8XR "bB+ *QM+2Tib - PDF Hull Dimension Over Finite Fields +)2.$%)#$!#3.'!4)$)&.!4).-+

*?Ti2` 8

6Q` i, j ∈ F4- H2iS_ij :=|{l ∈ {1,2, . . . , n} | β_l =

⎡

⎣i j

⎤

⎦}|X Ai Bb MQi /B{+mHi iQ b22 i?i i?2 ;2M2`iQ` Ki`Bt G M/ i?2 +Q/2 C `2 /2i2`KBM2/ 2tTHB+BiHv #v i?2 pHm2b Sij

7Q` HH i, j ∈F⁴X 6Q` +QMbi`m+iBQMb Q7 [mi2`M`v HBM2` +Q/2b- Bi bm{+2b iQ 2bi#HBb?2/

i?2 pHm2b SijX

Hi2`MiBp2Hv- H2i α1 =!

a11 a12 · · · a1n

M/ α2 =!

a21 a22 · · · a2n

X h?2M G M/ C +M #2 pB2r b G=

⎡

⎣α1

α2

⎤

⎦ M/

C ={0,α1,α2,ωα1,ω²α1,ωα2,ω²α2,α1+α2,ωα1+α2,ω²α1+α2,α1+ωα2,α1+ω²α2, ωα1+ωα2,ω²α1+ωα2,ωα1+ω²α2,ω²α1+ω²α2},

`2bT2+iBp2HvX

"v BMbT2+iBQM- Bi Bb 2bBHv b22M i?i

ri(α1) = ri(ωα1) = ri(ω²α1) = *

i∈F^∗4

j∈F⁴

Sij, U8XkV

ri(α2) = ri(ωα2) = ri(ω²α2) = *

i∈F4

j∈F^∗4

Sij, U8XjV

ri(α1+α₂) =ri(ωα1+ωα₂) =ri(ω²α₁ +ω²α₂) = *

i∈F4

j∈F4,i̸=j

S_ij, U8X9V

ri(α1+ωα2) = ri(ωα1+ω²α2) = ri(ω²α1+α2) = *

i∈F4

j∈F4,i̸=ωj

Sij, U8X8V M/

ri(α1+ω²α₂) = ri(ωα1+α₂) =ri(ω²α₁+ωα₂) =*

i∈F4

j∈F4,i̸=ω²j

S_ij. U8XeV

G2iy0 =S11+Sωω +S_ω²_ω², y1 =S_1ω² +Sω1+S_ω²_ω- M/ y2 =S_ω²₁+S1ω+S_ωω²X h?2M r2 ?p2

GG^† =

⎡

⎢⎢

⎣ '

i∈F^∗4

j∈F4

Sij y0+ωy1+ω²y2

y₀+ω²y₁+ωy₂ '

i∈F⁴

j∈F^∗4

S_ij

⎤

⎥⎥

⎦, U8XdV

r?2`2 i?2 +H+mHiBQM Bb /QM2 BM F⁴.

h?2 +QMbi`m+iBQM Bb BHHmbi`i2/ BM i?2 7QHHQrBM; 2tKTH2X

1tKTH2 8XRXRX G2iC #2 [mi2`M`v HBM2` +Q/2 Q7 H2M;i? 4/2i2`KBM2/ #vS10= S01 = S11 = S1ω = 1 M/ S00 = S0ω = S0ω² = S1ω² = Sω0 = Sω1 = Sωω = Sωω² = S_ω²₀ =S_ω²₁ =S_ω²_ω =S_ω²_ω² = 0X h?2M i?2 ;2M2`iQ` Ki`Bt Q7 C Bb Q7 i?2 7Q`K

⎡

⎣1 0 1 1 0 1 1 ω

⎤

⎦.

*QMb2[m2MiHv- r2 ?p2

y₀ = 1, y₁ = 0, M/ y₂ = 1.

