We extend the Calderbank-Shor-Steane (CSS) quantum code construction to Gaussian integer rings & Zopf;q[i], where q is a positive integer whose prime divisors all satisfy p = 1 (mod 4). Using nested principal ideal codes with Hermitian dual containment C perpendicular to H 2 C_ C1 C_ C2, we construct quantum stabilizer codes with dimension K = |C2|/|C1|. To address the challenges of decoding over rings with zero divisors, we introduce an algebraic error model where correctable errors are characterized by the coset structure of C2 relative to C1. This framework operates independently of specific physical noise assumptions, focusing on deterministic coset distinguishability. The associated coset-based decoder is shown to require exactly|epsilon | = [C2 : C1] membership tests, where the coset index is analytically determined by the norm quotient N(alpha 1)/N(alpha 2). Examples over & Zopf;25[i], & Zopf;85[i], and & Zopf;325[i] demonstrate codes encoding K E {5, 17, 5} logical states. Numerical simulations confirm the coding gain of the proposed scheme and provide benchmarks for its performance under additive noise. These results establish that ring-theoretic coset structures provide an algebraically rigorous and computationally efficient foundation for structured error correction over Gaussian integer rings. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The concept of perfect $\mathcal{E}$-error-correcting codes constitutes a fundamental benchmark in classical coding theory; however, a corresponding definition for these codes remains absent in the quantum domain. This study addresses this theoretical gap by formally defining perfect $\mathcal{E}$-error-correcting quantum codes and constructing a new family of codes that satisfy this definition. We derive these quantum codes from classical linear codes generated over residue class rings of prime Hurwitz integers. The newly constructed quantum codes are presented with their parameters, and those qualifying as perfect $\mathcal{E}$-error-correcting codes are identified. Additionally, this study designs quantum logic gates tailored for the proposed codes over Hurwitz integers. These findings extend the algebraic utility of Hurwitz integers in quantum information theory and provide a novel class of quantum codes equipped with their fundamental logical operators.

In this paper, a novel ideal-based linear code family defined over Gaussian integer rings and compatible with restricted error models is proposed. The algebraic construction of the proposed codes allows for the exact correction of specific error patterns. At the receiver side, as an alternative to the computational complexity caused by the traditional Maximum Likelihood (ML) rule at large constellation sizes, an S-Search algorithm that restricts the search space solely to a fixed-element error set is presented. Furthermore, a hybrid decoding architecture utilizing a dynamic reliability threshold has been developed to autonomously adapt to channel quality and preserve error performance for risky symbols near the decision boundaries. Simulation results obtained under the AWGN channel demonstrate that the proposed hybrid algorithm exhibits an error performance close to that of the ML algorithm, while significantly reducing the computational load by making it completely independent of the constellation size.

Ensuring secure communication in the digital era is of paramount importance. This study introduces a novel encryption model that combines w $$ w $$ -cyclic codes with the one-time pad (OTP) method over Eisenstein–Jacobi integers, achieving information-theoretic security. The method constructs a finite ring over Eisenstein–Jacobi integers, generates one-time keys from w $$ w $$ -cyclic codewords, and uses these keys to encrypt messages, guaranteeing each key is used only once. Multiple plausible ciphertexts are produced for each plaintext, making unauthorized decryption infeasible. Key findings show that the scheme provides high computational efficiency and robust resistance to cryptographic attacks, outperforming existing code-based and public-key systems. Security scales with prime selection, as longer codewords and keys increase protection. The algebraic structure of Eisenstein–Jacobi integers enables efficient key generation and encryption operations, making the model suitable for critical security applications. Existing OTP and cyclic code-based encryption systems are typically limited to binary fields or specific rings, constraining key diversity and scalability. By extending OTP to w $$ w $$ -cyclic codes over Eisenstein–Jacobi integers, the proposed model overcomes these limitations, providing a larger key space and more complex encryption structure. Code-based encryption, including cyclic codes, offers quantum-resistant security. This feature aligns the proposed OTP model with post-quantum cryptography (PQC). This work demonstrates that combining cyclic codes with OTP creates an unbreakable and practical encryption framework, providing resistance against both conventional and quantum attacks.

A quaternion, each component is in integers, is called a Lipschitz integer. A Lipschitz integer is called a primitive Lipschitz integer just if the greatest common divisor of its components is one. Complex-valued codes over Lipschitz integers are obtained accompanied by a respective modulo function. In this study, we demonstrate that some primitive Lipschitz integers, considered suitable for encoding, are inappropriate for constructing complex-valued codes over Lipschitz integers. To solve this problem, we investigate the primitive Lipschitz integers that have the “division with small remainder” property. We construct a named set “Encoder Lipschitz integers,” consists of primitive Lipschitz integers that have the “division with small remainder” property. In addition, we examine some algebraic properties of this set.

