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PureMathematicsnØêÆ,2020,10(11),1014-1023
PublishedOnlineNovemb er2020inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/pm.2020.1011120
äk`‘Hermitian•MDSèE
¸¸¸………¦¦¦§§§¤¤¤‰‰‰ÉÉÉ
∗
§§§©©©uuu
þ°ŒÆnÆ§þ°
Email:
∗
qqyytt@shu.edu.cn
ÂvFϵ2020c1014F¶¹^Fϵ2020c114F¶uÙFϵ2020c1111F
Á‡
ˆSingleton.è¡•4ŒålŒ©è({¡•MDSè)§ÙņUå•r§3ņè¥k
Xš~2•A^"©ïÄMDSèHermitian•§|^2ÂReed-Solomon èE
äk`(`≥1) ‘Hermitian•MDSè"
'…c
MDSè§2ÂReed-Solomonè§Hermitian•
ConstructionofMDSCodewith
`-DimensionalHermitianHull
YuhuiHan,YutingQiu
∗
,XiaohuaLu
CollegeofSciences,ShanghaiUniversity,Shanghai
Email:
∗
qqyytt@shu.edu.cn
Received:Oct.14
th
,2020;accepted:Nov.4
th
,2020;published:Nov.11
th
,2020
Abstract
ThecodesachievingtheSingletonboundarecalledmaximumdistanceseparable(for
∗ÏÕŠö"
©ÙÚ^:¸…¦,¤‰É,©u.äk`‘Hermitian•MDSèE[J].nØêÆ,2020,10(11):
1014-1023.
DOI:10.12677/pm.2020.1011120
¸…¦
shortMDS)codes,whichhavethestrongesterrorcorrectionabilityandarewidely
appliedinerror-correctingcode.Inthispaper,westudytheHermitianhullsofMDS
codes.WeusethegeneralizedReed-SolomoncodetoconstructMDScodeswith
`-dimensional(`≥1)Hermitianhull.
Keywords
MDSCodes,GeneralizedReed-SolomonCodes,HermitianHull
Copyright
c
2020byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
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E,Ý[1][2]ÚdCEÅþfņèƒ'ëê[3],©z[4][5][6][7]?ØEuclidean 
•ÚHermitian •‘êÚ²þ‘ê.•‘ê•è´‚5pÖéóè({¡•LCD
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Å9Ïþfņè¥kƒ'A^.4ŒålŒ©è({¡•MDS è)´ˆ Singleton .è.
MDS èÏÙņUå r,3ó§þk2•A^,ÏïÄMDS è••ÚåïÄ<
'5[8–13].
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n!Eäk1 ‘Hermitian •MDS è,¿‰Ñ~f,•31o!¥Eäk`-‘
Hermitian•MDS è.
DOI:10.12677/pm.2020.10111201015nØêÆ
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DOI:10.12677/pm.2020.10111201016nØêÆ
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α
1
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2
,···,α
n
´k••F
Q
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a= (α
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,···,α
n
)
Ú
v= (v
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n
) ∈(F
∗
Q
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GRS
k
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1
f(α
1
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2
f(α
2
),···,v
n
f(α
n
)) : f(x) ∈F
Q
[x];degf(x) <k}(2.1)
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Q
‚5
è[16],=GRS
k
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Ún2.7.[16]GRS
k
(a,v)Euclidean éóè´
GRS
n−k
(a,w),
Ù¥w= (w
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i
=
1
v
i
Q
1≤j≤n,j6=i
(α
i
−α
j
)
−1
(i= 1,2,···,n).
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Ún2.8.êk÷v1 ≤k≤n.GRS
k
(a,v)s-Galois éóè´
GRS
n−k
(a
0
,w),
DOI:10.12677/pm.2020.10111201017nØêÆ
¸…¦
Ù¥a
0
= (α
[s]
1
,α
[s]
2
,···,α
[s]
n
),w= (w
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2
,···,w
n
),ùp
w
i
=
1
v
[s]
i
Y
1≤j≤n,j6=i
(α
[s]
i
−α
[s]
j
)
−1
(i= 1,2,···,n).