AM F⁴, r2 ?p2

i∈F^∗4

j∈F4

S_ij = 1 M/ *

i∈F4

j∈F^∗4

S_ij = 1.

"v U8XdV- Bi 7QHHQrb i?i

GG^†=

⎡

⎣ 1 1 +ω² 1 +ω 1

⎤

⎦=

⎡

⎣1 ω ω² 1

⎤

⎦

r?B+? BKTHB2b i?i rank(GG^†) = 1 M/ dim(Hull_H(C)) = 2−rank(GG^†) = 1 #v S`QTQbBiBQM kXkX9X

"v b2iiBM; k = 2 M/ q = 4 BM G2KK kXjX9- i?2 M2ti H2KK 7QHHQrbX G2KK 8XRXkX D^H₄(n,2,1)≤1_4n

2 7Q` HH BMi2;2`b n≥3X

**8Xk PTiBKH Zmi2`M`v GBM2` *Q/2b rBi? >2`@**

KBiBM >mHH .BK2MbBQM PM2 rBi? n ≡ 1, 2, 4 (mod 5)

AM i?Bb b2+iBQM- r2 7Q+mb QM +QMbi`m+iBQMb Q7 QTiBKH [mi2`M`v HBM2` +Q/2b rBi?

/BK2MbBQM 2 M/ >2`KBiBM ?mHH /BK2MbBQM QM2- M/ 2bi#HBb? i?2 2t+i pHm2b Q7 D^H₄(n,2,1) 7Q` `#Bi``v H2M;i?b n≥3 bm+? i?in ≡1,2,4 (mod 5)X

h?2Q`2K 8XkXRX G2i n ≥3 #2 M BMi2;2`X A7 n ≡1,2,4 (mod 5)- i?2M D₄^H(n,2,1) =

-4n 5

. .

S`QQ7X bbmK2 i?in ≡1,2,4 (mod 5)X 6`QK G2KK 8XRXk- Bi 7QHHQrb i?iD^H₄(n,2,1)≤ 1_4n

2X Ai `2KBMb iQ b?Qr i?2 2tBbi2M+2 Q7 3

n,2,1_4n

4 +Q/2 r?Qb2 >2`KBiBM ?mHH /BK2MbBQM Bb QM2X

q2 +QMbB/2` i?2 +QMbi`m+iBQMb BM i?2 7QHHQrBM; 3 +b2bX

*b2 RX n ≡ 1 (mod 5)X h?2M n = 5t+ 1 7Q` bQK2 TQbBiBp2 BMi2;2` tX "b2/ QM i?2 T`Biv Q7 t- r2 +QMbB/2` i?2 7QHHQrBM; irQ bm#+b2bX

*b2 RXRX t Bb 2p2MX 6Q` 1≤r≤n- H2i βr #2 i?2 2×1 Ki`Bt Qp2`F4 /2}M2/ #v

βr =

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

[0 1]^T B7 1≤r≤ ₂^t, [0 ω]^T B7 ₂^t + 1≤r ≤ ^2t₂, [ω 0]^T B7 ^2t₂ + 1≤r≤ ^3t₂, [1 1]^T B7 ^3t₂ + 1≤r≤ ^4t₂, [ω² ω²]^T B7 ^4t₂ + 1≤r≤ ^5t₂ [1 ω²]^T B7 ^5t₂ + 1≤r≤ ^6t₂, [ω² ω]^T B7 ^6t₂ + 1≤r≤ ^7t₂, [ω² 1]^T B7 ^7t₂ + 1≤r≤ ^8t₂, [ω ω²]^T B7 ^8t₂ + 1≤r≤ ^9t₂, [1 0]^T B7 ^9t₂ + 1≤r≤ ^10t₂ + 1.

G2iC #2 M [n,2]4 +Q/2 ;2M2`i2/ #v G=!

