Quaternary Codes Research Articles

Quaternary Codes and Their Binary Images

Quaternary Codes and Their Binary Images

Construction on large four-level designs via quaternary codes

In this paper, two simple and effective construction methods are proposed to construct four-level design with large size via quaternary codes from some small two-level initial designs. Under the popular criteria for selecting optimal design, such as generalized minimum aberration, minimum moment aberration and uniformity measured by average Lee discrepancy, the close relationships between the constructed four-level design and its initial design are investigated, which provide the guidance for choosing the suitable initial design. Moreover, some lower bounds of average Lee discrepancy for the constructed four-level designs are obtained, which can be used as a benchmark for evaluating the uniformity of the constructed four-level designs. Some numerical examples show that the large four-level designs can be constructed with high efficiency.

A novel coding scheme for generating sixteen codes from quaternary codes with applications

A novel coding scheme for generating sixteen codes from quaternary codes with applications

A degressive quantum convolutional neural network for quantum state classification and code recognition

With the rapid development of quantum computing, a variety of quantum convolutional neural networks (QCNNs) are proposed. However, only features of an n-qubits input are transferred to the next layer in a quantum pooling layer, which results in the accuracy reduction. To solve this problem, a QCNN with a degressive circuit is proposed. In order to enhance the ability of extracting global features, we remove the parameters sharing strategy in the quantum convolutional layer and design a quantum convolutional kernel with global eyesight. In addition, to prevent a sharp feature reduction, a degressive parameterized quantum circuit is adopted to construct the pooling layer. Then the Z-basis measurement is only performed on the first qubit to control the operations on other qubits. Compared with the state-of-the-art QCNN, i.e., hybrid quantum-classical convolutional neural network, the accuracy of our model increased by 0.9%, 1%, and 3%, respectively, in three tasks: quantum state classification, binary code recognition, and quaternary code recognition.

Theory of [formula omitted]-characteristics of four-level designs under quaternary codes

The J-characteristics play an instrumental role in the development of generalized resolution and minimum aberration criteria for two-level designs. In this paper, the concept of J-characteristics is naturally generalized to four-level designs via quaternary codes, which maps the four-level designs to two-level designs. Based on the relationship between the minimum G2-aberration criterion of two-level design and the generalized minimum aberration criterion of its projection designs, the properties of J-characteristics of four-level designs are explored. The relationship between J-characteristics of four-level design and J-characteristics of corresponding effective two-level sub-designs is built. The generalized resolution, confounding frequency vector and B-vector of four-level design are respectively defined based on the J-characteristics and their upper bounds, and minimum G-aberration and minimum G2-aberration criteria of four-level design are proposed, which are useful to assess the goodness of four-level designs.

The Lee Weight Distributions of a Class of Quaternary Codes

Let Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></sub> be the integer ring of residue classes modulo <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> . A well known fact states that the Gray map from a code with Lee weight over Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> to a code with Hamming weight over Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> is weight-preserving. This leads that the study of Lee weight of quaternary codes plays an extremely important role in the research of coding theory. For any given positive integers <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , and δ ∈ {0, 1, 2, 3}, let <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">δ</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>k</i>₁,<i>k</i>₂</sub> be a quaternary code with the generator matrix <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">δ</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>k</i>₁,<i>k</i>₂</sub> whose columns are all distinct nonzero vectors of the form ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , ∙ ∙ ∙, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>k</i>₁</sub> , ∙ ∙ ∙, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>k</i>₁+<i>k</i>₂</sub> ) with Σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>k</i>₁</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>i</i>=1</sub> <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c_i</i> ≡ δ (mod 4), <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c_j</i> ∈ Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c_t</i> ∈ {0, 2} for each 1 ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</i> ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> + 1 ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> + <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> . The objective of this paper is to establish the Lee weight distribution of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">δ</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>k</i>₁,<i>k</i>₂</sub> for each δ. Moreover, the linearity and optimality for the Gray image of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">δ</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>k</i>₁,<i>k</i>₂</sub> are characterized, from which we derive a family of optimal binary codes. Our result generalizes a work of Shi and Wang in 2014.

