Complete weight enumerators of a new class of linear codes

Yiwei Liu; Zihui Liu

doi:10.1016/j.disc.2018.03.025

Complete weight enumerators of a new class of linear codes

Yiwei Liu, Zihui Liu^*

^*Corresponding author for this work

School of Mathematics and Statistics

Beijing Institute of Technology

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

For an odd prime p, let q=p^m, and denote Tr_e ^m the trace function from F_q onto F_p^e, where e is a divisor of m. For a positive integer t and a∈F_p^e, let D_a={(x₁,x₂,…,x_t)∈F_q ^t∖{(0,0,…,0)}:Tr_e ^m(x₁+x₂+⋯+x_t)=a}, and define a p-ary linear code C_{D_a} as C_{D_a}={c(x₁,x₂,…,x_t):(x₁,x₂,…,x_t)∈F_q ^t}, where c(x₁,x₂,…,x_t)=(Tr₁ ^m(x₁d₁ ²+x₂d₂ ²+⋯+x_td_t ²))_{(d₁,d₂,…,d_t)∈D_a}. The complete weight enumerators of linear codes C_{D_a} will be presented for any divisor e of m and a∈F_p^e, and this new result generalizes that of both Ahn et al. (2017) and Yang et al. (2017).

Original language	English
Pages (from-to)	1959-1972
Number of pages	14
Journal	Discrete Mathematics
Volume	341
Issue number	7
DOIs	https://doi.org/10.1016/j.disc.2018.03.025
Publication status	Published - Jul 2018

Keywords

Complete weight enumerators
Defining set
Exponential sum
Gauss sum
Trace function

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Liu, Y., & Liu, Z. (2018). Complete weight enumerators of a new class of linear codes. Discrete Mathematics, 341(7), 1959-1972. https://doi.org/10.1016/j.disc.2018.03.025

@article{36621cff613d47d8ac074a93b8006fde,

title = "Complete weight enumerators of a new class of linear codes",

abstract = "For an odd prime p, let q=pm, and denote Tre m the trace function from Fq onto Fpe , where e is a divisor of m. For a positive integer t and a∈Fpe , let Da={(x1,x2,…,xt)∈Fq t∖{(0,0,…,0)}:Tre m(x1+x2+⋯+xt)=a}, and define a p-ary linear code CDa as CDa ={c(x1,x2,…,xt):(x1,x2,…,xt)∈Fq t}, where c(x1,x2,…,xt)=(Tr1 m(x1d1 2+x2d2 2+⋯+xtdt 2))(d1,d2,…,dt)∈Da . The complete weight enumerators of linear codes CDa will be presented for any divisor e of m and a∈Fpe , and this new result generalizes that of both Ahn et al. (2017) and Yang et al. (2017).",

keywords = "Complete weight enumerators, Defining set, Exponential sum, Gauss sum, Trace function",

author = "Yiwei Liu and Zihui Liu",

note = "Publisher Copyright: {\textcopyright} 2018 Elsevier B.V.",

year = "2018",

month = jul,

doi = "10.1016/j.disc.2018.03.025",

language = "English",

volume = "341",

pages = "1959--1972",

journal = "Discrete Mathematics",

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T1 - Complete weight enumerators of a new class of linear codes

AU - Liu, Yiwei

AU - Liu, Zihui

PY - 2018/7

Y1 - 2018/7

N2 - For an odd prime p, let q=pm, and denote Tre m the trace function from Fq onto Fpe , where e is a divisor of m. For a positive integer t and a∈Fpe , let Da={(x1,x2,…,xt)∈Fq t∖{(0,0,…,0)}:Tre m(x1+x2+⋯+xt)=a}, and define a p-ary linear code CDa as CDa ={c(x1,x2,…,xt):(x1,x2,…,xt)∈Fq t}, where c(x1,x2,…,xt)=(Tr1 m(x1d1 2+x2d2 2+⋯+xtdt 2))(d1,d2,…,dt)∈Da . The complete weight enumerators of linear codes CDa will be presented for any divisor e of m and a∈Fpe , and this new result generalizes that of both Ahn et al. (2017) and Yang et al. (2017).

AB - For an odd prime p, let q=pm, and denote Tre m the trace function from Fq onto Fpe , where e is a divisor of m. For a positive integer t and a∈Fpe , let Da={(x1,x2,…,xt)∈Fq t∖{(0,0,…,0)}:Tre m(x1+x2+⋯+xt)=a}, and define a p-ary linear code CDa as CDa ={c(x1,x2,…,xt):(x1,x2,…,xt)∈Fq t}, where c(x1,x2,…,xt)=(Tr1 m(x1d1 2+x2d2 2+⋯+xtdt 2))(d1,d2,…,dt)∈Da . The complete weight enumerators of linear codes CDa will be presented for any divisor e of m and a∈Fpe , and this new result generalizes that of both Ahn et al. (2017) and Yang et al. (2017).

KW - Complete weight enumerators

KW - Defining set

KW - Exponential sum

KW - Gauss sum

KW - Trace function

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DO - 10.1016/j.disc.2018.03.025

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SN - 0012-365X

VL - 341

SP - 1959

EP - 1972

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 7

ER -

Complete weight enumerators of a new class of linear codes

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