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PureMathematicsnØêÆ,2023,13(5),1389-1402
PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.135142
Aap‚5èE
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‚5觽Â8§•êÚ§-þ©Ù
ConstructionofSeveralClassesofp-Ary
LinearCodes
WenhuiLiu
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Apr.21
st
,2023;accepted:May23
rd
,2023;published:May30
th
,2023
Abstract
Linear codes with a fewweightsare widelystudied dueto theirimportant applications
©ÙÚ^:4©Ÿ.Aap‚5èE[J].nØêÆ,2023,13(5):1389-1402.
DOI:10.12677/pm.2023.135142
4©Ÿ
insecretsharing schemes,authentication codes,associationschemes,stronglyregular
graphs,etc.Inthispaper,severalclassesofp-arylinearcodesareconstructedby
selectinganewdefinitionset,andtheparametersandweightdistributionsofthe
codesaredeterminedbyexponentialsums.Finally,itisshownthatthelinearcodes
constructedinthispaperareminimallinearcodesinmostcases,whichcanbeused
todesignsecretsharingschemesswithgoodaccessstructures.
Keywords
LinearCode,DefiningSet,ExponentialSum,WeightDistribution
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/pm.2023.1351421391nØêÆ
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DOI:10.12677/pm.2023.1351421392nØêÆ
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DOI:10.12677/pm.2023.1351421393nØêÆ
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X
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X
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a

= 0…Tr
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a

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2
= (p−1)G
m
η(a)
X
y∈F
∗
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ζ
(ω−
M
4
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p
¯η(y) = (p−1)G
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η(a)
X
y∈F
∗
p
ζ
(ω−
M
4
)y
p
¯η

ω−
M
4

y

¯η

ω−
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4

= (p−1)G
m+1
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
ω−
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4
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œ/2:Tr

ρ
2
a

= 0…Tr

bρ
a

6= 0.K
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2
= G
m
η(a)
X
y∈F
∗
p
ζ
(ω−
M
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p
¯η(y)
X
z∈F
∗
p
ζ
−
T
(
bp
a
)
z
2
p
= −G
m
η(a)
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y∈F
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(ω−
M
4
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p
¯η(y)
= −G
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ω−
M
4

.
DOI:10.12677/pm.2023.1351421394nØêÆ
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
ρ
2
a

6= 0…Tr

bρ
a

.
= 0.K
Ω
2
= G
m
η(a)
P
y∈F
∗
p
ζ
(ω−
M
4
)y
p
¯η(y)
P
z∈F
∗
p
ζ
−
Tr

ρ
2
a

z
2
4y
p
= G
m
η(a)
X
y∈F
∗
p
ζ
(ω−
M
4
)y
p
¯η(y)
X
z∈F
p
ζ
−
Tr

ρ
2
a

z
2
4y
p
−G
m
η(a)
X
y∈F
∗
p
ζ
(ω−
M
4
)y
p
¯η(y).
dÚn2Œ
Ω
2
= G
m+1
η(a)¯η

−Tr

ρ
2
a

P
y∈F
∗
p
ζ
(ω−
M
4
)y
p
−G
m
η(a)
P
y∈F
∗
p
ζ
(ω−
M
4
)y
p
¯η(y)
= −G
m+1
η(a)

¯η

−Tr

ρ
2
a

+ ¯η(ω−
M
4
)

.
œ/4:Tr

ρ
2
a

6= 0…Tr

bρ
a

6= 0.K
Ω
2
= G
m
η(a)
X
y∈F
∗
p
ζ
(ω−
M
4
)y
p
¯η(y)
X
z∈F
p
ζ
−
Tr

ρ
2
a

z
2
4y
−
Tr
(
bp
a
)
z
2
p
−G
m+1
η(a)¯η(ω−
M
4
).
dÚn2Œ
Ω
2
= G
m+1
η(a)
X
y∈F
∗
p
ζ

4Tr(
ρ
2
a
)ω+Tr
2
(
bp
a
)
−MTr

ρ
2
a

y
4Tr

ρ
2
a

p
¯η

−Tr

ρ
2
a

−G
m+1
η(a)¯η(ω−
M
4
).
e4Tr(
ρ
2
a
)ω+Tr
2

bp
a

= MTr

ρ
2
a

,K
Ω
2
= G
m+1
η(a)

(p−1)¯η

−Tr

ρ
2
a

−¯η(ω−
M
4
)

.
e4Tr(
ρ
2
a
)ω+Tr
2

bp
a

6= MTr

ρ
2
a

,K
Ω
2
= −G
m+1
η(a)¯η

−Tr

ρ
2
a

−G
m+1
η(a)¯η(ω−
M
4
)
= −G
m+1
η(a)

¯η(ω−
M
4
)+ ¯η

−Tr

ρ
2
a

.