β1 β2 · · · βn

h?2M S₀₁=S_0ω =S_ω0 =S₁₁ =S_ω²_ω² =S_1ω² =S_ω²_ω =S_ω²₁ =S_ωω² = ₂^t- S₁₀= ₂^t + 1- M/ Sij = 0 Qi?2`rBb2X Ai 7QHHQrb i?i

y0 =S11+Sωω +Sω²ω² =t, y1 =S1ω² +Sω1+Sω²ω =t, M/

y2 =Sω²1+S1ω+Sωω² =t.

AM F⁴- Bi +M #2 /2/m+2/ i?i

i∈F^∗4

j∈F4

Sij = 1 M/ *

i∈F4

j∈F^∗4

Sij = 0.

"v U8XdV- r2 ?p2

GG^†=

⎡

⎣1 0 0 0

⎤

⎦

r?B+? BKTHB2b i?i rank(GG^†) = 1X "v S`QTQbBiBQM kXkX9- C ?b >2`KBiBM ?mHH /BK2MbBQM 2−rank(GG^†) = 1X

"v i?2 /2}MBiBQM Q7 Sij M/ U8XkVĜU8XeV- Bi +M #2 /2/m+2/ i?i ri(α1) = ri(ωα1) =ri(ω²α1) = 4t+ 1, ri(α2) = ri(ωα2) =ri(ω²α₂) = 4t, ri(α1+α2) = ri(ωα1+ωα2) = ri(ω²α1+ω²α2) = 4t+ 1, ri(α1+ωα2) =ri(ωα1+ω²α2) =ri(ω²α1+α2) = 4t+ 1, M/

ri(α1+ω²α2) =ri(ωα1+α2) = ri(ω²α1+ωα2) = 4t+ 1.

>2M+2- C ?b KBMBKmK r2B;?i 4t = /

4(5t+1) 5

= 1_4n

2X h?2`27Q`2- C Bb M QTiBKH [n,2,1_4n

2]4 +Q/2 rBi? >2`KBiBM ?mHH /BK2MbBQM QM2X

*b2 RXkX t Bb Q//X 6Q` 1≤r ≤n- H2i βr #2 i?2 2×1Ki`Bt Qp2` F⁴ /2}M2/ #v

βr =

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

[0ω]^T B7 1≤r≤ ^t+1₂ ,

[1 0]^T B7 ^t+1₂ + 1≤r≤ ^2(t+1)₂ , [ω 0]^T B7 ^2(t+1)₂ + 1≤r ≤ ^3(t+1)₂ , [ω² ω²]^T B7 ^3(t+1)₂ + 1≤r ≤ ^4(t+1)₂ , [ω² ω]^T B7 ^4(t+1)₂ + 1≤r ≤ ^5(t+1)₂ , [ω ω²]^T B7 ^5(t+1)₂ + 1≤r ≤ ^6(t+1)₂ , [0 1]^T B7 ^6(t+1)₂ + 1≤r ≤ ^7t+5₂ , [1 1]^T B7 ^7t+5₂ + 1≤r ≤ ^8t+4₂ , [1ω²]^T B7 ^8t+4₂ + 1≤r ≤ ^9t+3₂ , [ω² 1]^T B7 ^9t+3₂ + 1≤r ≤ ^10t+2₂ . G2iC #2 M [n,2]4 +Q/2 ;2M2`i2/ #v

G=!

β1 β2 · · · βn

h?2M S_0ω =S₁₀=S_ω0 =S_ω²_ω² =S_ω²_ω =S_ωω² = ^t+1₂ - S₀₁=S₁₁ =S_1ω² =S_ω²₁ = ^t−1₂ - M/ Sij = 0 Qi?2`rBb2X Ai 7QHHQrb i?i

y0 =S11+Sωω +Sω²ω² =t, y1 =S1ω² +Sω1+Sω²ω =t, M/

y2 =S_ω²₁+S1ω+S_ωω² =t.

