On period polynomials of degree 2 and weight distributions of certain 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 ].

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Weight distributions of some irreducible cyclic codes

Mathematics of Computation ◽

10.1090/s0025-5718-1986-0815855-7 ◽

1986 ◽

Vol 46 (173) ◽

pp. 341-341 ◽

Cited By ~ 6

Author(s):

Robert Segal ◽

Robert L. Ward

Keyword(s):

Cyclic Codes ◽

Irreducible Cyclic Codes ◽

Weight Distributions

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The weight distributions of irreducible cyclic codes of length 2npm

Asian-European Journal of Mathematics ◽

10.1142/s1793557118500857 ◽

2018 ◽

Vol 11 (06) ◽

pp. 1850085

Author(s):

Monika Sangwan ◽

Pankaj Kumar

Keyword(s):

Finite Field ◽

Explicit Formula ◽

Weight Distribution ◽

Cyclic Codes ◽

Primitive Root ◽

Irreducible Cyclic Codes ◽

Weight Distributions ◽

Primitive Root Modulo ◽

Explicit Expressions

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].

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

Advances in Mathematics of Communications ◽

10.3934/amc.2015.9.277 ◽

2015 ◽

Vol 9 (3) ◽

pp. 277-289 ◽

Cited By ~ 1

Author(s):

Pankaj Kumar ◽

◽

Monika Sangwan ◽

Suresh Kumar Arora ◽

Keyword(s):

Cyclic Codes ◽

Irreducible Cyclic Codes ◽

Weight Distributions

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The weight distributions of a class of non-primitive cyclic codes with two nonzeros

Science China Mathematics ◽

10.1007/s11425-014-4888-x ◽

2014 ◽

Vol 58 (6) ◽

pp. 1285-1296

Author(s):

DaBin Zheng ◽

FengLi Zhou ◽

Lei Hu ◽

XiangYong Zeng

Keyword(s):

Cyclic Codes ◽

Weight Distributions

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Three-weight cyclic codes and their weight distributions

Discrete Mathematics ◽

10.1016/j.disc.2015.09.001 ◽

2016 ◽

Vol 339 (2) ◽

pp. 415-427 ◽

Cited By ~ 53

Author(s):

Cunsheng Ding ◽

Chunlei Li ◽

Nian Li ◽

Zhengchun Zhou

Keyword(s):

Cyclic Codes ◽

Weight Distributions

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Primitive Idempotents of Irreducible Cyclic Codes of Length n

Mathematical Problems in Engineering ◽

10.1155/2018/6962508 ◽

2018 ◽

Vol 2018 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Yuqian Lin ◽

Qin Yue ◽

Yansheng Wu

Keyword(s):

Positive Integer ◽

Finite Field ◽

Matrix Method ◽

Cyclic Codes ◽

Prime Divisors ◽

Primitive Idempotents ◽

Irreducible Cyclic Codes

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.

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