研究具有较少重量线性码的重量分布在编码理论中具有重要的意义. 通过选取适当的定义集,构造一类新的二重或三重线性码;利用有限域上的指数和确定此类码的重量分布;进一步给出了一些例子证明了结论的正确性,并根据Griesmer界验证了其中一些码是最优码.
It is of vital significance to explore the weight distributions of linear codes with a few weights in coding theory. By selecting an appropriate defining set, a new class of two-weight or three-weight linear codes is constructed. Then, their weight distributions are determined by employing exponential sums over finite fields. Furthermore, some examples to verify the correctness of our results were gived and some of them are optimal codes with respect to the Griesmer bound.
[1] DING C S, NIEDERREITER H. Cyclotomic linear codes of order 3[J]. IEEE Trans Inf Theory, 2007, 53(6): 2274-2277. DOI:10.1109/TIT.2007.896886.
[2] JIAN G P, LIN Z C, FENG R Q. Two-weight and three-weight linear codes based on Weil sums[J]. Finite Fields Appl, 2019, 57: 92-107. DOI:10.1016/j.ffa.2019.02.001.
[3] 杜小妮,吕红霞,王蓉.一类四重和六重线性码的构造[J].电子与信息学报, 2019, 41(12): 2995-2999.
[4] YANG S D, YAO Z A. Complete weight enumerators of a family of three-weight linear codes[J]. Des Codes Cryptogr, 2017, 82(3): 663-674. DOI:10.1007/s10623-016-0191-x.
[5] 杨淑娣,岳勤.一类线性码的完全重量分布[J].计算机工程与科学, 2019, 41(2): 281-285. DOI:10.3969/j.issn.1007-130X.2019.02.013.
[6] LU H, YANG S D. Two classes of linear codes from Weil sums[J]. IEEE Access, 2020, 8: 180471-180480. DOI:10.1109/ACCESS.2020.3028141.
[7] LIDL R, NIEDERREITER H. Finite Fields[M]. Cambridge: Cambridge University Press, 1997. DOI:10.1017/cbo9780511525926.
[8] DING K L, DING C S. A class of two-weight and three-weight codes and their applications in secret sharing[J]. IEEE Trans Inf Theory, 2015, 61(11): 5835-5842. DOI:10.1109/TIT.2015.2473861.
[9] COULTER R S. Explicit evaluations of some Weil sums[J]. Acta Arith, 1998, 83(3): 241-251. DOI:10.4064/aa-83-3-241-251.
[10] COULTER R S. Further evaluations of Weil sums[J]. Acta Arith, 1998, 86(3): 217-226. DOI:10.4064/aa-86-3-217-226.
[11] HUFFMAN W C, PLESS V. Fundamentals of Error-Correcting Codes[M]. Cambridge: Cambridge University Press, 2003. DOI:10.1017/cbo9780511807077.
[12] KONG X L, YANG S D. Complete weight enumerators of a class of linear codes with two or three weights[J]. Discrete Math, 2019, 342(11): 3166-3176. DOI:10.1016/j.disc.2019.06.025. (
