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QEC and EAQEC Codes from Hermitian Sums and Hulls of Cyclic Codes over $\mathbb{F}_2 \times (\mathbb{F}_2+v\mathbb{F}_2)$

This paper constructs quantum and entanglement-assisted quantum error-correcting codes by determining the generator polynomials for Hermitian hulls and sums of cyclic codes over the composite ring $\mathbb{F}_2 \times (\mathbb{F}_2+v\mathbb{F}_2)$ and applying Quantum Construction X and matrix product code methods to these structures.

Original authors: Rabia Zengin, Mehmet Emin Köroğlu

Published 2026-06-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rabia Zengin, Mehmet Emin Köroğlu

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

Imagine you are trying to send a secret message across a stormy sea using a fragile boat. In the world of quantum computers, this "storm" is called decoherence—it's the noise that scrambles information and causes errors. To keep the message safe, scientists build "lifeboats" called Quantum Error-Correcting (QEC) codes. These are special mathematical structures designed to detect and fix mistakes before they ruin the data.

This paper is like a blueprint for building better, sturdier lifeboats using a specific type of mathematical "wood" and a new set of construction tools.

Here is a simple breakdown of what the authors did:

1. The Special Building Material: A "Composite Ring"

Most people build with standard bricks (like simple numbers). These authors decided to build with a more complex material called a composite ring (specifically $F_2 \times (F_2 + vF_2)$ ).

The Analogy: Think of this ring not as a single brick, but as a double-layered brick. One layer is a standard binary brick (0 or 1), and the second layer is a slightly more complex brick that has a special "twist" (represented by $v$ ).
By stacking these double-layered bricks, they create codes that are more flexible and powerful than those made from single-layer bricks.

2. The "Hull" and the "Sum": Finding the Core and the Total

The authors focused on two specific ways to organize these bricks, which they call Cyclic Codes (codes that shift around like a carousel).

The Hull (The Core): Imagine you have a pile of bricks and you want to find the "perfect overlap"—the specific bricks that fit perfectly into both your original pile and its mirror image (the "dual"). This overlap is the Hull. It represents the most stable, central part of the code.
The Sum (The Total): Imagine taking your original pile and adding it to its mirror image to see how much "stuff" you have in total when you combine them. This is the Sum.
The Twist: Usually, scientists look at these shapes using a standard mirror (Euclidean). These authors used a Hermitian mirror, which is a special kind of reflection that accounts for the "twist" ( $v$ ) in their double-layered bricks. This revealed new shapes and overlaps that standard mirrors couldn't see.

3. The "Gray Map": Translating the Blueprint

The double-layered bricks are great for design, but quantum computers speak a different language (binary 0s and 1s).

The Analogy: The authors used a tool called the Gray Map. Think of this as a translator or a 3D printer. It takes one of their complex, double-layered bricks and prints out three standard binary bricks.
This translation is crucial because it preserves the "weight" (how heavy or important the code is) and the "distance" (how far apart the codes are from each other), ensuring the safety features don't get lost in translation.

4. Building the Lifeboats (QEC Codes)

Once they had the translated binary codes, they used a construction method called Quantum Construction X.

The Process: They took the "Hull" (the stable core) and the "Sum" (the total collection) they found earlier. By applying a specific mathematical recipe to these, they constructed new Quantum Error-Correcting (QEC) codes.
The Result: They found that using their "Hermitian" (twisted) method produced lifeboats with different, and often better, dimensions than the old "Euclidean" (standard) method. They listed many specific examples (like a boat that can hold 12 people with a safety rating of 2) that are among the best currently known.

5. The "Entanglement" Upgrade (EAQEC Codes)

The paper also tackled a more advanced type of lifeboat called Entanglement-Assisted QEC (EAQEC).

The Analogy: Imagine two boats that are magically tethered together (entangled). If one boat hits a wave, the tether helps the other boat stabilize it. This "tether" allows the boats to be more efficient.
To build these, the authors used Matrix Product Codes. Think of this as taking two different sets of bricks (two different cyclic codes) and weaving them together into a single, massive, super-strong mat.
They specifically looked for codes where the "Hull" was empty (called LCD codes). An empty hull is like a building with no internal walls blocking the view; it makes the structure very rigid and easy to check. By weaving these rigid structures together, they created new families of entangled lifeboats that can carry more data or fix more errors than previous designs.