AM F⁴- r2 ?p2

i∈F^∗4

j∈F4

Sij = 1 M/ *

i∈F4

j∈F^∗4

Sij = 0.

6`QK U8XdV- Bi +M #2 +QM+Hm/2/ i?i

GG^†=

⎡

⎣1 0 0 0

⎤

⎦.

6`QK S`QTQbBiBQM kXkX9- Bi +M #2 +QM+Hm/2/ i?i C ?b >2`KBiBM ?mHH /BK2MbBQM 2−rank(GG^†) = 2−1 = 1X

lbBM; i?2 /2}MBiBQM Q7 Sij M/ U8XkVĜU8XeV- r2 ?p2

ri(α1) = ri(ωα1) =ri(ω²α1) = 4t+ 1, ri(α2) = ri(ωα2) =ri(ω²α₂) = 4t, ri(α1+α2) = ri(ωα1+ωα2) = ri(ω²α1+ω²α2) = 4t+ 1, ri(α1+ωα2) =ri(ωα1+ω²α2) =ri(ω²α1+α2) = 4t+ 1, M/

ri(α1+ω²α2) =ri(ωα1+α2) = ri(ω²α1+ωα2) = 4t+ 1.

>2M+2- C ?b KBMBKmK r2B;?i 4t=/

4(5t+1) 5

=1_4n

2X *QMb2[m2MiHv- C Bb M QTiBKH [n,2,1_4n

2]₄ +Q/2 rBi? >2`KBiBM ?mHH /BK2MbBQM QM2X 6`QK i?2 irQ bm#+b2b- Bi 7QHHQrb i?i

D^H₄(n,2,1) = -4n

5 .

7Q` HH n ≡1 (mod 5)X

*b2 kX n ≡ 2 (mod 5)X h?2M n = 5t + 2 7Q` bQK2 TQbBiBp2 BMi2;2` tX aBM+2 [mi2`M`v HBM2` +Q/2 C Bb /2i2`KBM2/ #v i?2 pHm2b Sij 7Q` HH i, j ∈ F⁴- BM i?2

`2KBMBM; T`ib- r2 ;Bp2 +QMbi`m+iBQMb Q7 HBM2` +Q/2b BM i2`Kb Q7 SijX >Qr2p2`- M 2tTHB+Bi 7Q`K Q7 Bib ;2M2`iQ` Ki`Bt G +M #2 /2i2`KBM2/ BM i?2 bK2 rv b *b2 RX q2 +QMbB/2` i?2 7QHHQrBM; irQ bm#+b2bX

*b2 kXR t Bb 2p2MX G2i S01 =Sω0 =Sω²ω² =S1ω² =Sω²ω =Sωω² = ₂^t, S0ω = ₂^t −1, S₁₀=S₁₁=S_ω²₁ = ₂^t + 1, M/ S_ij = 0 Qi?2`rBb2. Ai +M #2 +QM+Hm/2/ i?i

y0 =S11+Sωω +S_ω²_ω² =t+ 1, y₁ =S_1ω² +S_ω1+S_ω²_ω =t, M/

y2 =Sω²1+S1ω+Sωω² =t+ 1.

AM F4- Bi +M #2 /2/m+2/ i?i

i∈F^∗4

j∈F4

Sij = 1 M/ *

i∈F4

j∈F^∗4

Sij = 1.

"v U8XdV- r2 ?p2

GG^† =

⎡

⎣1 ω ω² 1

⎤

⎦

r?B+? BKTHB2b i?i rank(GG^T) = 1 M/ dim(>mHHH(C)) = 1 #v S`QTQbBiBQM kXkX9X

"b2/ QM i?2 /2}MBiBQM Q7 Sij M/ U8XkVĜU8XeV- Bi +M #2 /2/m+2/ i?i ri(α1) = ri(ωα1) =ri(ω²α1) = 4t+ 3, ri(α2) = ri(ωα2) =ri(ω²α2) = 4t+ 1, ri(α1+α2) = ri(ωα1+ωα2) = ri(ω²α1+ω²α2) = 4t+ 1, ri(α1+ωα2) =ri(ωα1+ω²α2) =ri(ω²α1+α2) = 4t+ 2, M/

ri(α1+ω²α₂) =ri(ωα1+α₂) = ri(ω²α₁+ωα₂) = 4t+ 1.

