The explicit expressions of generator polynomials cyclic codes length [Formula: see text] over finite fields are obtained. coefficients these and check obtained through modular Lucas sequences. Further, using polynomials, self-dual, reversible self-orthogonal classified.

Let p, q and l be distinct odd primes such that is a primitive root modulo pn as well qm with g.c.d. (φ(p),φ(q)) = 2. Then the explicit expressions for complete set of 2mn + m n 1 idempotents minimal cyclic codes length pnqm over GF(l) are obtained. An algorithm also given to factorise polynomial (xn - 1) GF(k), where an integer (n, k) 1. Using generator polynomials above can computed numerically. Some bounds on minimum distance these

Now a days, major stress in steganography is to hide the maximum information possible alongwith consideration of minimum bits used such way that image not significantly degraded after embedding and embedded immune modifications from intelligent attacks or manipulations.In present paper we are proposing mode multiple methods which shows an improvement over existing digital logic method with comparatively lesser number insert extract hidden image.

Sufficient conditions in terms of distance for the existence binary 2-frameproof codes are obtained. A new class $q$-ary 2-IPP codes has been explicitly constructed using latin square designs.

Generalized cyclotomic numbers of order [Formula: see text] with respect to an odd prime power are obtained. Hence, explicit expressions for primitive idempotents in the ring obtained two cases, when multiplicative 2 modulo is and text], where prime. Orthogonality self-duality some cyclic codes also discussed. Further, a method obtaining self-dual/isodual length over given.

Cyclotomic classes of order 2 with respect to a product two distinct odd primes [Formula: see text] and are represented in some specific forms using these an alternate proof Theorem 3 [C. Ding T. Helleseth, New generalized cyclotomy its applications, Finite Fields Appl. 4 (1998) 140–166] is given, when text]. Further, it observed that related text]-cyclotomic cosets, where such gcd([Formula: Finally, arithmetic properties families hence studied.

In this paper we propose an explicit construction of a new class binary 2-frameproof code size 4(9+9n) and length 4(4+3n), n≥1. For code, show that the is almost three times code. By definition frameproof in [2,4] for C being 2-frameproof, minimum distance d given by , n Moreover, as Plotkin Bound [18] relates with q = 2 defined if . Here large than obtained Bound[18].

Let $\mathbb{F}_q$ be a finite field with $q$ elements and $n$ positive integer. In this paper, we determine the weight distribution of class cyclic codes length $2^n$ over whose parity check polynomials are either binomials or trinomials $2^l$ zeros $\mathbb{F}_q$, where integer $l\ge 1$. addition, constant two-weight linear constructed when $q\equiv3\pmod 4$.

A new class of cyclic codes length 2n over GF(q) is proposed, where q a prime the form 8m ± 3 and n > an integer. These are defined in terms their generator polynomials. have many properties analogous to those duadic codes. Generator polynomials some pn also discussed, p odd prime, integer = ρ or ρ2 for ρ.

Necessary and Sufficient Conditions for an Equidistant Code to be 2-TA code have been obtained in [1,2]. In the present paper we revisit these using pictorial approach. Further discuss conditions again under which is not a Code.

Arithmetic properties of some families in [Formula: see text] and hence are obtained by using the cyclotomic classes order 2 with respect to text], where is primitive root modulo text]. The form these enables us generalize results [S. Jain S. Batra, Cyclotomy arithmetic Asian-Eur. J. Math. 13(4) (2020) 2050077].

Explicit expressions for primitive idempotent in the ring are obtained, where p and q distinct odd primes with multiplicative order of 2 modulo being ϕ(q) respectively. Hence generators cyclic self dual codes length pq over Z4 obtained. Further, it is observed that when = 8k - 1 8m 3, extension each these augmented all-ones vector a Type I code.