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[s]
Euclidean éóè,±9NF
s
´F
Q
þF
q
g
Ó,¤±
[GRS
k
(a,v)]
⊥
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n
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[s]
1
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(α
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1
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[s]
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[s]
(α
[s]
n
)) : f(x) ∈F
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⊥
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[s]
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f(α
[s]
1
),···,v
[s]
n
f(α
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n
)) : f(x) ∈F
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[x];degf(x) <k
o
⊥
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k
(a
0
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0
)]
⊥
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0
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[s]
1
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[s]
2
,···,α
[s]
n
),v
0
= (v
[s]
1
,v
[s]
2
,···,v
[s]
n
).dÚn2.7 =Œy²T(Ø.
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2
,F
Q
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∗
Q
L«F
Q
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n
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Hermitianéóè•ŒP•
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⊥
H
=
(
(b
1
,b
2
,···,b
n
) ∈F
n
Q
:
n
X
i=1
b
i
c
q
i
= 0,∀(c
1
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2
,···,c
n
) ∈C
)
.
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q
=
{α
1
,α
2
,···,α
q
}ÚF
Q
= {α
1
,α
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,···,α
q
,β
1
,β
2
,···,β
Q−q
},Ù¥β
i
∈F
Q
\F
q
(i= 1,2,···,Q−q).
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a= (α
1
,α
2
,···,α
q
,β
1
,β
2
,···,β
t
)
Ú
v= (v
1
,v
2
,···,v
n
) ∈(F
∗
Q
)
n
.
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k
(a,v)Hermitian éóè.
íØ3.1.GRS
k
(a,v)Hermitian éóè´
GRS
n−k
(a
0
,w),
Ù¥a
0
= (α
1
,α
2
,···,α
q
,β
q
1
,β
q
2
,···,β
q
t
),w= (w
1
,w
2
,···,w
n
),ùp
w
i
=
1
v
q
i
Y
1≤j≤q,j6=i
(α
i
−α
j
)
−1
Y
1≤m≤t
(α
i
−β
q
m
)
−1
,i= 1,2,···,q,
w
i
=
1
v
q
i
Y
1≤j≤q
(β
q
i−q
−α
j
)
−1
Y
1≤m≤t,m6=i−q
(β
q
i−q
−β
q
m
)
−1
,i= q+1,q+2,···,n.
DOI:10.12677/pm.2020.10111201018nØêÆ
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½n3.2.k≤qÚ(n−k−1)q<n.eéu?¿÷v1 ≤i≤qêiÑk
Y
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(α
i
−α
j
)
−1
Y
1≤m≤t
(α
i
−β
q
m
)
−1
= v
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i
,(3.1)
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Y
1≤j≤q
(β
q
i−q
−α
j
)
−1
Y
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(β
q
i−q
−β
q
m
)
−1
= v
q +1
i
,(3.2)
KGRS
k
(a,v)´äk1 ‘Hermitian •MDS è,…Ùëê•[n,k,n−k+1]
Q
.
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[GRS
k
(a,v)]
⊥
H
= GRS
n−k
(a
0
,w),
Ù¥a
0
= (α
1
,α
2
,···,α
q
,β
q
1
,β
q
2
,···,β
q
t
),w= (w
1
,w
2
,···,w
n
),ùp
w
i
=
1
v
q
i
Y
1≤j≤q,j6=i
(α
i
−α
j
)
−1
Y
1≤m≤t
(α
i
−β
q
m
)
−1
,i= 1,2,···,q;
w
i
=
1
v
q
i
Y
1≤j≤q
(β
q
i−q
−α
j
)
−1
Y
1≤m≤t,m6=i−q
(β
q
i−q
−β
q
m
)
−1
,i= q+1,q+2,···,n.