>2M+2- i?2 KBMBKmK r2B;?i Q7 C Bb 4t+ 1 = /

4(5t+2) 5

0 = 1_4n

2X b /2bB`2/- C Bb M QTiBKH[n,2,1_4n

2]4 +Q/2 rBi? >2`KBiBM ?mHH /BK2MbBQM QM2X

*b2 kXkX t Bb Q//X G2i S₁₀ = S₁₁ = S_ω²_ω² = S_1ω² = S_ω²_ω = S_ω²₁ = S_ωω² = ^t+1₂ , S01=S0ω =Sω0 = ^t−1₂ , M/ Sij = 0 Qi?2`rBb2. h?2M

y0 =S11+Sωω+Sω²ω² =t+ 1, y1 =S1ω² +Sω1+Sω²ω =t+ 1, M/

y2 =Sω²1+S1ω+Sωω² =t+ 1.

AM F⁴- r2 ?p2

i∈F^∗4

j∈F4

Sij = 1 M/ *

i∈F4

j∈F^∗4

Sij = 0.

"v U8XdV- Bi 7QHHQrb i?i

GG^†=

⎡

⎣1 0 0 0

⎤

⎦.

AM i?Bb +b2-rank(GG^T) = 1M/dim(>mHHH(C)) = 2−rank(GG^T) = 1#v S`QTQbBiBQM kXkX9X

lbBM; i?2 /2}MBiBQM Q7 Sij M/ U8XkVĜU8XeV- r2 ?p2

ri(α1) = ri(ωα1) =ri(ω²α1) = 4t+ 3, ri(α2) = ri(ωα2) =ri(ω²α2) = 4t+ 2, ri(α1+α₂) = ri(ωα1+ωα₂) = ri(ω²α₁+ω²α₂) = 4t+ 1, ri(α1+ωα2) =ri(ωα1+ω²α2) =ri(ω²α1+α2) = 4t+ 1, M/

ri(α1+ω²α2) =ri(ωα1+α2) = ri(ω²α1+ωα2) = 4t+ 1 r?B+? BKTHB2b i?iC ?b KBMBKmK r2B;?i 4t+ 1 =/

4(5t+2) 5

=1_4n

2X AM i?Bb +b2-C Bb M QTiBKH[n,2,1_4n

2]4 +Q/2 rBi? >2`KBiBM ?mHH /BK2MbBQM QM2X 6`QK i?2 irQ bm#+b2b- Bi 7QHHQrb i?i

D^H₄(n,2,1) = -4n

5 .

7Q` HH n ≡2 (mod 5)X

*b2 jX n ≡ 4 (mod 5)X h?2M n = 5t+ 4 7Q` bQK2 TQbBiBp2 BMi2;2` tX q2 +QMbB/2`

i?2 7QHHQrBM; irQ +b2b #b2/ QM i?2 T`Biv Q7 tX

*b2 jXRX t Bb 2p2MX G2i S0ω =S10 =Sω0 =Sω²ω² = Sωω² =Sω²ω = ₂^t, S01 = S11 = S1ω² =Sω²1 = ₂^t + 1, M/ Sij = 0 Qi?2`rBb2.h?2M

y₀ =S₁₁+S_ωω+S_ω²_ω² =t+ 1, y1 =S1ω² +Sω1+Sω²ω =t+ 1, M/

y2 =Sω²1+S1ω+Sωω² =t+ 1.

