Irreducible Cyclic Codes Of Length Research Articles

Some combinatorial identities

In this paper, it is shown that a positive integer [Formula: see text] can be written as [Formula: see text]. It is also proved that the two expressions for number of codewords of weight [Formula: see text], obtained in Theorem 1[P. Kumar, M. Sangwan and S. K. Arora, The weight distributions of some irreducible cyclic codes of length [Formula: see text] and [Formula: see text], Adv. Math. Commun. 9 (2015) 277–289] and in Theorem 1 Case (i)[G. Vega, The weight distribution for any irreducible cyclic code of length [Formula: see text], Appl. Algebra Eng. Commun. Comput. 29(4) (2018) 363–370] are same.

Weight enumerators of all irreducible cyclic codes of length n

Let [Formula: see text], where [Formula: see text] are distinct odd primes, be an integer and [Formula: see text] be a finite field of order [Formula: see text] with [Formula: see text]. We determine the weight enumerators of all irreducible cyclic codes of length [Formula: see text] over [Formula: see text] when multiplicative order of [Formula: see text] modulo [Formula: see text] is [Formula: see text]; [Formula: see text] and [Formula: see text]; [Formula: see text], where [Formula: see text].

Weight distributions of some irreducible cyclic codes of length n

Weight distributions of some irreducible cyclic codes of length n

The weight distributions of irreducible cyclic codes of length 2npm

Let [Formula: see text] be a primitive root modulo [Formula: see text], where [Formula: see text] and [Formula: see text] are distinct odd primes. Let [Formula: see text] be a finite field. For such pair of [Formula: see text] and [Formula: see text], the explicit expressions of minimal and generating polynomials over [Formula: see text] are obtained for all irreducible cyclic codes of length [Formula: see text]. In Sec. 4, it is observed that the weight distributions of all irreducible cyclic codes of length [Formula: see text] over [Formula: see text] can be computed easily with the help of the results obtained in [P. Kumar, M. Sangwan and S. K. Arora, The weight distribution of some irreducible cyclic codes of length [Formula: see text] and [Formula: see text], Adv. Math. Commun. 9 (2015) 277–289]. An explicit formula is also given to compute the weight distributions of irreducible cyclic codes of length [Formula: see text] over [Formula: see text].

Primitive Idempotents of Irreducible Cyclic Codes of Length n

Let Fq be a finite field with q elements and n a positive integer. In this paper, we use matrix method to give all primitive idempotents of irreducible cyclic codes of length n, whose prime divisors divide q-1.

Primitive idempotents of irreducible cyclic codes and self-dual cyclic codes over Galois rings

Let R be the Galois ring GR(pk,s) of characteristic pk and cardinality psk. Firstly, we give all primitive idempotent generators of irreducible cyclic codes of length n over R, and a p-adic integer ring with gcd(p,n)=1. Secondly, we obtain all primitive idempotents of all irreducible cyclic codes of length rlm over R, where r,l, and t are three primes with 2∤l, r|(qt−1), lv∥(qt−1) and gcd(rl,q(q−1))=1. Finally, as applications, weight distributions of all irreducible cyclic codes for t=2 and generator polynomials of self-dual cyclic codes of length lm and rlm over R are given.

The minimum Hamming distances of the irreducible cyclic codes of length

Let be a finite field with q elements and , where are distinct odd primes and . In this paper, we give all irreducible factors of over all primitive idempotents in the ring . Furthermore, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length over .

A class of minimal cyclic codes over finite fields

Let Fq be a finite field of odd order q and n=2ap1a1p2a2, where a,a1,a2 are positive integers, p1,p2 are distinct odd primes and 4p1p2|q−1. In this paper, we study the irreducible factorization of xn−1 over Fq and all primitive idempotents in the ring Fq[x]∕〈xn−1〉.Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length n over Fq.

The weight distributions of some irreducible cyclic codes of length $p^n$ and $2p^n$

In this paper, an algorithm is given for computing the weight distributions of all irreducible cycliccodes of dimension $p^jd$ generated by $x^{p^j}-1$, where $p$ is anodd prime, $j\geq 0 $ and $d > 1$. Then the weight distributions ofall irreducible cyclic codes of length $p^n$ and $ 2p^n $ over$F_q$, where $n$ is a positive integer, $p$, $q$ are distinct oddprimes and $q$ is primitive root modulo $ p^n$, are obtained. Theweight distributions of all the irreducible cyclic codes of length$3^{n+1}$ over $F_5$ are also determined explicitly.

A note on weight distributions of irreducible cyclic codes

Irreducible cyclic codes form an important family of cyclic codes and have applications in space communications. Their weight distributions measure their error performance relative to several channels, and hence have been an interesting object of study for a long time. In this note, we provide a method to determine the weight distributions of q-ary irreducible cyclic codes of length n, where q is a prime power and n is a positive integer coprime to q. This method is more effective for irreducible cyclic codes of some special lengths. We also list some optimal irreducible cyclic codes, which attain the distance bounds given in Grassl's Table [Code Tables: Bounds on the parameters of various types of codes, http://www.codetables.de ].

The weight distribution of some irreducible cyclic codes

Let F q be the finite field with q elements, p be an odd prime co-prime to q and m ⩾ 1 be an integer. In this paper, we explicitly determine the weight distribution of all the irreducible cyclic codes of length p m over F q from their generating polynomials in three different cases, when (i) the multiplicative order of q modulo p m is ϕ ( p m ) , (ii) the multiplicative order of q modulo p m is a power of p, and (iii) the multiplicative order of q modulo p m is twice a power of p.

Fast computation of Gauss sums and resolution of the root of unity ambiguity

We present an algorithm for computing Gauss sums over Fq for large prime powers q. This allows us to find the exact values of Gauss sums in many previously intractable cases. The efficient computation of such Gauss sums up to multiplication with a root of unity is achieved by using Stickelberger’s factorization of Gauss sums, an application of the Fincke-Pohst algorithm, and fast arithmetic for subfields of cyclotomic fields. This still leaves the crucial problem of resolving the root of unity ambiguity, which we settle by recursively computing “H-polynomials”. These are integer polynomials which interpolate Gauss sums at certain roots of unity. As an application, we compute the weight distribution of all binary irreducible cyclic codes of length n and dimension k with 2 k−1 n ≤ 5, 000. 2000 Mathematics Subject Classification: 11T24 (primary), 11Y40 (secondary)

The weight distributions of irreducible cyclic codes of length [formula omitted

Let m be a positive integer and q be an odd prime power. In this paper, the weight distributions of all the irreducible cyclic codes of length 2 m over F q are determined explicitly.

Two recursive algorithms for computing the weight distribution of certain irreducible cyclic codes

Two recursive algorithms for computing the weight distributions of certain binary irreducible cyclic codes of length n in the so-called index 2 case are presented. The running times of these algorithms are smaller than O(log/sup 2/r) where r=2/sup m/ and n is a factor of r-1.