5¿?¿ρ∈F
∗
p
m
,wt(c
ρ
) 6= 0,KC
D
‘ê•m.nܱþ(JŒXe(Ø.
·K1.m•Ûê,½Â8 •(2)ª.K(1) ª½Â‚5èC
D
è••n, ‘ê•m. š"è
iDz-þ•
wt(c
ρ
) = (p−1)p
m−2
+N.
(1)eω−
M
4
= 0,Kn= p
m−1
.…
DOI:10.12677/pm.2023.1351421395nØêÆ
4©Ÿ
N=





















0,Tr

ρ
2
a

= 0,
−
p−1
p
2
G
m+1
η(a),Tr

bρ
a

= 0…¯η

Tr

ρ
2
a

= −1,
p−1
p
2
G
m+1
η(a),Tr

bρ
a

= 0…¯η

Tr

ρ
2
a

= 1,
1
p
2
G
m+1
η(a),Tr

bρ
a

6= 0…¯η

Tr

ρ
2
a

= −1,
−
1
p
2
G
m+1
η(a),Tr

bρ
a

6= 0…¯η

Tr

ρ
2
a

= 1.
(2)eω−
M
4
6= 0,Kn= p
m−1
+
1
p
G
m+1
η(a)¯η(ω−
M
4
).…
N=



















































0,Tr

ρ
2
a

= Tr

bρ
a

= 0,
1
p
G
m+1
η(a)¯η(ω−
M
4
),Tr

ρ
2
a

= 0…Tr

bρ
a

6= 0,
1
p
2
G
m+1
η(a)(p¯η

ω−
M
4
)+1

,

Tr

ρ
2
a

6= 0,Tr

bρ
a

= 0…¯η

Tr

ρ
2
a

= −1

½ö

Tr

ρ
2
a

6= 0,Tr

bρ
a

6= 0,4Tr(
ρ
2
a
)ω+Tr
2

bp
a

6=
MTr

ρ
2
a

…¯η

Tr

ρ
2
a

= −1

,
1
p
2
G
m+1
η(a)(p¯η(ω−
M
4
)−1),

Tr

ρ
2
a

6= 0,Tr

bρ
a

= 0…¯η

Tr

ρ
2
a

= 1

½ö

Tr

ρ
2
a

6= 0,Tr

bρ
a

6= 0,4Tr(
ρ
2
a
)ω+Tr
2

bp
a

6=
MTr

ρ
2
a

…¯η

Tr

ρ
2
a

= 1

,
1
p
2
G
m+1
η(a)¯η(ω−
M
4
),Tr

ρ
2
a

6= 0,Tr

bρ
a

6= 0…4Tr(
ρ
2
a
)ω+Tr
2

bp
a

=
MTr

ρ
2
a

.
y²d(4),(5)Ú(6)ªŒ
wt(c
ρ
) = p
m−1
+
1
p
Ω
1
−

p
m−2
+
1
p
2
(Ω
1
+Ω
2
)

= (p−1)p
m−2
+
p−1
p
2
Ω
1
−
1
p
2
Ω
2
,
K
N=
p−1
p
2
Ω
1
−
1
p
2
Ω
2
.
2(Ü(7)ªÚÚn5w,Œ(Ø.
5P1.éu·K1(1)¥èC
D
,m≥5 ž:
eG
m+1
η(a) = (−1)
(p−1)(m+1)
4
p
m+1
2
η(a) = ±p
m+1
2
,K
w
min
w
max
=
(p−1)(p
m−2
−p
m−3
2
)
(p−1)(p
m−2
+p
m−3
2
)
>
p−1
p
.
éu·K1(2)¥èC
D
,m≥5 ž:
eG
m
η(a) = (−1)
m−1
√
−1
(p−1)
2
m
4
√
p
m
η(a)·¯η(ω−
M
4
) = p
m
2
,K
w
min
w
max
=
(p−1)p
m−2
(p−1)p
m−2
+(p+1)p
m−4
2
>
p−1
p
.
DOI:10.12677/pm.2023.1351421396nØêÆ
4©Ÿ
eG
m
η(a) = (−1)
m−1
√
−1
(p−1)
2
m
4
√
p
m
η(a)·¯η(ω−
M
4
) = −p
m
2
,K
w
min
w
max
=
(p−1)p
m−2
−(p+1)p
m−4
2
(p−1)p
m−2
>
p−1
p
.
dÚn3Œ•,!E‚5è3õꜹe´4è.
e¡m©OŽ-þ©Ù.Äk,‰Ñü‡OŽªÇI‡Ún,Ù¥mþ•Ûê.
Ún6.[17]t∈F
p
.eN(t) = ]
n
x∈F
p
m
|Tr