AM F⁴- r2 ?p2

i∈F^∗4

j∈F4

S_ij = 1 M/ *

i∈F4

j∈F^∗4

S_ij = 0.

"v U8XdV- Bi +M #2 +QM+Hm/2/ i?i

GG^†=

⎡

⎣1 0 0 0

⎤

⎦.

Ai K2Mb i?irank(GG^T) = 1M/C ?b >2`KBiBM ?mHH /BK2MbBQM QM2 #v S`QTQbBiBQM kXkX9X

6`QK i?2 /2}MBiBQM Q7Sij M/ U8XkVĜU8XeV- Bi +M #2 /2/m+2/ i?i ri(α1) = ri(ωα1) =ri(ω²α1) = 4t+ 3, ri(α2) = ri(ωα2) =ri(ω²α₂) = 4t+ 4, ri(α1+α2) = ri(ωα1+ωα2) = ri(ω²α1+ω²α2) = 4t+ 3, ri(α1+ωα2) =ri(ωα1+ω²α2) =ri(ω²α1+α2) = 4t+ 3, M/

ri(α1+ω²α2) =ri(ωα1+α2) = ri(ω²α1+ωα2) = 4t+ 3.

>2M+2- C ?b KBMBKmK r2B;?i 4t+ 3 = /

4(5t+4) 5

01_4n

2X h?2`27Q`2- C Bb M QTiBKH [mi2`M`v HBM2` [n,2,1_4n

2]4 +Q/2 rBi? >2`KBiBM ?mHH /BK2MbBQM QM2X

*b2 jXkX t Bb Q//X G2i S₀₁ = S_0ω = S_ω0 = S₁₁ = S_ω²_ω² = S_1ω² = S_ω²_ω = S_ω²₁ = Sωω² = ^t+1₂ , S10= ^t⁻₂¹, M/ Sij = 0 Qi?2`rBb2.h?2M

y0 =S11+Sωω+S_ω²_ω² =t+ 1, y1 =S_1ω² +Sω1+S_ω²_ω =t+ 1, M/

y₂ =S_ω²₁+S_1ω+S_ωω² =t+ 1.

AM F4- Bi +M #2 /2/m+2/ i?i

i∈F^∗4

j∈F4

Sij = 1 M/ *

i∈F4

j∈F^∗4

Sij = 0.

"v U8XdV- r2 ?p2

GG^†=

⎡

⎣1 0 0 0

⎤

⎦

r?B+? BKTHB2b i?i rank(GG^T) = 1 M/ dim(>mHHH(C)) = 2−rank(GG^T) = 1X TTHvBM; i?2 /2}MBiBQM Q7Sij M/ U8XkVĜU8XeV- r2 ?p2

ri(α1) = ri(ωα1) =ri(ω²α1) = 4t+ 3, ri(α2) = ri(ωα2) =ri(ω²α2) = 4t+ 4, ri(α1+α2) = ri(ωα1+ωα2) = ri(ω²α1+ω²α2) = 4t+ 3, ri(α1+ωα₂) =ri(ωα1+ω²α₂) =ri(ω²α₁+α₂) = 4t+ 3, M/

ri(α1+ω²α2) =ri(ωα1+α2) = ri(ω²α1+ωα2) = 4t+ 3 98

r?B+? BKTHB2b i?i wt(C) = 4t+ 3 = /

4(5t+4) 5

01_4n

2X *QMb2[m2MiHv- C Bb M QTiBKH [n,2,1_4n

2]4 +Q/2 rBi? >2`KBiBM ?mHH /BK2MbBQM QM2X 6`QK i?2 irQ bm#+b2b- Bi +M #2 +QM+Hm/2/ i?i

D^H₄(n,2,1) = -4n

5 .

