Linear Codes Constructed From Two Weakly Regular Plateaued Functions with Index $(p-1)/2$ ¹¹footnotemark: 1

Shudi Yang yangshudi@qfnu.edu.cn Tonghui Zhang zhangthvvs@126.com Zheng-An Yao mcsyao@mail.sysu.edu.cn School of Mathematical Sciences, Qufu Normal University, Shandong 273165, P.R.China School of Mathematics and Statistics,Fujian Normal University, Fujian 350117, P.R.China School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P.R.China

Abstract

Linear codes are the most important family of codes in cryptography and coding theory. Some codes have only a few weights and are widely used in many areas, such as authentication codes, secret sharing schemes and strongly regular graphs. By setting $p\equiv 1\pmod{4}$ , we construct an infinite family of linear codes using two distinct weakly regular unbalanced (and balanced) plateaued functions with index $(p-1)/2$ . Their weight distributions are completely determined by applying exponential sums and Walsh transform. As a result, most of our constructed codes have a few nonzero weights and are minimal.

keywords:

linear code , plateaued function , the weight distribution , Walsh transform.

MSC:

[2010] 94B15 , 11T71

^†^†journal: Finite Fields and Their Applications

TheweightdistributionofthiscaseissummarizedinTable

4.∎

Table 1: TheweightdistributionofCDf,ginTheoremLABEL:th:s+toddwhenlf=p-12.

weight	frequency
$0$	$1$
$(p-1)p^{2m-2}$	$p^{2m}-(p-1)p^{\gamma-1}-1\qquad$
$(p-1){\big{(}{p^{2m-2}-\sqrt{p}^{\tau-3}}\big{)}}$	$\frac{p-1}{2}{\big{(}{p^{\gamma-1}+\sqrt{p}^{\gamma-1}}\big{)}}$
$(p-1){\big{(}{p^{2m-2}+\sqrt{p}^{\tau-3}}\big{)}}$	$\frac{p-1}{2}{\big{(}{p^{\gamma-1}-\sqrt{p}^{\gamma-1}}\big{)}}$

Table 2: TheweightdistributionofCDf,ginTheoremLABEL:th:s+toddwhenlf=p-1.

weight	frequency
$0$	$1$
$(p-1)p^{2m-2}$	$p^{2m}-1-E_{1}-E_{2}-B_{S_{q}}-B_{N_{sq}}$
$(p-1){\big{(}{p^{2m-2}-\frac{1}{2}\varepsilon_{f}\varepsilon_{g}\eta(2)\sqrt{p}^{\tau-3}}\big{)}}$	$E_{1}$
$(p-1){\big{(}{p^{2m-2}+\frac{1}{2}\varepsilon_{f}\varepsilon_{g}\eta(2)\sqrt{p}^{\tau-3}}\big{)}}$	$E_{2}$
$(p-1){\big{(}{p^{2m-2}-\varepsilon_{f}\varepsilon_{g}\sqrt{p}^{\tau-3}}\big{)}}$	$B_{S_{q}}$
$(p-1){\big{(}{p^{2m-2}+\varepsilon_{f}\varepsilon_{g}\sqrt{p}^{\tau-3}}\big{)}}$	$B_{N_{sq}}$

Table 3: TheweightdistributionofCDf,ginTheoremLABEL:th:s+toddwhenlf=2andp≡1(mod8).

weight	frequency
$0$	$1$
$(p-1)p^{2m-2}$	$p^{2m}-1-E_{3}-E_{4}-\frac{(p-1)^{2}}{4}{\big{(}{\mathcal{N}_{f}(i)\mathcal{N}_{g}(i)+\mathcal{N}_{f}(j)\mathcal{N}_{g}(j)}\big{)}}$
$(p-1){\big{(}{p^{2m-2}-\varepsilon_{f}\varepsilon_{g}\sqrt{p}^{\tau-3}}\big{)}}$	$E_{3}$
$(p-1){\big{(}{p^{2m-2}+\varepsilon_{f}\varepsilon_{g}\sqrt{p}^{\tau-3}}\big{)}}$	$E_{4}$
$(p-1){\big{(}{p^{2m-2}+2\varepsilon_{f}\varepsilon_{g}\sqrt{p}^{\tau-3}}\big{)}}$	$\frac{(p-1)^{2}}{4}\mathcal{N}_{f}(i)\mathcal{N}_{g}(i)$
$(p-1){\big{(}{p^{2m-2}-2\varepsilon_{f}\varepsilon_{g}\sqrt{p}^{\tau-3}}\big{)}}$	$\frac{(p-1)^{2}}{4}\mathcal{N}_{f}(j)\mathcal{N}_{g}(j)$