Quaternary Linear Codes and Related Binary Subfield Codes

In this paper, we mainly study quaternary linear codes and their binary subfield codes. First we obtain a general explicit relationship between quaternary linear codes and their binary subfield codes in terms of generator matrices and defining sets. Second, we construct quaternary linear codes via simplicial complexes and determine the weight distributions of these codes. Third, the weight distributions of the binary subfield codes of these quaternary codes are also computed by employing the general characterization. Furthermore, we present two infinite families of optimal linear codes with respect to the Griesmer Bound, and a class of binary almost optimal codes with respect to the Sphere Packing Bound. We also need to emphasize that we obtain at least 9 new quaternary linear codes.

The completion of optimal cyclic quaternary codes of weight 3 and distance 3

The completion of optimal cyclic quaternary codes of weight 3 and distance 3

On the structure of 1-generator quasi-polycyclic codes over finite chain rings

Quasi-polycyclic (QP for short) codes over a finite chain ring R are a generalization of quasi-cyclic codes, and these codes can be viewed as an R[x]-submodule of \({\mathcal {R}}_m^{\ell }\), where \({\mathcal {R}}_m:= R[x]/\langle f\rangle \), and f is a monic polynomial of degree m over R. If f factors uniquely into monic and coprime basic irreducibles, then their algebraic structure allow us to characterize the generator polynomials and the minimal generating sets of 1-generator QP codes as R-modules. In addition, we also determine the parity check polynomials for these codes by using the strong Gröbner bases. In particular, via Magma system, some quaternary codes with new parameters are derived from these 1-generator QP codes.

A Quaternary Code Correcting a Burst of at Most Two Deletion or Insertion Errors in DNA Storage.

Due to the properties of DNA data storage, the errors that occur in DNA strands make error correction an important and challenging task. In this paper, a new code design of quaternary code suitable for DNA storage is proposed to correct at most two consecutive deletion or insertion errors. The decoding algorithms of the proposed codes are also presented when one and two deletion or insertion errors occur, and it is proved that the proposed code can correct at most two consecutive errors. Moreover, the lower and upper bounds on the cardinality of the proposed quaternary codes are also evaluated, then the redundancy of the proposed code is provided as roughly .

Two-level parallel flats designs

Regular 2n−p designs are also known as single flat designs. Parallel flats designs (PFDs) consisting of three parallel flats (3-PFDs) are the most frequently utilized PFDs, due to their simple structure. Generalizing to f-PFD with f>3 is more challenging. This paper aims to study the general theory for the f-PFD for any f≥3. We propose a method for obtaining the confounding frequency vectors for all nonequivalent f-PFDs, and to find the least G-aberration (or highest D-efficiency) f-PFD constructed from any single flat. PFDs are particularly useful for constructing nonregular fraction, split-plot or randomized block designs. We also characterize the quaternary code design series as PFDs. Finally, we show how designs constructed by concatenating regular fractions from different families may also have a parallel flats structure. Examples are given throughout to illustrate the results.

Optimal Additive Quaternary Codes of Low Dimension

An additive quaternary [n,k,d]-code (length n, quaternary dimension k, minimum distance d) is a 2k-dimensional \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -vector space of n-tuples with entries in \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ⊕\mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (the 2-dimensional vector space over \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) with minimum Hamming distance d. We determine the optimal parameters of additive quaternary codes of dimension k ≤ 3. The most challenging case is dimension k=2.5. We prove that an additive quaternary [n,2.5,d]-code where d <; n-1 exists if and only if 3(n-d) ≥ d/2+d/4+d/8 articular, we construct new optimal 2.5-dimensional additive quaternary codes. As a by-product, we give a direct proof for the fact that a binary linear [3m,5,2e] <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -code for e <; m-1 exists if and only if the Griesmer bound 3(m-e) ≥ e/2+e/4+e/8 is satisfied.