x
2
a

= t
o
.K
N(t) =
(
p
m−1
,t= 0,
p
m−1
+
1
p
G
m+1
η(a)¯η(−t),t6= 0.
Ún7.[17]t∈F
p
,…a,b∈F
∗
p
m
.KN(t,0) = ]
n
x∈F
p
m
|Tr

x
2
a

= t,Tr

bx
a

= 0
o
.
et= 0,K
N(0,0) =
(
p
m−2
,p|M,
p
m−2
+
p−1
p
2
G
m+1
η(a)¯η(−M),p-M.
et6= 0,K
N(t,0) =
(
p
m−2
+
1
p
G
m+1
η(a)¯η(−t),p|M,
p
m−2
−
1
p
2
G
m+1
η(a)¯η(−M),p-M.
±þ(Jƒ,Kke¡(Ø.
½n1.p•Ûƒê,m•Ûê,½Â8X(2)ª¤«…ω−
M
4
= 0.K(1)ª½ÂèC
D
•
[p
m−1
,m]‚5è,Ù-þ©ÙXL1 ¤«.
Table1.TheweightdistributionofcodeC
D
inTheorem1
L1.½n1¥èC
D
-þ©Ù
-þªê
01
(p−1)p
m−2
A
ω
1
(p−1)

p
m−2
+
1
p
2
G
m+1
η(a)

A
ω
2
(p−1)

p
m−2
−
1
p
2
G
m+1
η(a)