Table 4: TheweightdistributionofCDf,ginTheoremLABEL:th:s+toddwhenlf=2andp≡5(mod8).

weight	frequency
$0$	$1$
$(p-1)p^{2m-2}$	$p^{2m}-1-E_{3}-E_{4}-\sum_{u,v\in\mathbb{F}_{p}^{*}}\mathcal{N}_{f}(u)\mathcal{N}_{g}(v)$
$(p-1){\big{(}{p^{2m-2}-\varepsilon_{f}\varepsilon_{g}\sqrt{p}^{\tau-3}}\big{)}}$	$E_{3}$
$(p-1){\big{(}{p^{2m-2}+\varepsilon_{f}\varepsilon_{g}\sqrt{p}^{\tau-3}}\big{)}}$	$E_{4}$
$(p-1)p^{2m-2}-\varepsilon_{f}\varepsilon_{g}\sqrt{p}^{\tau-3}\eta(u){\Big{(}{I_{4}{\big{(}{\frac{v}{u}}\big{)}}-\eta{\big{(}{\frac{v}{u}}\big{)}}}\Big{)}}$ for all $u,v\in\mathbb{F}_{p}^{*}$	$\mathcal{N}_{f}(u)\mathcal{N}_{g}(v)$

Theorem 2.

Supposethatf,g∈WRPwithlg=p-12.Lets+tbeeven.Iflf=p-12,thenCDf,gisathree-weight[n,2m]linearcodewithitsweightdistributionlistedinTable

LABEL:evenc1.Iflf=p-1,thenCDf,gisafour-weight[n,2m]linearcodewithitsweightdistributionlistedinTableLABEL:evenc2.Otherwiseiflf=2,thenCDf,gisafour-weight[n,2m]linearcodewithitsweightdistributionlistedinTableLABEL:evenc3whenp≡1(mod8),andinTableLABEL:evenc4whenp≡5(mod8).Herewesetn=p2m-1-1+(p-1)εfεgpτ-2forbrevity.

Proof.

ThelengthofthecodeCDf,gcomesfromLemma

LABEL:lem:length.For(a,b)∈Fq2\{(0,0)},theweightwt(c(a,b))=n+1-N0canbeobtainedfromLemmaLABEL:lem:N0_even.Tobemoreexplicit,when(a,b)∉Sf×Sg,wehavewt(c(a,b))=(p-1)(p2m-2+(p-1)εfεgpτ-4).ByLemmaLABEL:lem:car,thefrequencyofsuchcodewordsequalsp2m-pγsincef,g∈WRP.When(a,b)∈Sf×Sg\{(0,0)},wewilldiscussthefollowingfourdifferentcases.

(1)Whenlf=p-12,wehavewt(c(a,b))={(p-1)p2m-2,T(0)-1 times,(p-1)(p2m-2+εfεgpτ-2),(p-1)T(c) times,whereT(0)andT(c)aregiveninLemma

LABEL:lem:Tforc≠0.ThisgivestheweightdistributioninTableLABEL:evenc1.

(2)Whenlf=p-1,wehavewt(c(a,b))={(p-1)p2m-2,Nf(0)Ng(0)-1 times,(p-1)(p2m-2+12εfεgpτ-2),F1 times,(p-1)(p2m-2+εfεgpτ-2),F2 times,wherewedefineF1=#{(a,b)∈Sf×Sg:f⋆(a)≠0,g⋆(b)=±f⋆(a)}=2∑c∈Fp∗Nf(c)Ng(c),F2=pγ-Nf(0)Ng(0)-F1.ThusweobtaintheweightdistributioninTable

LABEL:evenc2.

(3)Whenlf=2andp≡1(mod8),wehavewt(c(a,b))={(p-1)p2m-2,Nf(0)Ng(0)-1 times,(p-1)p2m-2+(p-3)εfεgpτ-2,F3 times,(p-1)(p2m-2+εfεgpτ-2),F4 times,whereF3=#{(a,b)∈Sf×Sg:f⋆(a)g⋆(b)∈Sq}=(p-1)24(Nf(i)Ng(i)+Nf(j)Ng(j)),F4=pγ-Nf(0)Ng(0)-F3,fori∈Sqandj∈Nsq.ThisimpliestheweightdistributionlistedinTable

LABEL:evenc3.