in the rank, kernel dimension and the kernel-block rank profile, but a two-block refinement separates them.

cs.IT↗
arXiv2026

Random Quantum LDPC Codes Approaching the Gilbert-Varshamov Bound

We show that for any $R,ε>0$ and any prime $p\geq 2$, there exists an infinite family of $p$-ary quantum low-density parity-check (QLDPC) codes, rate $R$, checks of weight $O_ε(1)$, and normalized distance at least $δ_{\mathrm{GV}}(p,R)-ε$. Here, $δ_{\mathrm{GV}}(p,R)$ denotes the quantum Gilbert-Varshamov (GV) bound for $p$-ary stabilizer codes. In fact, we construct an ensemble of such QLDPC codes, such that a random code from this ensemble is close to the GV bound with high probability. Moreover, this ensemble matches the performance of random stabilizer codes on several quantum channels. Specifically, it approaches the quantum capacity of the erasure channel and the hashing bound for memoryless Pauli channels, including the depolarizing channel. A significant challenge in working with QLDPC codes is that they are necessarily \emph{degenerate}, i.e., contain many low-weight stabilizers. A key contribution of our work is a construction of QLDPC codes with quantitative control on their degeneracy. These codes are obtained by combining known constructions of asymptotically good QLDPC codes with the expander-based distance amplification procedure of Alon, Edmonds, and Luby [FOCS'95]. Our random ensemble is constructed by starting with these low-degeneracy QLDPC codes near the quantum Singleton bound and concatenating each coordinate with a random inner code. This can be viewed as a quantum analogue of Thommesen's construction, and as an LDPC version of a result of Ouyang, with an appropriately designed outer code.

cs.IT↗