arXiv subjects

Rajen Kumar

Publications and source records attributed to Rajen Kumar.

5 recordsLinked to original sources

arXiv2024

A New Construction of Optimal Symmetrical ZCCS

We propose new constructions for a two-dimensional ($2$D) perfect array, complete complementary code (CCC), and multiple CCCs as an optimal symmetrical $Z$-complementary code set (ZCCS). We propose a method to generate a two-dimensional perfect array and CCC. By utilising mutually orthogonal sequences, we developed a method to extend the length of a CCC without affecting the set or code size. Additionally, this concept is extended to include the development of multiple CCCs, and the correlation characteristics of these multiple CCCs are identical with the characteristics of optimal symmetrical ZCCS.

cs.IT↗

arXiv2024

Multiple Spectrally Null Constrained Complete Complementary Codes of Various Lengths Over Small Alphabet

Complete complementary codes (CCCs) are highly valuable in the fields of information security, radar and communication. The spectrally null constrained (SNC) problem arises in radar and modern communication systems due to the reservation or prohibition of specific spectrums from transmission. The literature on SNC-CCCs is somewhat limited in comparison to the literature on traditional CCCs. The main objective of this paper is to discover several configurations of SNC-CCCs that possess more flexibility in their parameters. The proposed construction utilised the existing CCCs and mutually orthogonal sequences. The proposed construction can cover almost all lengths with the smallest alphabets $\{-1,0,1\}$. Further, the idea of SNC-CCC is extended to multiple SNC-CCC with an inter-set zero cross-correlation zone (ZCCZ). Based on our construction, we can also control the correlation value outside the ZCCZ. The beauty of the obtained codes have aperiodic and periodic inter-set ZCCZ and low cross-correlation side-lobs.

cs.IT↗

arXiv2023

A Construction of Arbitrarily Large Type-II $Z$ Complementary Code Set

For a type-I $(K,M,Z,N)$-ZCCS, it follows $K \leq M \left\lfloor \frac{N}{Z}\right\rfloor$. In this paper, we propose a construction of type-II $(p^{k+n},p^k,p^{n+r}-p^r+1,p^{n+r})$-$Z$ complementary code set (ZCCS) using an extended Boolean function, its properties of Hamiltonian paths and the concept of isolated vertices, where $p\ge 2$. However, the proposed type-II ZCCS provides $K = M(N-Z+1)$ codes, where as for type-I $(K,M,N,Z)$-ZCCS, it is $K \leq M \left\lfloor \frac{N}{Z}\right\rfloor$. Therefore, the proposed type-II ZCCS provides a larger number of codes compared to type-I ZCCS. Further, as a special case of the proposed construction, $(p^k,p^k,p^n)$-CCC can be generated, for any integral value of $p\ge2$ and $k\le n$.

cs.IT↗

arXiv2022

A Construction of Type-II ZCCS for the MC-CDMA System with Low PMEPR

In this letter, we propose a novel construction of type-II $Z$-complementary code set (ZCCS) having arbitrary sequence length using the Kronecker product between a complete complementary code (CCC) and mutually orthogonal uni-modular sequences. In this construction, Barker sequences are used to reduce row sequence peak-to-mean envelope power ratio (PMEPR) for some specific lengths sequence and column sequence PMEPR for some specific sizes of codes. The column sequence PMEPR of the proposed type-II ZCCS is upper bounded by a number smaller than $2$. The proposed construction also contributes new lengths of type-II $Z$-complementary pair (ZCP) and type-II $Z$-complementary set (ZCS). Furthermore, the PMEPR of these new type-II ZCPs is also lower than existing type-II ZCPs.

cs.IT↗

arXiv2020

The real non-attractive fixed point conjecture and beyond

Is it always true that every polynomial P with the degree at least two has a fixed point z0, the real part of whose multiplier is bigger than or equal to 1, i.e., Real part of (P'(z0))> 1? This question, raised by Coelho and Kalantari in How many real attractive fixed points can a polynomial have? Math. Gaz. 103 (2019), no. 556, 65{76. [3] is answered affirmatively not only for all polynomials but also for all rational functions with a super attracting fixed point. However, this is not true for all rational functions. Some further investigation on distribution of multipliers of fixed points is made. Quadratic and cubic polynomials, all of whose multipliers have real part 1 are characterized. A necessary and sufficient condition is found for cubic and quartic polynomials, all of whose multipliers are equidistant from Is it always true that every polynomial P with the degree at least two has a fixed point z0, the real part of whose multiplier is bigger than or equal to 1, i.e., Real part of P'(z0)> 1? This question, raised by Coelho and Kalantari in How many real attractive fixed points can a polynomial have? Math. Gaz. 103 (2019), no. 556, 65{76. [3] is answered affirmatively not only for all polynomials but also for all rational functions with a super attracting fixed point. However, this is not true for all rational functions. Some further investigation on the distribution of multipliers of fixed points is made. Quadratic and cubic polynomials, all of whose multipliers have real part 1 are characterized. A necessary and sufficient condition is found for cubic and quartic polynomials, all of whose multipliers are equidistant from 1.

math.CV↗