Integer-valued Coefficients Research Articles

A Novel 3D Reversible Data Hiding Scheme Based on Integer-Reversible Krawtchouk Transform for IoMT.

To avoid rounding errors associated with the limited representation of significant digits when applying the floating-point Krawtchouk transform in image processing, we present an integer and reversible version of the Krawtchouk transform (IRKT). This proposed IRKT generates integer-valued coefficients within the Krawtchouk domain, seamlessly aligning with the integer representation commonly utilized in lossless image applications. Building upon the IRKT, we introduce a novel 3D reversible data hiding (RDH) algorithm designed for the secure storage and transmission of extensive medical data within the IoMT (Internet of Medical Things) sector. Through the utilization of the IRKT-based 3D RDH method, a substantial amount of additional data can be embedded into 3D carrier medical images without augmenting their original size or compromising information integrity upon data extraction. Extensive experimental evaluations substantiate the effectiveness of the proposed algorithm, particularly regarding its high embedding capacity, imperceptibility, and resilience against statistical attacks. The integration of this proposed algorithm into the IoMT sector furnishes enhanced security measures for the safeguarded storage and transmission of massive medical data, thereby addressing the limitations of conventional 2D RDH algorithms for medical images.

The Approximate Sum Capacity of the Symmetric Gaussian <inline-formula> <tex-math notation="TeX">$K$ </tex-math></inline-formula>-User Interference Channel

Interference alignment has emerged as a powerful tool in the analysis of multiuser networks. Despite considerable recent progress, the capacity region of the Gaussian $K$ -user interference channel is still unknown in general, in part due to the challenges associated with alignment on the signal scale using lattice codes. This paper develops a new framework for lattice interference alignment, based on the compute-and-forward approach. Within this framework, each receiver decodes by first recovering two or more linear combinations of the transmitted codewords with integer-valued coefficients and then solving these linear combinations for its desired codeword. For the special case of symmetric channel gains, this framework is used to derive the approximate sum capacity of the Gaussian interference channel, up to an explicitly defined outage set of the channel gains. The key contributions are the capacity lower bounds for the weak through strong interference regimes, where each receiver should jointly decode its own codeword along with part of the interfering codewords. As part of the analysis, it is shown that decoding $K$ linear combinations of the codewords can approach the sum capacity of the $K$ -user Gaussian multiple-access channel up to a gap of no more than ${K}/{2}\log{K}$ bits.

Compute-and-Forward on a Multiaccess Relay Channel: Coding and Symmetric-Rate Optimization

We consider a system in which two users communicate with a destination with the help of a half-duplex relay. Based on the compute-and-forward scheme, we develop and evaluate the performance of coding strategies that are of network coding spirit. In this framework, instead of decoding the users' information messages, the destination decodes two integer-valued linear combinations that relate the transmitted codewords. Two decoding schemes are considered. In the first one, the relay computes one of the linear combinations and then forwards it to the destination. The destination computes the other linear combination based on the direct transmissions. In the second one, accounting for the side information available at the destination through the direct links, the relay compresses what it gets using lattice-based Wyner-Ziv compression and conveys it to the destination. The destination then computes the two linear combinations, locally. For both coding schemes, we discuss the design criteria, and derive the allowed symmetric-rate. Next, we address the power allocation and the selection of the integer-valued coefficients to maximize the offered symmetric-rate; an iterative coordinate descent method is proposed. The analysis shows that the first scheme can outperform standard relaying techniques in certain regimes, and the second scheme, while relying on feasible structured lattice codes, can at best achieve the same performance as regular compress-and-forward for the multiaccess relay network model that we study. The results are illustrated through some numerical examples.

Cyclic-Coded Integer-Forcing Equalization

A discrete-time intersymbol interference (ISI) channel with additive Gaussian noise is considered, where only the receiver has knowledge of the channel impulse response. An approach for combining decision-feedback equalization with channel coding is proposed, where decoding precedes the removal of ISI. The proposed approach involves equalizing the channel impulse response to a response with integer-valued coefficients in conjunction with utilizing cyclic block codes. Leveraging the property that a cyclic code is closed under cyclic integer-valued convolution allows us to perform decoding prior to applying decision feedback. Explicit bounds on the performance of the proposed scheme are derived.

The density of integral points on hypersurfaces of degree at least four

Let f be a polynomial of degree at least four with integer-valued coefficients. We establish new bounds for the density of integer solutions to the equation f=0, using an iterated version of Heath-Browns q-analogue of van der Corput's method of exponential sums.

Optimization of Polynomial Expressions by Using the Extended Dual Polarity

Reed-Muller expressions and their various extensions and generalizations for binary and multiple-valued logic functions are an important class of discrete function representations that are often used in practical applications. These expressions can be uniformly viewed as discrete polynomial expressions over finite fields GF(2) and GF(q) or the field of rational numbers in the case of expressions with integer-valued coefficients. The optimization of them in the number of product terms count is performed by selecting either positive or negative literals (polarities) for variables in the functions to be represented. Since there are no ways to select in advance the polarity for variables that will result in most compact expression for a given function, all possible expressions have to be generated and the simplest of them selected. This is a task computationally very demanding, the complexity of which is O(q^n \times C), where C is the time to calculate a particular polarity. Since the reduction of the first factor may lead to missing the most compact expression, the reduction of C is the single option to speed up the procedure. In this paper, we propose an approach to the solution of this problem by exploiting the notion of extended dual polarity, which provides a simple way of ordering polarities to obtain an effective way of finding the optimal one by reducing the time to move between them. The method still implies exhaustive search, but it is an optimized search, which may be expressed in very simple rules resulting in efficient implementation. Experimental results illustrate the effectiveness of the proposed method.

Odd Ramanujan Sums of Complex Roots of Unity

A special class of odd-symmetric length-4N periodic signals is studied, and it is shown how the odd Ramanujan sums are used as weighting coefficients to compute their pure imaginary discrete Fourier transform (DFT) integer-valued coefficients. The odd Ramanujan sum, being the sums of complex roots of unity, can be calculated either using closed-form formulas or computed recursively through the impulse response of a derived infinite impulse response (IIR) filter. This special class of odd-symmetric signals and odd Ramanujan sums can be combined together with the previous even-symmetric special class signals and the well-known even Ramanujan sums as a useful tool for signal processing

Ramanujan sums and discrete Fourier transforms

A special class of even-symmetric periodic signals is introduced. The most distinctive feature of these signals is that their real-valued Fourier coefficients can be calculated by forming a weighted average of the signal values using integer-valued coefficients. The signals arise from number-theoretic concepts concerning a class of functions called even arithmetical functions. The integer-valued weighting coefficients, being sums of complex roots of unity, are the Ramanujan sums and may be computed recursively or through closed-form arithmetical relations. The recursive method of computation is based on the cyclotomic polynomials and is described in detail. If the signal values are integers, the computation of the discrete Fourier transform (DFT) coefficients of this class of signals can be performed in an exact quantization-error-free manner by performing arithmetical operations on integers. The theoretical development is supplemented by concrete examples.

The method of solving systems of linear equations with integer-valued coefficients by extraction and estimation of unknowns

The method of solving systems of linear equations with integer-valued coefficients by extraction and estimation of unknowns