e¡û½GRS
k
(a,v)Hermitian •,Ø”
(v
1
f(α
1
),···,v
q
f(α
q
),v
q +1
f(β
1
),···,v
n
f(β
t
)) ∈GRS
k
(a,v)∩[GRS
k
(a,v)]
⊥
H
,
Ù¥degf(x) ≤k−1.Ïd,•3˜‡gê؇Ln−k−1õ‘ªg(x), ¦
(v
1
f(α
1
),···,v
q
f(α
q
),v
q +1
f(β
1
),···,v
n
f(β
t
)) = (w
1
g(α
1
),···,w
q
g(α
q
),w
q+1
g(β
q
1
),···,w
n
g(β
q
t
)),

(
v
i
f(α
i
) = w
i
g(α
i
),i= 1,2,···,q;
v
i
f(β
i−q
) = w
i
g(β
q
i−q
),i= q+1,q+2,···,n.
dª(3.1)Ú(3.2) Œw
i
= v
i
6= 0,é1 ≤i≤n,ÏdkXeª:
(
f(α
i
) = g(α
i
),i= 1,2,···,q;
f(β
i−q
) = g(β
q
i−q
),i= q+1,q+2,···,n.
þªL²µg(x
q
) −f(x)=0–kn‡ØÓŠ,g(x) −f(x)=0–kq‡ØÓŠ.Ï•
(n−k−1)q<n,¤±deg[g(x
q
)−f(x)]<n,Ïdg(x
q
)=f(x).dždegf(x)=q·degg(x),•
Ò´`degf(x)≥degg(x).qÏ•k≤q,¤±deg[g(x)−f(x)]≤degf(x)≤k−1≤q−1.d
ug(x)−f(x) =0 –kq‡ØÓŠ,Ïdg(x)=f(x),?kdegf(x) =degg(x)=0.Ïd,
GRS
k
(a,v)Hermitian •´d•þv)¤1‘‚5˜m.½ny.
e¡·‚‰ÑA‡äN~f.
~3.1.q= 4,Q= q
2
= 16,γ´k••F
Q
,K
F
16
= {0,1,γ,γ
2
,···,γ
14
}
DOI:10.12677/pm.2020.10111201019nØêÆ
¸…¦
Ú
F
4
= {0,1,γ
5
,γ
10
}.
-n= 6 Úk= 4,džn≥q+1,k≤q…(n−k−1)q<n.À
a= (α
1
,α
2
,α
3
,α
4
,β
1
,β
2
)
Ú
v= (v
1
,v
2
,v
3
,v
4
,v
5
,v
6
) ∈(F
∗
16
)
6
,
Ù¥α
1
=0,α
2
=1,α
3
=γ
5
,α
4
=γ
10
,β
1
=γ,β
2
=γ
4
;v
1
=γ
14
,v
2
=1,v
3
=γ
14
,v
4
=1,
v
5
= γ
13
,v
6
= γ
13
.
Œ±y,éu?¿÷v1≤i≤4 êi,(3.1)ª¤á.éu?¿÷v5≤i≤6 êi,
(3.2)ª¤á.d½n3.2Œ•µGRS
4
(a,v) ´äk1 ‘Hermitian •MDS è,…Ùëê•
[6,4,3]
16
.
~3.2.q= 3,Q= q
2
= 9,γ´k••F
Q
,K
F
9
= {0,1,γ
1
,γ
2
,γ
3
,γ
4
,γ
5
,γ
6
,γ
7
}
Ú
F
3
= {0,1,γ
4
}.
Ù¥,γ
4
= 2.
-n= 5 Úk= 3,džn≥q+1,k≤qÚ(n−k−1)q<n.À
a= (α
1
,α
2
,α
3
,α
4
,α
5
)
Ú
v= (v
1
,v
2
,v
3
,v
4
,v
5
) ∈(F
∗
9
)
5
,
Ù¥α
1
= 0,α
2
= 1,α
3
= 2,α
4
= γ
2
,α
5
= γ
6
;v
1
= γ,v
2
= 1,v
3
= 1,v
4
= 1,v
5
= 1.