Summary

In short, the authors:

Built codes using a special double-layered mathematical material.
Used a special mirror (Hermitian) to find the most stable parts (Hulls) and total combinations (Sums) of these codes.
Translated these complex codes into standard binary language using a Gray Map.
Used these translations to build new, improved quantum lifeboats (QEC codes) and super-efficient tethered lifeboats (EAQEC codes).

They didn't just build one boat; they provided a whole new catalog of blueprints (tables of parameters) that other scientists can use to build better quantum computers.

Technical Summary: QEC and EAQEC Codes from Hermitian Sums and Hulls of Cyclic Codes over $\mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)$

Problem Statement
The paper addresses the construction of Quantum Error-Correcting (QEC) and Entanglement-Assisted Quantum Error-Correcting (EAQEC) codes. While previous research has established methods for deriving these codes using Euclidean sums and hulls of cyclic codes over the composite ring $\mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)$ , this work investigates whether utilizing Hermitian structures (specifically Hermitian sums and hulls) over the same ring yields distinct and potentially superior code parameters. The goal is to determine the generator polynomials for these Hermitian structures and apply them to quantum code construction.

Methodology
The authors employ a multi-step algebraic framework:

Ring Structure and Definitions: The study operates over the finite commutative ring $R = \mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)$ , where $v^2 = v$ . The authors define Euclidean and Hermitian inner products, dual codes, and the Gray map $\tau: R \to \mathbb{F}_2^3$ .
Gray Map Properties: A critical component is the Gray map $\tau$ , which is shown to be an orthogonality-preserving and weight-preserving map. Specifically, the paper establishes that the Gray image of the Hermitian dual of a code over $R$ corresponds to the Euclidean dual of the Gray image of the code over $\mathbb{F}_2$ .
Generator Polynomials: The authors derive explicit generator polynomials for the Hermitian hulls ( $Hull_H(C)$ ) and Hermitian sums ( $Sum_H(C)$ ) of cyclic codes over $R$ . These are expressed in terms of the least common multiple (LCM) and greatest common divisor (GCD) of the generator polynomials of the code and its Hermitian dual.
Quantum Construction X: To construct QEC codes, the authors apply the Hermitian dual version of "Quantum Construction X" to the derived Hermitian hulls and sums. This method transforms classical linear codes into quantum codes.
Matrix Product Codes for EAQEC: For EAQEC codes, the authors utilize matrix product code methods applied to Linear Complementary Dual (LCD) codes. They establish conditions under which cyclic codes over $R$ are Hermitian LCD codes (i.e., their Hermitian hull is trivial) and use these to generate EAQEC codes.

Key Contributions and Results

Theoretical Derivation: The paper provides theorems (Theorem 8) that explicitly define the generator polynomials and dimensions of Hermitian hulls and sums for cyclic codes over $R$ .
New QEC Codes: By applying Quantum Construction X to these Hermitian structures, the authors construct binary QEC codes. Theorem 10 presents a general formula for the parameters $[[N, K, D]]$ $[[N, K, D]]$ of these codes.
- The authors demonstrate that the parameters obtained via the Hermitian approach differ from those obtained via the Euclidean approach (as seen in Theorem 3.5 of reference [22]).
- Example 1 and Table 1 provide specific instances of binary QEC codes (e.g., $[[26, 12, \ge 2]]$ ). The authors note that many of these parameters match or improve upon the best-known codes listed in the online database codetables.de.
New EAQEC Codes: The paper constructs families of EAQEC codes using matrix products of LCD codes.
- Theorem 14 establishes the existence of binary EAQEC codes with parameters derived from two cyclic codes $C_1$ and $C_2$ over $R$ .
- Examples 2 and Tables 2–4 list specific new EAQEC codes for lengths $n=3, 5$ and other values. The authors explicitly state that these codes are not currently included in the EAQECC section of codetables.de.

Significance and Claims
The paper claims that the proposed method yields new QEC and EAQEC code parameters that are distinct from those derived using Euclidean sums and hulls. The significance lies in:

Expansion of Code Families: The work expands the library of known quantum codes by leveraging the specific algebraic properties of the Hermitian inner product over the ring $\mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)$ .
Optimality: Several constructed codes are reported to be optimal with respect to the quantum Singleton bound or are among the best-known codes.
Generalization Potential: The authors modestly suggest that the theory presented could be generalized for primes $p > 2$ , which might yield further new and good parameters for both QECCs and EAQECCs.

The work relies on computational verification using the MAGMA computer algebra system to generate the specific code parameters listed in the tables.