A
ω
3
(p−1)p
m−2
+
1
p
2
G
m+1
η(a)A
ω
4
(p−1)p
m−2
−
1
p
2
G
m+1
η(a)A
ω
5
Ù¥,A
ω
1
= p
m−1
−1.
DOI:10.12677/pm.2023.1351421397nØêÆ
4©Ÿ
e¯η(−M) = −1,KA
ω
2
= A
ω
3
=
1
2
(p−1)(p
m−2
+
1
p
2
G
m+1
η(a)).
A
ω
4
=
1
2
1
p
2
G
m+1
η(a)
((1−p)G
2
m+1
η
2
(a)+(p
m
+p
m+2
−2p
m+1
)G
m+1
η(a)+(p
m+3
−p
m+2
)),
A
ω
5
=
1
2
1
p
2
G
m+1
η(a)
((1−p)G
2
m+1
η
2
(a)+(p
m
+p
m+2
−2p
m+1
)G
m+1
η(a)+(p
m+2
−p
m+3
)).
e¯η(−M) = 1,KA
ω
2
= A
ω
3
=
1
2
(p−1)(p
m−2
−
1
p
2
G
m+1
η(a)).
A
ω
4
=
1
2
1
p
2
G
m+1
η(a)
((p−1)G
2
m+1
η
2
(a)+(p
m
+p
m+2
−2p
m+1
)G
m+1
η(a)+(p
m+3
−p
m+2
)),
A
ω
5
=
1
2
1
p
2
G
m+1
η(a)
((p−1)G
2
m+1
η
2
(a)+(p
m
+p
m+2
−2p
m+1
)G
m+1
η(a)+(p
m+2
−p
m+3
)).
y²d·K1(1)Œ•,C
D
¥š"èiDz-þ•L1¥11 ¤«. òL1¥11 
Šlþ–e•gP•w
1
,w
2
,w
3
,w
4
,w
5
,KéAªêP•A
w
1
,A
w
2
,A
w
3
,A
w
4
,A
w
5
.
d·K1ÚÚn6Œ
A
w
1
= N(0)−1 = p
m−1
−1.
Ï•ω−
M
4
= 0, ¤±M6= 0, =p-M.d·K1ÚÚn7 ŒA
ω
2
= A
ω
3
=
1
2
(p−1)(p
m−2
−
1
p
2
G
m+1
η(a)¯η(−M)).
dPless˜Ýúª[19]Œ
(
A
w
1
+A
w
2
+A
w
3
+A
w
4
+A
w
5
= p
m
−1,
w
1
A
w
1
+w
2
A
w
2
+w
3
A
w
3
+w
4
A
w
5
+w
5
A
w
5
= (p−1)np
m−1
.
(8)
Ïd,e¯η(−M)) = −1,KA
ω
2
= A
ω
3
=
1
2
(p−1)(p
m−2
+
1
p
2
G
m+1
η(a)). 2(Ü(8) ªŒ
A
ω
4
=
1
2
1
p
2
G
m+1
η(a)
((1−p)G
2
m+1
η
2
(a)+(p
m
+p
m+2
−2p
m−1
)G
m+1
η(a)+(p
m+3
−p
m+2
)),
A
ω
5
=
1
2
1
p
2
G
m+1
η(a)
((1−p)G
2
m+1
η
2
(a)+(p
m
+p
m+2
−2p
m−1
)G
m+1
η(a)+(p
m+2
−p
m+3
)).
e¯η(−M) = 1,KA
ω
2
= A
ω
3
=
1
2
(p−1)(p
m−2
−
1
p
2
G
m+1
η(a)). 2(Ü(8) ªŒ
A
ω
4
=
1
2
1
p
2
G
m+1
η(a)
((p−1)G
2
m+1
η
2
(a)+(p
m
+p
m+2
−2p
m−1
)G
m+1
η(a)+(p
m+3
−p
m+2
)),
A
ω
5
=
1
2
1
p
2
G
m+1
η(a)
((p−1)G
2
m+1
η
2
(a)+(p
m
+p
m+2
−2p
m−1
)G
m+1
η(a)+(p
m+2
−p
m+3
)).
ÏdèC
D
-þ©Ù.
e¡~fŒdMagma§Sy.
~1.½Â8X(2) ª¤«,…p= 5,m= 3,a= 2,b= 4,ω= Tr(2). KC
D
•[25,3] ‚5
DOI:10.12677/pm.2023.1351421398nØêÆ
4©Ÿ
è,Ù-þOêì•1+24z
20
+12z
24
+12z
16
+48z
21
+28z
19
,†½n1(ؘ—.
~2.½Â8X(2) ª¤«,…p= 7,m= 5,a=1,b=2,ω= Tr(1). KC
D
•[2401,5] ‚
5è,Ù-þOêì•1+2400z
2058
+1050z
2016
+1050z
2100
+6006z
2051
+6300z
2065
,†½n1(
ؘ—.
3.2.m•óêžèC
D
ëê9Ù-þ©Ù
!?Øm•óêž, Äu(2) ªE(1)ª‚5èC
D
è•, ‘ê±9-þ©Ù. du
y²•{Úm•Ûêœ/aq,¤±Ø2Kã.
Ún8.C
D
è••n,K
n=
(
p
m−1
+
p−1
p
G
m
η(a),ω−
M
4
= 0,
p
m−1
−
1
p
G
m
η(a),ω−
M
4
6= 0.
·K2.m•óê, ½Â8•(2)ª. K(1) ª½Â‚5èC
D
è••n, ‘ê•m. š"è
iDz-þ•
wt(c
ρ
) = (p−1)p
m−2
+N.
(1)eω−
M
4
= 0,Kn= p
m−1
+
p−1
p
G
m
η(a). …
N=









0,Tr

ρ
2
a

= Tr(
bρ
a
) = 0,
p−1
p
G
m
η(a),Tr

ρ
2
a

·Tr(
bρ
a
) = 0…Tr

bρ
a

6= Tr

ρ
2
a

,
p−2
p
G
m
η(a),Tr

ρ
2
a

·Tr(
bρ
a
) 6= 0.
(1)eω−
M
4
6= 0,Kn= p
m−1
−
1
p
G
m
η(a)). …
N=













































0,(Tr

ρ
2
a

= Tr

bρ
a

= 0)½ö(Tr

ρ
2
a

6= 0,Tr

bρ
a

= 0…
¯η(−(ω−
M
4
)Tr

ρ
2
a

) = −1))½ö(Tr
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ρ
2
a

·Tr
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a
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6= 0,
4Tr(
ρ
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a
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2

bp
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6= MTr

ρ
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a
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…¯η(4ωTr
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ρ
2
a

