Abstract It is known that entanglement-assisted quantum error-correcting codes (EAQECCs), a type of quantum error correction codes, can be easily constructed using linear codes that satisfy the property called linear complementary duals (LCD). For quasi-cyclic (QC) codes, which are a class of linear codes, we have already published the methods for constructing the codes with properties such as self-orthogonality, self-duality and reversibility according to the prime-factor decomposition of $$-1+x^m$$ - 1 + x m , which reduce the amount of calculation by assembling several small generator polynomial matrices into a large generator polynomial matrix. In this paper, we propose a method to construct LCD–QC codes according to the prime-factor decomposition. The main idea of this method is to decompose the generator polynomial matrix of the QC code into several small generator polynomial matrices corresponding to the prime factors and perform LCD determination, which leads to a reduction in the amount of calculation. As an application of our construction method, we create EAQECCs from the constructed LCD–QC codes, compare those minimum weights with the maximum values of the minimum weights of existing EAQECCs and find 15 EAQECCs with larger minimum weights than the existing ones.

In this study, we construct a cyclic DNA code that satisfies the Watson-Crick model, i.e., the reverse and the complement of any codeword are codewords. Similar to irreducible cyclic codes, we use the trace function over a finite field extension to generate our DNA code. MacWilliams-Seery algorithm to evaluate the weight distribution of binary irreducible cyclic codes is used to observe the GC-content distribution of our code. Then, a simple algorithm is proposed to select a sub-code with constant GC-content. Therefore, this sub-code would be a cyclic, reverse-complement, and constant GC-content DNA code. A few illustrative examples show the appropriateness of our algorithm for generating over 100000 DNA strands of length n > 1000 and a predefined constant GC-content.

In this paper, we establish a lemma in algebraic coding theory that frequently appears in the encoding and decoding of, e.g., Reed-Solomon codes, algebraic geometry codes, and affine variety codes. Our lemma corresponds to the nonsystematic encoding of affine variety codes, and can be stated by giving a canonical linear map as the composition of an extension through linear feedback shift registers from a Gröbner basis and a generalized inverse discrete Fourier transform. We clarify that our lemma yields the error-value estimation in the fast erasure-and-error decoding of a class of dual affine variety codes. Moreover, we show that systematic encoding corresponds to a special case of erasure-only decoding. The lemma enables us to reduce the computational complexity of error-evaluation from O(n3) using Gaussian elimination to O(qn2) with some mild conditions on n and q, where n is the code length and q is the finite-field size.

In this paper, we study no 4-cycle, high-rate LDPC codes based on finite geometries for use in data storage devices and prove that these codes cannot be classified as quasi-cyclic (QC) codes but should be considered as broader generalized quasi-cyclic (GQC) codes. Because of the GQC structure of such codes, they can be systematically encoded using Groebner bases and their encoder can be implemented using simple feedback-shift registers. In order to demonstrate the efficiency of the encoder, we show that the hardware complexity of the serial-in serial-out encoder architecture of these codes is of linear order O(n). To encode a binary codeword of length n, less than 2n adders and 3n memory elements are required. Furthermore, we evaluated the error performances of these codes with sum product algorithm (SPA) decoding over additive white Gaussian noise (AWGN) channels. At a bit error rate (BER) of 10^-5, they perform 1-dB away from the Shannon limit after 10 decoding iterations.

In this paper, a fundamental lemma in algebraic coding theory is established, which is frequently appeared in the encoding and decoding for algebraic codes such as Reed-Solomon and algebraic geometry codes. This lemma states that two vector spaces, one corresponds to information symbols and the other is indexed by the support of Gröbner basis, are canonically isomorphic, and moreover, the isomorphism is given in terms of extension by linear feedback shift registers from Gröbner basis and discrete Fourier transforms. Next, we apply the lemma to unified system of encoding and decoding erasure-errors in algebraic geometry codes. Finally, we comment on an improved bound for the generic erasure-error correcting capabilities.

We define generalized quasi-cyclic (GQC) codes as linear codes with nontrivial automorphism groups. Therefore, GQC codes, unlike quasi-cyclic codes, can include many important codes such as Hermitian and projective geometry (PG) codes; this capability is important in practical applications. Further, we propose the echelon canonical form algorithm for computing Grobner bases from their parity check matrices. Consequently, by applying Grobner base theory, GQC codes can be systematically encoded and implemented with simple feedback shift registers. Our algorithm is based on Gaussian elimination and requires a sufficiently small number of finite-field operations, which is related to the third power of code-length. In order to demonstrate our encoder's efficiency, we prove that the number of circuit elements in the encoder architecture is proportional to the code-length for finite geometry (FG) LDPC codes (a class of GQC codes). We show that the hardware complexity of a serial-in-serial-out encoder architecture for FG-LDPC codes is related to the linear order of the code-length; less than 2n adder and 2n memory elements are required to encode a binary codeword of length n.

We propose a novel encoding scheme for algebraic codes such as codes on algebraic curves, multidimensional cyclic codes, and hyperbolic cascaded Reed-Solomon codes and present numerical examples. We employ the recurrence from the Grobner basis of the locator ideal for a set of rational points and the two- dimensional inverse discrete Fourier transform. We generalize the functioning of the generator polynomial for Reed-Solomon codes and develop systematic encoding for various algebraic codes.

The authors have already proposed small-scale decoder for codes on algebraic curves, which updates decoding data serially and has the same number of calculators for the finite-field computations as the decoders for Reed-Solomon codes except for one calculator for inverse. In this research, we eliminate divisions of the finite field from error-location algorithm, and propose an inverse-free architecture for the error-location of codes on algebraic curves which reduces Kotter's architecture

A lot of non-photorealistic rendering methods have been proposed for creating an artistic image from an image. In this paper, we propose a method for creating colored pencil style images. The feature of colored pencil drawings is that, though colored pencil drawings are drawn with limited number of colors of pencils, we can express a lot of colors and gentle textures by changing the strengths when drawing strokes and by overlapping strokes of different colors. In order to realize this feature, we determine which colors of pencils to use and how deep to push the pencils (equivalent to the strength when drawing strokes), then draw several types of strokes, such as strokes for outlines, basecoats, and shading, allowing the strokes to overlap each other. When we create strokes for shading, we make their directions to align along the boundaries of regions, resulting in images that are more like drawings made by human.

We construct a two-dimensional systolic array implementing the Berlekamp-Massey-Sakata (BMS) algorithm to provide error-locator polynomials for codes on selected algebraic curves. This array is constructed by introducing some new polynomials in order to increase the parallelism of the algorithm. The introduced polynomials are used in the majority logic scheme by Sakata et al. to correct errors up to the designed minimum distance without affecting its high speed. The arrangement of the nearest local connection of processing units in the systolic array is obtained for the general case. Furthermore, shortened systolic arrays that reduce the circuit scale and have the same function are constructed with only a slight modification of the connections and controls; this enables the adjustment of the circuit scale for different types of systems.