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Computations in Macaulay2 to construct an algebraic system associated to a given cyclic code and a positive integer ?w? , whose solutions are in bijection with codewords of weight less than or equal to ?w? .
Authors
Dr. Arunkumar Patil,Pooja Rajani
Downlaod Center
Keywords
linear codes, Cyclic code, BCH code, Mattson-Solomon polynomial, locators of a codeword
Abstract
This paper aims at constructing a user defined functions in Macaulay2 which accepts values of ?n
?
and ?q
?with
gcd(n, q) = 1 and displays all q - cyclotomic cosets modulo n . This list of cyclotomic cosets can help in
describinga cyclic code in Macaulay2 by selecting a representative from each cyclotomic coset in its defining set.
For a given cyclic code ?C
?
and a positive integer?w?, there is a well-known algebraic system constructed from
Newton's identities which are satisfied by elementary symmetric functions of locators of a codeword of weight
?w? and coefficients of its Mattson-Solomon polynomial. This paper aims at constructing a user defined function
in Macaulay2, which accepts a cyclic code (in terms of list of elements from distinct cyclotomic cosets in its
defining set) and a positive integer ?w? and it returns the algebraic system described above. Further, the simplified
form of this system is also constructed.In a special case when integer w is equal to BCH bound of C, the simplified
system is used for computing number of codewords of minimum weight in C, using Gro?bner basis.
IJCRT's Publication Details
Unique Identification Number - IJCRTBM02015
Paper ID - 300494
Author type - Indian Author
Page Number(s) - 108-116
Pubished in - Volume 14 | Issue 2 | February 2026
DOI (Digital Object Identifier) -
No Of Downloads - 54
Author Country - India, -, -, -, -, Science and Technology
Publisher Name - IJPUBLICATION | www.ijcrt.org | ISSN : 2320-2882
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Dr. Arunkumar Patil,Pooja Rajani, "Computations in Macaulay2 to construct an algebraic system associated to a given cyclic code and a positive integer ?w? , whose solutions are in bijection with codewords of weight less than or equal to ?w? .", International Journal of Creative Research Thoughts (IJCRT), ISSN:2320-2882, Volume.14, Issue 2, pp.108-116, February 2026, Available at :http://www.ijcrt.org/papers/IJCRTBM02015.pdf
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