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Alphabet-Dependent Bounds for Pure Quantum $(r,\rho)$ -Locally Recoverable Codes

This paper derives three new alphabet-dependent upper bounds (Griesmer-like, Plotkin-like, and sphere-packing-like) for pure quantum $(r,\rho)$ -locally recoverable codes using the Hermitian CSS construction, establishing their asymptotic hierarchy and identifying the specific relative-distance regions where each bound provides the tightest rate constraint.

Original authors: Vijay Kumar, Ramakrishna Bandi

Published 2026-09-01

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CC BY 4.0

Original authors: Vijay Kumar, Ramakrishna Bandi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast, invisible architecture of the digital world, information is constantly at risk. Whether it is a photo stored in the cloud or a critical medical record on a server, data can vanish if a single drive fails or a connection drops. To protect against this, engineers use error-correcting codes, which are like adding redundant copies of a message so that if part of it is lost, the rest can be used to rebuild the missing pieces. For decades, these codes have been designed for classical computers, which process information as simple on-off switches. However, the next generation of computing relies on quantum mechanics, where information is stored in delicate states that can exist in multiple possibilities at once. These quantum systems are far more fragile, and the rules for protecting them are different. A specific type of code, known as a locally recoverable code, has emerged as a vital tool for these systems. Its unique strength is that if a piece of data is lost, it can be repaired by looking at only a small, nearby group of other pieces, rather than having to scan the entire massive dataset. This efficiency is crucial for the massive storage systems of the future.

Researchers Vijay Kumar and Ramakrishna Bandi have now taken a closer look at the theoretical limits of these quantum codes. While previous studies had established general rules for how much data these codes could hold, those rules treated the size of the data alphabet as a constant, ignoring the specific dimensions of the quantum units involved. The authors realized that for smaller or moderate-sized quantum systems, these general rules were too loose to be truly useful. They set out to find tighter, more precise limits that account for the specific size of the quantum alphabet. By focusing on a particular construction method that links classical codes to quantum ones, they derived three new mathematical boundaries. These boundaries act like a set of fences, defining exactly how much information can be packed into a quantum code before it becomes impossible to recover from errors, depending on the specific size of the quantum system being used.

The team discovered that the old, general rules were not the most restrictive ones available. Instead, they found that three new types of limits, which they named after famous concepts in coding theory, provide a much sharper picture of reality. One of these limits, which they call a Plotkin-like bound, proved to be the strictest of all for certain types of quantum codes. It essentially says that if you want to correct a specific number of errors, there is a hard ceiling on how much information you can store, and this ceiling is lower than what the older, more general formulas suggested. Another limit, based on the idea of packing spheres in a high-dimensional space, showed that for very small error rates, the constraints change in a different way, creating a distinct boundary where the efficiency of the code drops off. The researchers mapped out exactly where each of these new limits applies, showing that for many practical scenarios, the old rules were overly optimistic.

What makes this work significant is that it moves beyond abstract theory to provide concrete, usable constraints for engineers building these systems. The authors did not just suggest these limits; they proved them mathematically using a specific method that connects classical linear codes to quantum ones. They showed that for codes with certain properties, the new Plotkin-like bound is strictly tighter than the previously accepted best limits. This means that anyone designing a quantum storage system with these specific parameters must now plan for a lower capacity than they might have thought possible. The study also clarified the relationship between the size of the quantum alphabet and the code's ability to recover from errors, revealing that smaller alphabets impose stricter limits on performance. By establishing this hierarchy of limits, the researchers have provided a more accurate map for the landscape of quantum error correction, ensuring that future designs are built on a foundation of precise, rather than approximate, understanding.

The implications of these findings are immediate for the field of quantum information. By identifying the exact regions where different limits apply, the work helps researchers avoid wasting effort trying to build codes that violate these fundamental boundaries. The authors noted that while they have defined these upper limits, the actual construction of codes that reach these limits remains a task for future work. Their contribution is the rigorous definition of the walls within which these codes must operate. In doing so, they have refined the understanding of how much data can be safely stored and recovered in a quantum environment, ensuring that the path toward reliable quantum storage is guided by the most accurate constraints available.

Technical Summary: Alphabet-Dependent Bounds for Pure Quantum $(r, \rho)$ -Locally Recoverable Codes

Problem Statement
Quantum Locally Recoverable Codes (qLRCs) are designed to allow the recovery of an erased qudit from a small subset of other qudits, a property essential for efficient error correction in distributed quantum storage. Specifically, an $(r, \rho)$ -qLRC ensures that any single qudit can be recovered from a set of at most $r + \rho - 1$ qudits, even if up to $\rho - 1$ additional erasures occur within that recovery set.