Œ±y,éu?¿÷v1≤i≤3 êi,(3.1)ª¤á.éu?¿÷v4≤i≤5 êi,
(3.2)ª¤á.d½n3.2Œ•µGRS
3
(a,v) ´äk1 ‘Hermitian •MDS è,…Ùëê•
[5,3,3]
9
.
4.äk`(`≥1)‘Hermitian•MDSèE
!¥,F
Q
= {α
1
,α
2
,···,α
Q
},n´÷v1 ≤n≤Qê,êk÷v1 ≤k≤n.-
a= (α
1
,α
2
,···,α
n
)
Ú
v= (v
1
,v
2
,···,v
n
) ∈(F
∗
Q
)
n
.
dÚn2.8=ŒGRS
k
(a,v)Hermitian éóè.
DOI:10.12677/pm.2020.10111201020nØêÆ
¸…¦
íØ4.1.GRS
k
(a,v)Hermitian éóè´
GRS
n−k
(a
0
,w),
Ù¥a
0
= (α
q
1
,α
q
2
,···,α
q
n
),w= (w
1
,w
2
,···,w
n
),ùp
w
i
=
1
v
q
i
Y
1≤j≤n,j6=i
(α
q
i
−α
q
j
)
−1
,i= 1,2,···,n.
½n4.2.(n−k−1)q<n.eéu?¿÷v1 ≤i≤nêiÑk
Y
1≤j≤n,j6=i
(α
q
i
−α
q
j
)
−1
= v
q +1
i
,(4.1)
KGRS
k
(a,v)´äkd
k
q
e‘Hermitian•MDSè,…Ùëê•[n,k,n−k+1]
Q
.
y²díØ4.1Œ•:
[GRS
k
(a,v)]
⊥
H
= GRS
n−k
(a
0
,w)
Ù¥a
0
= (α
q
1
,α
q
2
,···,α
q
n
),w= (w
1
,w
2
,···,w
n
),ùp
w
i
=
1
v
q
i
Y
1≤j≤n,j6=i
(α
q
i
−α
q
j
)
−1
,i= 1,2,···,n.
e¡û½GRS
k
(a,v)Hermitian •,Ø”
(v
1
f(α
1
),v
2
f(α
2
),···,v
n
f(α
n
)) ∈GRS
k
(a,v)∩[GRS
k
(a,v)]
⊥
H
,
Ù¥degf(x) ≤k−1.Ïd,•3˜‡gê؇Ln−k−1õ‘ªg(x), ¦
(v
1
f(α
1
),v
2
f(α
2
),···,v
n
f(α
n
)) = (w
1
g(α
q
1
),w
2
g(α
q
2
),···,w
n
g(α
q
n
)),

v
i
f(α
i
) = w
i
g(α
q
i
),i= 1,2,···,n.
dª(4.1)Œw
i
= v
i
6= 0,é1 ≤i≤n,ÏdkXeª:
f(α
i
) = g(α
q
i
),i= 1,2,···,n.
þªL²g(x
q
)−f(x) = 0–kn‡ØÓŠ. Ï•(n−k−1)q<n,¤±deg[g(x
q
)−f(x)] <n,
Ïdg(x
q
)=f(x).dždegf(x)=q·degg(x),qdegf(x)≤k−1,¤±degg(x)≤
k−1
q
.=
degg(x) ≤b
k−1
q
c.¤±Hermitian •‘ê•b
k−1
q
c+1 = d
k
q
e.½ny.
e¡‰ÑA‡äN~f
~4.1.q= 4,Q= q
2
= 16,γ´k••F
Q
,K
F
16
= {0,1,γ,γ
2
,···,γ
14
}.