7Q` HH n ≡4 (mod 5)X

1tKTH2 8XkXkX G2i t = 2X "v h?2Q`2K 8XkXR M/ t Bb 2p2M- r2 ?p2 i?i S01 = S0ω = Sω0 = S11 = S_ω²_ω² = S_1ω² = S_ω²_ω = S_ω²₁ = S_ωω² = 1, S10 = 2 M/ Sij = 0 Qi?2`rBb2X G2iC #2 [mi2`M`v HBM2` +Q/2 Q7 H2M;i? 11 rBi? ;2M2`iQ` Ki`Bt

⎡

⎣1 1 0 0 ω 1 ω² 1 ω² ω² ω 0 0 1 ω 0 1 ω² ω² ω 1 ω²

⎤

⎦.

h?2M "v h?2Q`2K 8XkXR- Bi +M #2 /2/m+2/ i?i

y0 =t, y1 =t, M/ y2 =t.

AM F⁴- r2 ?p2

i∈F^∗4

j∈F4

Sij = 1 M/ *

i∈F4

j∈F^∗4

Sij = 0.

"v U8XdV- Bi 7QHHQrb i?i

GG^†=

⎡

⎣1 0 0 0

⎤

⎦

r?B+? BKTHB2b i?i rank(GG^†) = 1 M/ dim(HullH(C)) = 2−rank(GG^†) = 1 #v S`QTQbBiBQM kXkX9X h?2M C Bb M QTiBKH[11,2,8]4 +Q/2 rBi? ?mHH /BK2MbBQM QM2X

8Xj "QmM/b QM D

₄^H

(n, 2, 1) rBi? n ≡ 0, 3 (mod 5)

AM i?Bb b2+iBQM- r2 7Q+mb QM i?2 irQ `2KBMBM; +b2b r?2`2 n ≡ 0,3 (mod 5)X 6Q` i?2b2 +b2b- r2 T`QpB/2 ;QQ/ HQr2` M/ mTT2` #QmM/b QM i?2 KBMBKmK r2B;?i Q7 [n,2]4 +Q/2b rBi? >2`KBiBM ?mHH /BK2MbBQM QM2X S`2+Bb2Hv- mTT2` M/ HQr2` #QmM/b QMD^H₄(n,2,1)`2 ;Bp2M rBi? i?2 ;T QM2X

h?2Q`2K 8XjXRX G2i n #2 TQbBiBp2 BMi2;2`X A7 n ≡0,3 (mod 5)- i?2M i?2`2 2tBbib M [n,2,1_4n

2−1]4 +Q/2 rBi? >2`KBiBM ?mHH /BK2MbBQM QM2X

S`QQ7X bbmK2 i?i n ≡0,3 (mod 5)X q2 +QMbB/2` i?2 7QHHQrBM; irQ +b2bX

*b2 RX n ≡0 (mod 5)X h?2Mn−1≡4 (mod 5)M/n = 5t7Q` bQK2 TQbBiBp2 BMi2;2`

tX "v h?2Q`2K 8XkXR- i?2`2 2tBbib M [n−1,2,/

4(n−1) 5

0]4 +Q/2C rBi? >2`KBiBM ?mHH /BK2MbBQM QM2X G2i G#2 ;2M2`iQ` Ki`Bt 7Q` C M/ H2i C^′ #2 [mi2`M`v HBM2`

+Q/2 ;2M2`i2/ #v G^′ = [0 G]X aBM+2 -4(n−1)

5 .

-4(5t−1) 5

= 4t−1 =

-4(5t) 5

−1 = -4n

5 .