Existing theoretical limits for these codes, specifically the Singleton-like bound and the GG Singleton-like bound, are alphabet-independent. They rely solely on parameters such as block length ( $n$ ), dimension ( $\kappa$ ), locality ( $r, \rho$ ), and minimum distance ( $\delta$ ), ignoring the dimension of the underlying qudit alphabet ( $q$ ). Consequently, these bounds are often loose for small-to-moderate qudit dimensions. The paper addresses the gap in deriving alphabet-dependent upper bounds that more accurately reflect the trade-offs between rate and distance for pure $(r, \rho)$ -qLRCs.

Methodology
The authors derive new bounds by leveraging the Hermitian CSS construction, which maps classical linear codes over $\mathbb{F}_{q^2}$ to quantum codes over $\mathbb{F}_q$ .

Classical Foundation: The work builds upon recent results for classical $(r, \rho)$ -LRCs by Gao et al., who established alphabet-dependent bounds (Griesmer-like, sphere-packing-like, and Plotkin-like) for classical codes.
Translation to Quantum: The authors apply these classical bounds to the underlying Hermitian dual-containing classical codes required by the CSS construction. By substituting the quantum parameters ( $n, \kappa, \delta$ ) with their classical counterparts ( $n, k, d$ ) where $k = (n+\kappa)/2$ and the alphabet size is $q^2$ , they derive corresponding quantum bounds.
Asymptotic Analysis: The paper analyzes the behavior of these bounds as the code length $n$ tends to infinity. It compares the asymptotic forms of the new bounds against existing Singleton-like bounds to establish a hierarchy of tightness.
Optimization: For the sphere-packing-like bound, the authors perform a rigorous optimization over the shortening index $\tau$ (represented asymptotically by a ratio $\lambda$ ) to determine the tightest constraint across different relative distance ( $\Delta$ ) regimes.

Key Contributions and Results
The paper establishes three new alphabet-dependent upper bounds for pure $(r, \rho)$ -qLRCs (where the quantum minimum distance equals the minimum distance of the underlying classical code):

Pure Griesmer-like Bound: An upper bound on the block length $n$ as a function of dimension and distance, incorporating a summation over powers of $q^2$ .
Pure Sphere-Packing-like Bound: An upper bound on the rate $\kappa/n$ involving the volume of Hamming balls in $\mathbb{F}_{q^2}$ . The authors derive an asymptotic form that splits into two regimes based on a critical relative distance $\Delta_{crit}$ . Below this threshold, the bound depends explicitly on the locality parameters and the alphabet size; above it, it converges to the standard asymptotic quantum sphere-packing bound.
Pure Plotkin-like Bound: An upper bound on the relative distance $\delta/n$ that is strictly tighter than the Griesmer-like bound for $r \ge 3$ .

Asymptotic Hierarchy
A central result of the paper is the establishment of a strict hierarchy among the bounds for pure $(r, \rho)$ -qLRCs with $r \ge 3$ . In the asymptotic regime, the bounds satisfy the following ordering of tightness (from tightest to loosest):
$\text{Pure Plotkin-like} \succ \text{Pure Griesmer-like} \succ \text{Pure Singleton-like} \succ \text{GG Singleton-like}$
This demonstrates that the alphabet-dependent Plotkin-like bound provides the most restrictive (and thus most accurate) constraint on the rate-distance trade-off among the derived bounds.

Significance
The paper claims significance in filling a theoretical gap by providing the first alphabet-dependent bounds for pure $(r, \rho)$ -qLRCs. By explicitly incorporating the qudit dimension $q$ , these bounds offer tighter constraints than previously known alphabet-independent limits, particularly for codes with small-to-moderate alphabet sizes. The work characterizes the specific regions of the rate-distance plane where each bound yields the tightest constraint, providing a more nuanced understanding of the fundamental limits of quantum locally recoverable codes. The authors note that constructions attaining these bounds are left for future work.

Technical Summary: Alphabet-Dependent Bounds for Pure Quantum (r,ρ)(r, \rho)(r,ρ)-Locally Recoverable Codes

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Technical Summary: Alphabet-Dependent Bounds for Pure Quantum $(r, \rho)$ -Locally Recoverable Codes