Cryptology is a part of mathematics as encryption and decryption. The purpose of encryption is to make information incomprehensible when it is in the hands of unauthorized people. The receiver can decrypt the message that encrypted by the sender with helping of the key. The important point is that the key cannot be decrypted by other people. One Time Pad method solves this problem. The key is used only once each encryption in this method. So, the key becomes harder to guess. If the key is solved by unauthorized people, the message cannot be solved. Because of with each decryption, many meaningful messages are obtained. Every cyclic shift in a cyclic code constructs a new key and in each encryption is used the new key. Many keys are generated thanks to cyclic codes. In this paper, we improve the new encryption scheme by using the cyclic codes with One Time Pad method.

In this paper, the goal is to obtain constacyclic codes over Lipschitz integers in terms of Lipschitz metric. A decoding procedure is proposed for these codes, some of which have been shown to be perfect codes. Performance of constacyclic codes over Lipschitz integers is investigated over Additive White Gaussian Channel (AWGN) by means of symbol error rates and coding gain. According to the achieved results, these codes can be used in coded modulation schemes based on Quadrature Amplitude Modulation (QAM)-type constellations. Furthermore, it is shown that the Lipschitz metric is more suitable than Hamming metric and Lee metric for QAM type two dimensional constellations.

Let $\alpha$ be a prime Hurwitz integer. $\mathcal{H}_{\alpha}$, which is the set of residual class with respect to related modulo function in the rings of Hurwitz integers, is a subset of $\mathcal{H},$ which is the set of all Hurwitz integers. In this study, we present an algebraic construction technique, which is a modulo function formed depending on two modulo operations, for codes over Hurwitz integers. We consider left congruent modulo $\alpha,$ and the domain of related modulo function is $\mathbb{Z}_{N(\alpha)},$ which is residual class ring of ordinary integers with $N(\alpha)$ elements. Therefore, we obtain the residue class rings of Hurwitz integers with $N(\alpha)$ size. In addition, we present some results for mathematical notations used in two modulo functions, and for the algebraic construction technique formed depending upon two modulo functions. Moreover, we presented graphs obtained by graph layout methods, such as spring, high-dimensional, and spiral embedding, for the set of the residual class obtained with respect to the related modulo function in the rings of Hurwitz integers.

In this paper, we study on linear, cyclic, self-dual and self-orthogonal codes over the ring Rq. We especially focus on extremal self-dual codes over $\mathbb {F}_{q}$ . Our method is easy and constructing self-dual codes over $\mathbb {F}_{\boldsymbol {q}}$ . We define a distance-invariant Gray map from Rq to the finite field $\mathbb {F}_{\boldsymbol {q}}$ . It is proved that the images of self-dual codes of length n over Rq under this Gray map correspond to self-dual codes of length 2n over $\mathbb {F}_{\boldsymbol {q}}$ . Using all of these codes, we construct stabilizer quantum codes over $\mathbb {F}_{\boldsymbol {q}}$ . We obtain some self-dual codes and some optimal quantum codes.

Encryption is the process of scrambling a message and can provide a means of securing information. Information security is becoming more important in data storage and transmission. It is known that most encryption method in the literature. In this paper we propose two new encryption schemes by using cyclic codes. Our method is based on the One Time Pad system. We use the properties of cyclic codes to provide its security.

This is a survey on the theory of skew‐cyclic codes over the ring F2+vF2$$ {\mathbbm{F}}_2+v{\mathbbm{F}}_2 $$ , where v2 = v. Coding theory plays an important role in the constructing cryptosystems. In this survey, we propose a public‐key cryptosystem using skew‐cyclic codes. The encryption is done by the Gray map ψ and the projection map φ is used for decryption. This approach is different from the others. More importantly, it is a secure system.

In this paper, a new family of t−error correcting perfect codes over Hurwitz integers is presented. To obtain these perfect codes, the perfect t−dominating sets over the circulant graphs are used. The codewords of such perfect codes are generated by the elements of a subgroup of the considered group.

Abstract The modulo function is used to construct signal constellations over high-dimensional vector spaces such as Gaussian integers, Einstein-Jacobi integers, and quaternion integers. There is a one-to-one relationship between the Euclid division and the modulo function. If the Euclid division works for any quaternion integers, then it has the ”division with small remainder” property. The Hurwitz integers form a subring of ring of quaternions. Hurwitz constellations are constructed by using modulo with primitive Hurwitz integers whose norm is a prime integer. If the norm of Hurwitz integers is not a prime integer, then we can not set up an isomorphism between the residual class ring of ordinary integers and the left (right) equivalence class of Hurwitz integers since the size of sets is to be different. In this study, to solve this problem, we define a new set, named encoder Hurwitz integers, consisting of primitive Hurwitz integers that have the ”division with small remainder” property as an alternative instead of the set of primitive Hurwitz integers. Also, we investigate their performance over additive Gaussian noise (AWGN) channel by means of constellation figure of merit (CFM), average energy, minimum square Euclidean distance, and signal-to-noise ratio (SNR).