Few-weight quaternary codes via simplicial complexes

<abstract> In this paper, we construct quaternary linear codes via simplicial complexes and we also determine the weight distributions of these codes. Moreover, we present an infinite family of minimal quaternary linear codes, which also meet the Griesmer bound. </abstract>

Symmetric Block Designs and Optimal Equidistant Codes

We prove that any symmetric block design (v, k, λ) generates optimal ternary and quaternary constant-weight equidistant codes, whose parameters n, N, w, d, q are uniquely determined by the parameters of the block design. For one rather special case, we construct symbolwise uniform equidistant codes of the minimum length.

A New Method of Constructing Binary Quantum Codes From Arbitrary Quaternary Linear Codes

In this letter, a method of constructing Hermitian dual containing quaternary linear codes from general quaternary linear codes is proposed. Based on this result, a construction method of binary linear quantum codes is obtained. Using three cyclic codes and one linear code over $\mathbb {F}_{4}$ that are not Hermitian dual containing, we construct four record breaking binary linear quantum codes. Furthermore, ten new record breaking binary additive quantum codes are derived from these four linear quantum codes by propagation rule.

Additive Quaternary Codes Related to Exceptional Linear Quaternary Codes

We study additive quaternary codes whose parameters are close to those of the extended cyclic $[{12,6,6}]_{4}$ -code or to the quaternary linear codes generated by the elliptic quadric in $PG(3,4)$ or its dual. In particular we characterize those codes in the category of additive codes and construct some additive codes whose parameters are better than those of any linear quaternary code. Our new code parameters are $[{22,17.5,4}]_{4}$ .

Quaternary Census Transform Based on the Human Visual System for Stereo Matching

The census transform is a non-parametric local transform that is widely used in stereo matching. This transform encodes the structural information of a local patch into a binary code stream representing the relative intensity ordering of the pixels within the patch. Despite its high performance in stereo matching, the census transform often generates identical binary code streams for two different patches because it simply thresholds the pixels within the patch at the center pixel intensity. To overcome this problem, we introduce a quaternary census transform that encodes the local structural information into a quaternary code stream by employing both the relative intensity ordering and the minimum visibility threshold of the human eye known as the just-noticeable difference. Moreover, because the human eye activates different areas of the retina based on brightness, the patch size for the proposed quaternary census transform adaptively varies depending on the luminance of each pixel. Experimental results on well-known Middlebury stereo datasets prove that the proposed transform outperforms the other census transform-based methods in terms of the accuracy of stereo matching.

New Quaternary Codes Derived from Posets of the Disjoint Union of Two Chains

Based on the generic construction of linear codes, we construct linear codes over the ring $\Bbb Z_4$ via posets of the disjoint union of two chains. We determine the Lee weight distributions of the quaternary codes. Moreover, we obtain some new linear quaternary codes and good binary codes by using the Gray map.

Quaternary Hermitian Linear Complementary Dual Codes

The largest minimum weights among quaternary Hermitian linear complementary dual codes are known for dimension $2$. In this paper, we give some conditions for the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum weights. As an application, we completely determine the largest minimum weights for dimension $3$, by using a classification of some quaternary codes. In addition, for a positive integer $s$, a maximal entanglement entanglement-assisted quantum $[[21s+5,3,16s+3;21s+2]]$ codes is constructed for the first time from a quaternary Hermitian linear complementary dual $[26,3,19]$ code.

A dictionary-based text compression technique using quaternary code

Improving encoding and decoding time in compression technique is a great demand to modern users. In bit level compression technique, it requires more time to encode or decode every single bit when a binary code is used. In this research, we develop a dictionary-based compression technique where we use a quaternary tree instead of a binary tree for construction of Huffman codes. Firstly, we explore the properties of quaternary tree structure mathematically for construction of Huffman codes. We study the terminology of new tree structure thoroughly and prove the results. Secondly, after a statistical analysis of English language, we design a variable length dictionary based on quaternary codes. Thirdly, we develop the encoding and decoding algorithms for the proposed technique. We compare the performance of the proposed technique with the existing popular techniques. The proposed technique performs better than the existing techniques with respect to decompression speed while the space requirement increases insignificantly.