+Tr
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
bρ
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−MTr
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ρ
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a
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) = −1),
−1
p
G
m
η(a),(Tr

ρ
2
a

= 0,Tr

bρ
a

6= 0)½ö(Tr

ρ
2
a

·Tr

bρ
a

6= 0…
4Tr(
ρ
2
a
)ω+Tr
2

bp
a

= MTr

ρ
2
a

),
−2
p
G
m
η(a),(Tr

ρ
2
a

6= 0,Tr

bρ
a

= 0,¯η(−(ω−
M
4
)Tr

ρ
2
a

) = 1))½ö
(Tr

ρ
2
a

·Tr

bρ
a

6= 0,4ωTr(
ρ
2
a
)+Tr
2

bp
a

6= MTr

ρ
2
a

,
¯η

4ωTr

ρ
2
a

+Tr
2
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bρ
a
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−MTr

ρ
2
a
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= 1).
5P2.éu·K2 ü«œ/eèC
D
,m≥6ž,C
D
´4‚5è. y²•{†5P1 ˜
,ùp2ØKã.
DOI:10.12677/pm.2023.1351421399nØêÆ
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Ún9.[17]t∈F
p
,eN(t) = ]
n
x∈F
p
m
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x
2
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= t
o
.K
N(t) =
(
p
m−1
+
p−1
p
G
m
η(a),t= 0,
p
m−1
−
1
p
G
m
η(a),t6= 0.
Ún10.[17]t∈F
p
,…a,b∈F
∗
p
m
.N(0,t) = ]
n
x∈F
p
m
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
x
2
a

= 0,Tr

bx
a

= t
o
.
et= 0,K
N(0,0) =
(
p
m−2
+
p−1
p
G
m
η(a),p|M,
p
m−2
,p-M.
et6= 0,K
N(t,0) =
(
p
m−2
,p|M,
p
m−2
+
1
p
G
m
η(a),p-M.
nܱþ(Jw,ŒXe½n.
½n2.p•Ûƒê,m•óê,½Â8X(2)ª¤«…ω−
M
4
= 0.K(1)ª½ÂèC
D
•
h
p
m−1
+
p−1
p
G
m
η(a),m
i
‚5è,Ù-þ©ÙXL2¤«.
Table2.TheweightdistributionofcodeC
D
inTheorem2
L2.½n2¥èC
D
-þ©Ù
-þªê
01
(p−1)p
m−2
p
m−2
−1
(p−1)p
m−2
+
p−1
p
G
m
η(a)2(p
m−1
−p
m−2
)+
1
G
m
η(a)
(p
m
−p
m−1
)
(p−1)p
m−2
+
p−2
p
G
m
η(a)(p
m
+p
m−2
−2p
m−1
)−
1
G
m
η(a)
(p
m
−p
m−1
)
5P3.dL2w,Œ, m=2ž, -þ•(p−1)p
m−2
èiÑygê•p
m−2
−1= 0
g.džC
D
•-‚5è.
e¡~fŒdMagma§Sy.
~3.½Â8X(2)ª¤«, …p=5,m= 4,a=1,b= 2,ω=Tr(1).KC
D
•[105,4]‚
5è,Ù-þOêì•1+24z
100
+180z
80
+420z
85
,†½n2(ؘ—.
~4.½Â8X(2)ª¤«, …p= 7,m= 2,a= 2,b= 4,ω= Tr(2).KC
D
•[13,2]‚5
è,Ù-þOêì•1+18z
12
+30z
11
,†½n2(ؘ—.
4.o(
©í2©z[15–17] ¥‚5èE,Äu(2)ª…©m•ÛêÚm•óêü«œ/é(1)
ª‚5è?1ïÄ, EAa#ëêp‚5è. è•Ý, ‘ê±9-þ©Ùëꆮk
DOI:10.12677/pm.2023.1351421400nØêÆ
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(JþØÓ.¿^Magma §Sy(Ø(5.•, ÏLAschikhmin-Barg^‡`²,©
E‚5èõꜹe´4‚5è, Œ±^5OäkûЖ¯(“—••Y. ω−
M
4
6= 0
žC
D
ªÇ©Ù3‰a,Öö.
ë•©z
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