Optimal quinary negacyclic codes with minimum distance four

<p style='text-indent:20px;'>Based on solutions of certain equations over finite yields, a necessary and sufficient condition for the quinary negacyclic codes with parameters <inline-formula><tex-math id="M1">\begin{document}$ [\frac{5^m-1}{2},\frac{5^m-1}{2}-2m,4] $\end{document}</tex-math></inline-formula> to have generator polynomial <inline-formula><tex-math id="M2">\begin{document}$ m_{\alpha^3}(x)m_{\alpha^e}(x) $\end{document}</tex-math></inline-formula> is provided. Several classes of new optimal quinary negacyclic codes with the same parameters are constructed by analyzing irreducible factors of certain polynomials over finite fields. Moreover, several classes of new optimal quinary negacyclic codes with these parameters and generator polynomial <inline-formula><tex-math id="M3">\begin{document}$ m_{\alpha}(x)m_{\alpha^e}(x) $\end{document}</tex-math></inline-formula> are also presented.</p>

XOR-FREE Implementation of Convolutional Encoder for Reconfigurable Hardware

This paper presents a novel XOR-FREE algorithm to implement the convolutional encoder using reconfigurable hardware. The approach completely removes the XOR processing of a chosen nonsystematic, feedforward generator polynomial of larger constraint length. The hardware (HW) implementation of new architecture uses Lookup Table (LUT) for storing the parity bits. The design implements architectural reconfigurability by modifying the generator polynomial of the same constraint length and code rate to reduce the design complexity. The proposed architecture reduces the dynamic power up to 30% and improves the hardware cost and propagation delay up to 20% and 32%, respectively. The performance of the proposed architecture is validated in MATLAB Simulink and tested on Zynq-7 series FPGA.

Generalizing Krawtchouk Polynomials Using Hadamard Matrices

We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged in a square matrix; in particular, the case where this matrix is a Hadamard matrix is considered. Orthogonality relations and recurrence relations are established, and coefficients for the expansion of any polynomial in terms of m-polynomials are obtained. We conclude this paper by an implementation of m-polynomials and some of the results obtained for them in Mathematica.