−1,

Bi 7QHHQrb i?i C^′ Bb M [n,2,/

4(n−1) 5

0]2 = [n,2,1_4n

2−1]2 +Q/2 rBi? >2`KBiBM ?mHH /BK2MbBQM QM2X

*b2 kX n ≡ 3 (mod 5)X h?2M n−1≡2 (mod 5) M/ n = 5t+ 37Q` bQK2 TQbBiBp2 BMi2;2`tX "v h?2Q`2K 8XkXR- i?2`2 2tBbib M[n−1,2,/

4(n−1) 5

0]4+Q/2CrBi? >2`KBiBM

?mHH /BK2MbBQM QM2X G2i G #2 ;2M2`iQ` Ki`Bt 7Q` C M/ H2i C^′ #2 [mi2`M`v HBM2` +Q/2 ;2M2`i2/ #v G^′ = [0 G]X aBM+2

-4(n−1) 5

-4(5t+ 2) 5

= 4t+ 1 =

-4(5t+ 3) 5

−1 = -4n

5 .

−1,

Bi +M #2 2bBHv b22M i?i C^′ Bb M [n,2,/

4(n−1) 5

]2 = [n,2,1_4n

2−1]2 +Q/2 rBi? >2`@

KBiBM ?mHH /BK2MbBQM QM2X

1tKTH2 8XjXkX "v 1tKTH2 8XkXk- r2 ?p2 i?iC Bb HBM2` +Q/2 rBi? T`K2i2`b [11,2,8]4 ;2M2`i2/ #v

⎡

⎣1 1 0 0 ω 1 ω² 1 ω² ω² ω 0 0 1 ω 0 1 ω² ω² ω 1 ω²

⎤

⎦.

G2i G^′ = [0 G]X "v h?2Q`2K 8XkXR- r2 ?p2 i?i G^′ ;2M2`i2b M QTiBKH [10,2,7]3

+Q/2 rBi? ?mHH /BK2MbBQM QM2X

*Q`QHH`v 8XjXjX G2i n #2 TQbBiBp2 BMi2;2`X A7 n ≡0,3 (mod 5) i?2M -4n

5 .

−1≤D₄^H(n,2,1)≤ -4n

5 .

. 9d

S`QQ7X q?BH2 i?2 HQr2` #QmM/ Bb ;Bp2M BM h?2Q`2K 8XjXR- i?2 mTT2` #QmM/ Bb ;m`M@

i22/ #v G2KK 8XRXkX

*QMbi`m+iBQMb M/ #QmM/b QM [mi2`M`v HBM2` +Q/2b rBi? /BK2MbBQM irQ M/

>2`KBiBM ?mHH /BK2MbBQM QM2 ?p2 #22M bim/B2/X PTiBKH [mi2`Mv [n,2]4 +Q/2b rBi? >2`KBiBM ?mHH /BK2MbBQM QM2 ?p2 #22M +QMbi`m+i2/ 7Q` HH H2M;i?b n ≥3 bm+?

i?i n ≡ 1,2,4 (mod 5)X :QQ/ HQr2` M/ mTT2` #QmM/b QM i?2 KBMBKmK r2B;?i Q7 [mi2`Mv [n,2]4 +Q/2b rBi? >2`KBiBM ?mHH /BK2MbBQM QM2 ?p2 #22M ;Bp2M 7Q` HH H2M;i?b n≡0,3 (mod 5)X h?2 #Qp2 `2bmHib `2 bmKK`Bx2/ b 7QHHQrbX

h?2Q`2K 8XjX9X G2i n ≥3 #2 M BMi2;2`X h?2M D^H₄(n,2,1) =

-4n 5

. 7Q` HH n≡1,2,4 (mod 5)

M/ -

4n 5

−1≤D^H₄(n,2,1)≤ -4n

. 7Q` HH n≡0,3 (mod 5).

"b2/ QM Qm` BMbT2+iBQM- r2 T`QTQb2 i?2 7QHHQrBM; +QMD2+im`2X

*QMD2+im`2 8XjX8X D^H₄(n,2,1) =1_4n

2−1 7Q` HH TQbBiBp2 BMi2;2`b n≡0,3 (mod 5)X

*?Ti2` e

Dalam dokumen PDF Hull Dimension Over Finite Fields +)2.$%)#$!#3.'!4)$)&.!4).-+ (Halaman 44-57)