-n= 8 Úk= 6,dž(n−k−1)q<n,d
k
q
e= 2 .À
a= (α
1
,α
2
,α
3
,α
4
,α
5
,α
6
,α
7
,α
8
)
DOI:10.12677/pm.2020.10111201021nØêÆ
¸…¦
Ú
v= (v
1
,v
2
,v
3
,v
4
,v
5
,v
6
,v
7
,v
8
) ∈(F
∗
16
)
8
,
Ù¥α
1
=0,α
2
=1,α
3
=γ
5
,α
4
=γ
10
,α
5
=γ,α
6
=γ
4
, α
7
=γ
3
,α
8
=γ
12
; v
1
=γ
14
,v
2
=γ
14
,
v
3
= γ
14
,v
4
= γ
14
,v
5
= γ
14
,v
6
= γ
8
,v
7
= γ
14
,v
8
= γ
5
.
Œ±y,éu?¿÷v1≤i≤8 êi,(4.1)ª¤á.d½n4.2Œ•µGRS
6
(a,v)´ä
k2 ‘Hermitian •MDSè,…Ùëê•[8,6,3]
16
.
~4.2.q= 4,Q= q
2
= 16,γ´k••F
Q
,K
F
16
= {0,1,γ,γ
2
,···,γ
14
}.
-n= 12 Úk= 9,dž(n−k−1)q<n,d
k
q
e= 3 .À
a= (α
1
,α
2
,α
3
,α
4
,α
5
,α
6
,α
7
,α
8
,α
9
,α
10
,α
11
,α
12
)
Ú
v= (v
1
,v
2
,v
3
,v
4
,v
5
,v
6
,v
7
,v
8
,v
9
,v
10
,v
11
,v
12
) ∈(F
∗
16
)
12
,
Ù ¥α
1
=0,α
2
=1,α
3
=γ
5
,α
4
=γ
10
,α
5
=γ,α
6
=γ
4
, α
7
=γ
2
,α
8
=γ
8
,α
9
=γ
3
,α
10
=γ
12
,
α
11
= γ
11
,α
12
= γ
14
; v
1
= v
2
= v
3
= v
4
= γ,v
5
= v
6
= v
7
= v
8
= γ
2
,v
9
= v
10
= v
11
= v
12
= 1.
Œ±y,éu?¿÷v1≤i≤12 êi,(4.1)ª¤á.d½n4.2Œ•µGRS
9
(a,v) ´
äk3 ‘Hermitian •MDSè,…Ùëê•[12,9,4]
16
.
~4.3.q= 5,Q= q
2
= 25,γ´k••F
Q
,K
F
25
= {0,1,γ,γ
2
,···,γ
23
}.
Ù¥§γ
6
= 2,γ
12
= 4,γ
18
= 3.
-n= 9 Úk= 7,dž(n−k−1)q<n,d
k
q
e= 2 .À
a= (α
1
,α
2
,α
3
,α
4
,α
5
,α
6
,α
7
,α
8
,α
9
)
Ú
v= (v
1
,v
2
,v
3
,v
4
,v
5
,v
6
,v
7
,v
8
,v
9
) ∈(F
∗
25
)
9
,
Ù¥α
1
=0,α
2
=1,α
3
=2,α
4
=3,α
5
=4,α
6
=γ
2
,α
7
=γ
10
,α
8
=γ
7
,α
9
=γ
11
;v
1
=γ
5
,
v
2
= γ
6
,v
3
= γ
5
,v
4
= γ
5
,v
5
= γ
5
,v
6
= γ
5
,v
7
= γ
5
,v
8
= γ
5
,v
8
= γ
5
.
Œ±y,éu?¿÷v1≤i≤9 êi,(4.1)ª¤á.d½n4.2Œ•µGRS
7
(a,v)´ä
k2 ‘Hermitian •MDSè,…Ùëê•[9,7,3]
25
.
ë•©z
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[2]Sendrier,N.andSkersys,G.(2001)OntheComputationoftheAutomorphismGroupofa
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DOI:10.12677/pm.2020.10111201023nØêÆ

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