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Cyclic codes from the first class two-prime Whiteman's generalized cyclotomic sequence with order 6
Pramod Kumar Kewat,Priti Kumari 대한수학회 2019 대한수학회보 Vol.56 No.2
Let $p_1$ and $p_2$ be two distinct odd primes with $\mathrm{gcd}(p_1-1,p_2-1)=6$. In this paper, we compute the linear complexity of the first class two-prime Whiteman's generalized cyclotomic sequence (WGCS-I) of order $d=6$. Our results show that their linear complexity is quite good. So, the sequence can be used in many domains such as cryptography and coding theory. This article enrich a method to construct several classes of cyclic codes over $\mathrm{GF}(q)$ with length $n=p_1p_2$ using the two-prime WGCS-I of order 6. We also obtain the lower bounds on the minimum distance of these cyclic codes.
Kewat, Pramod Kumar,Kushwaha, Sarika Korean Mathematical Society 2018 대한수학회보 Vol.55 No.1
Let $R_{u{^2},v^2,w^2,p}$ be a finite non chain ring ${\mathbb{F}}_p[u,v,w]{\langle}u^2,\;v^2,\;w^2,\;uv-vu,\;vw-wv,\;uw-wu{\rangle}$, where p is a prime number. This ring is a part of family of Frobenius rings. In this paper, we explore the structures of cyclic codes over the ring $R_{u{^2},v^2,w^2,p}$ of arbitrary length. We obtain a unique set of generators for these codes and also characterize free cyclic codes. We show that Gray images of cyclic codes are 8-quasicyclic binary linear codes of length 8n over ${\mathbb{F}}_p$. We also determine the rank and the Hamming distance for these codes. At last, we have given some examples.
CYCLIC CODES FROM THE FIRST CLASS TWO-PRIME WHITEMAN'S GENERALIZED CYCLOTOMIC SEQUENCE WITH ORDER 6
Kewat, Pramod Kumar,Kumari, Priti Korean Mathematical Society 2019 대한수학회보 Vol.56 No.2
Let $p_1$ and $p_2$ be two distinct odd primes with gcd($p_1-1$, $p_2-1$) = 6. In this paper, we compute the linear complexity of the first class two-prime Whiteman's generalized cyclotomic sequence (WGCS-I) of order d = 6. Our results show that their linear complexity is quite good. So, the sequence can be used in many domains such as cryptography and coding theory. This article enrich a method to construct several classes of cyclic codes over GF(q) with length $n=p_1p_2$ using the two-prime WGCS-I of order 6. We also obtain the lower bounds on the minimum distance of these cyclic codes.
Cyclic codes over the ring ${\mathbb F}_p[u,v,w]/\langle u^2, v^2, w^2, uv-vu, vw-wv, uw-wu \rangle$
Pramod Kumar Kewat,Sarika Kushwaha 대한수학회 2018 대한수학회보 Vol.55 No.1
Let $R_{u^2, v^2, w^2, p}$ be a finite non chain ring $ \F_p[u,v,w]\langle u^2,$ $v^2, w^2$, $uv-vu, vw-wv, uw-wu \rangle$, where $p$ is a prime number. This ring is a part of family of Frobenius rings. In this paper, we explore the structures of cyclic codes over the ring $R_{u^2, v^2, w^2, p}$ of arbitrary length. We obtain a unique set of generators for these codes and also characterize free cyclic codes. We show that Gray images of cyclic codes are 8-quasicyclic binary linear codes of length $8n$ over $\F_p$. We also determine the rank and the Hamming distance for these codes. At last, we have given some examples.