ON ℤpℤp[u]/k>-CYCLIC CODES AND THEIR WEIGHT ENUMERATORS
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In this paper we study the algebraic structure of ℤpℤp[u]/k>-cyclic codes, where uk = 0 and p is a prime. A ℤpℤp[u]/k>-linear code of length (r + s) is an Rk-submodule of ℤrp × Rsk with respect to a suitable scalar multiplication, where Rk = ℤp[u]/k>. Such a code can also be viewed as an Rk-submodule of ℤp[x]/r - 1> × Rk[x]/s - 1>. A new Gray map has been defined on ℤp[u]/k>. We have considered two cases for studying the algebraic structure of ℤpℤp[u]/k>-cyclic codes, and determined the generator polynomials and minimal spanning sets of these codes in both the cases. In the first case, we have considered (r, p) = 1 and (s, p) ≠ 1, and in the second case we consider (r, p) = 1 and (s, p) = 1. We have established the MacWilliams identity for complete weight enumerators of ℤpℤp[u]/k>-linear codes. Examples have been given to construct ℤpℤp[u]/k>-cyclic codes, through which we get codes over ℤp using the Gray map. Some optimal p-ary codes have been obtained in this way. An example has also been given to illustrate the use of MacWilliams identity.
Abstract
In this paper we study the algebraic structure of ℤpℤp[u]/k>-cyclic codes, where uk = 0 and p is a prime. A ℤpℤp[u]/k>-linear code of length (r + s) is an Rk-submodule of ℤrp × Rsk with respect to a suitable scalar multiplication, where Rk = ℤp[u]/k>. Such a code can also be viewed as an Rk-submodule of ℤp[x]/r - 1> × Rk[x]/s - 1>. A new Gray map has been defined on ℤp[u]/k>. We have considered two cases for studying the algebraic structure of ℤpℤp[u]/k>-cyclic codes, and determined the generator polynomials and minimal spanning sets of these codes in both the cases. In the first case, we have considered (r, p) = 1 and (s, p) ≠ 1, and in the second case we consider (r, p) = 1 and (s, p) = 1. We have established the MacWilliams identity for complete weight enumerators of ℤpℤp[u]/k>-linear codes. Examples have been given to construct ℤpℤp[u]/k>-cyclic codes, through which we get codes over ℤp using the Gray map. Some optimal p-ary codes have been obtained in this way. An example has also been given to illustrate the use of MacWilliams identity.
참고문헌 (12)
- -additive codes, "International Journal of Computer Mathematics", 2014, 1806~1814, pp.Informa UK Limited, 10.1080[DOI]
- I. Aydogdu, "The structure of Z2Z2s -additive codes: bounds on the minimum distance", Appl. Math. Inf. Sci., 7(6), pp.2271~2278 (2013)[DOI]
- Ismail Aydogdu, "On Zpr Zps -additive codes", Linear and Multilinear Algebra, 63(10), pp.2089~2102 (2014), Informa UK Limited[DOI]
- -linear and cyclic codes, "Finite Fields and Their Applications", 2017, 241~260, pp.Elsevier BV, 10.1016[DOI]
- Joaquim Borges, "Z2Z4-additive cyclic codes, generator polynomials, and dual codes", IEEE Transactions on Information Theory, 62(11), pp.6348~6354 (2016), Institute of Electrical and Electronics Engineers (IEEE)[DOI]
- J. Borges, "Z2-double cyclic codes", Des. Codes Cryptogr, 86(3), pp.463~479 (2018)
- -additive cyclic codes, "International Journal of Information and Coding Theory", 2018, 1~, pp.Inderscience Publishers, 10.1504[DOI]
- A.R. Hammons, "The Z", sub 4, IEEE Transactions on Information Theory, pp.1994, 40(2)[DOI]
- Liqin Qian, "Bounds and optimal q-ary codes derived from the ZqR-cyclic codes", IEEE Transactions on Information Theory, 66(2), pp.923~935 (2020), Institute of Electrical and Electronics Engineers (IEEE)[DOI]
- J. Rif`a, "Product perfect Z2Z4-linear codes in stegonography", International Symposium on Information Theory and Its Applications, pp.696~701 (2010)
- Minjia Shi, "On ZpZpk -additive codes and their duality", IEEE Transactions on Information Theory, 65(6), pp.3841~3847 (2019), Institute of Electrical and Electronics Engineers (IEEE)[DOI]
- "hu ki", Designs, Codes and Cryptography, 74(1), pp.1~13 (2015), Springer Science and Business Media LLC[DOI]