The residue class set of a Lipschitz integer is constructed by modulo function with primitive Lipschitz integer whose norm is a prime integer, i.e. prime Lipschitz integer. In this study, we consider primitive Lipschitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Lipschitz integer is less than the norm of the primitive Lipschitz integer used to construct the residue class set of the Lipschitz integer, then, the Euclid division algorithm works for this primitive Lipschitz integer. The Euclid division algorithm always works for prime Lipschitz integers. In other words, the prime Lipschitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Lipschitz residue class set that lies on primitive Lipschitz integers whose norm is not a prime integer. In this study, we solve this problem by defining Lipschitz integers that have the ”division with small remainder” property, namely, encoder Lipschitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Also, we investigate the performances of Lipschitz signal constellations (the left residue class set) obtained by modulo function with Lipschitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by agency of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

The residue class set of a Hurwitz integer is constructed by modulo function with primitive Hurwitz integer whose norm is a prime integer, i.e. prime Hurwitz integer. In this study, we consider primitive Hurwitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Hurwitz integer is less than the norm of the primitive Hurwitz integer used to construct the residue class set of the Hurwitz integer, then, the Euclid division algorithm works for this primitive Hurwitz integer. The Euclid division algorithm always works for prime Hurwitz integers. In other words, the prime Hurwitz integers and halves-integer primitive Hurwitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Hurwitz residue class set that lies on primitive Hurwitz integers that their norms are not a prime integer and their components are in integers set. In this study, we solve this problem by defining Hurwitz integers that have the ”division with small remainder” property, namely, encoder Hurwitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Especially, Euclidean metric. Also, we investigate the performances of Hurwitz signal constellations (the left residue class set) obtained by modulo function with Hurwitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by means of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

In this paper, we aim to obtain quantum error correcting codes from codes over a nonlocal ring R_q=𝔽_q+α𝔽_q . We first define a Gray map φ from R_q^n to 𝔽_q^2n preserving the Hermitian orthogonality in R_q^n to both the Euclidean and trace-symplectic orthogonality in 𝔽_q^2n . We characterize the structure of cyclic codes and their duals over R_q and derive the condition of existence for cyclic codes containing their duals over R_q . By making use of the Gray map φ , we obtain two classes of q -ary quantum codes. We also determine the structure of additive cyclic codes over R_p^2 and give a condition for these codes to be self-orthogonal with respect to Hermitian inner product. By defining and making use of a new map δ , we construct a family of p -ary quantum codes.

Let $i,j,k$ be elements of real quaternions $\mathbb{H}$. Let $\alpha , \beta , \gamma$ be the elements corresponding to $1+i, 1+j, 1+k,$ respectively. In this study,  quantum codes from classical codes over $\mathbb{F}_{2^m}+\alpha \mathbb{F}_{2^m}+\beta \mathbb{F}_{2^m}+ \gamma \mathbb{F}_{2^m}$ are obtained.

Let $i,j,k$ be elements of real quaternions $\mathbb{H}$. Let $\alpha , \beta , \gamma$ be the elements corresponding to $1+i, 1+j, 1+k,$ respectively. In this study, quantum codes from classical codes over $\mathbb{F}_{2^m}+\alpha \mathbb{F}_{2^m}+\beta \mathbb{F}_{2^m}+ \gamma \mathbb{F}_{2^m}$ are obtained.

In this paper, we aim to obtain quantum error correcting codes from codes over a nonlocal ring $$R_q={\mathbb {F}}_q+\alpha {\mathbb {F}}_q$$. We first define a Gray map $$\varphi $$ from $$R_q^n$$ to $${\mathbb {F}}_q^{2n}$$ preserving the Hermitian orthogonality in $$R_q^n$$ to both the Euclidean and trace-symplectic orthogonality in $${\mathbb {F}}_q^{2n}$$. We characterize the structure of cyclic codes and their duals over $$R_q$$ and derive the condition of existence for cyclic codes containing their duals over $$R_q$$. By making use of the Gray map $$\varphi $$, we obtain two classes of q -ary quantum codes. We also determine the structure of additive cyclic codes over $$R_{p^2}$$ and give a condition for these codes to be self-orthogonal with respect to Hermitian inner product. By defining and making use of a new map $$\delta $$, we construct a family of p -ary quantum codes.

The article considers linear codes over Hurwitz integers. The codes are considered with respect to a new Hurwitz metric. This metric is more suitable for (QAM)-type constellations than the Hamming Metric and the Lee metric. Also, one error correcting perfect codes with respect to the Hurwitz metric are defined. The decoding algorithm of these codes is obtained. Moreover, a simple comparison in respect to the average energy for the transmitted signal and the bandwidth occupancy is given.