PDF Hull Dimension Over Finite Fields +)2.$%)#$!#3.'!4)$)&.!4).-+

CONSTRUCTION AND BOUNDS ON LINEAR CODES WITH PERSCRIBED HULL DIMENSION OVER FINITE FIELD

By

Mr. Todsapol MANKEAN

A Thesis Proposal Submitted in Partial Fulfillment of the Requirements for Doctor of Philosophy (MATHEMATICS)

Department of MATHEMATICS Graduate School, Silpakorn University

Academic Year 2020

Copyright of Graduate School, Silpakorn University

CONSTRUCTIONS AND BOUNDS ON LINEAR CODES WITH PRESCRIBED HULL DIMENSION OVER FINITE FIELDS

การสร้างและค่าขอบเขตของรหัสเชิงเส้นบนฟีลด์จ ากัดซึ่งก าหนดมิติของเปลือกหุ้ม

โดย

Mr.Todsapol MANKEAN

โครงร่างวิทยานิพนธ์นี้เป็นส่วนหนึ่งของการศึกษาตามหลักสูตรปรัชญาดุษฎีบัณฑิต สาขาวิชาคณิตศาสตร์ แบบ 2.1 ปรัชญาดุษฎีบัณฑิต นานาชาติ

ภาควิชาคณิตศาสตร์

บัณฑิตวิทยาลัย มหาวิทยาลัยศิลปากร ปีการศึกษา 2563

ลิขสิทธิ์ของบัณฑิตวิทยาลัย มหาวิทยาลัยศิลปากร

CONSTRUCTION AND BOUNDS ON LINEAR CODES WITH PERSCRIBED HULL DIMENSION OVER FINITE FIELD

By

Mr. Todsapol MANKEAN

A Thesis Proposal Submitted in Partial Fulfillment of the Requirements for Doctor of Philosophy (MATHEMATICS)

Department of MATHEMATICS Graduate School, Silpakorn University

Academic Year 2020

Copyright of Graduate School, Silpakorn University

CONSTRUCTIONS AND BOUNDS ON LINEAR CODES WITH PRESCRIBED HULL DIMENSION OVER FINITE FIELDS

Title CONSTRUCTION AND BOUNDS ON LINEAR CODES WITH PERSCRIBED HULL DIMENSION OVER FINITE FIELD

By Todsapol MANKEAN

Field of Study (MATHEMATICS)

Advisor Associate Professor SOMPHONG JITMAN , Ph.D.

Graduate School Silpakorn University in Partial Fulfillment of the Requirements for the Doctor of Philosophy

(Associate Professor Jurairat Nunthanid, Ph.D.) Dean of graduate school Approved by

(Assistant Professor Ratana Srithus , Ph.D.) Chair person

(Associate Professor SOMPHONG JITMAN , Ph.D.)

Advisor

(Professor Patanee Udomkavanich , Ph.D.) External Examiner

Title CONSTRUCTIONS AND BOUNDS ON LINEAR CODES WITH PRESCRIBED HULL DIMENSION OVER FINITE FIELDS By Todsapol MANKEAN

Title CONSTRUCTIONS AND BOUNDS ON LINEAR CODES WITH PRESCRIBED HULL DIMENSION OVER FINITE FIELDS

By Todsapol MANKEAN

C

60305805 : MAJOR : MATHEMATICS

KEY WORDS : LINEAR CODES / BOUNDS / OPTIMAL CODES / HULLS OF LINEAR CODES / EUCLIDEAN INNER PRODUCT / HERMITIAN INNER PRODUCT

TODSAPOL MANKEAN : CONSTRUCTIONS AND BOUNDS ON LINEAR CODES WITH PRESCRIBED HULL DIMENSION OVER FINITE FIELDS. THESIS ADVISOR : ASSOCIATE PROFESSOR SOMPHONG JINMAN, Ph.D.

Due to their wide applications, hulls of linear codes have been of interest and extensively studied. In this thesis, bounds on the parameters and constructions of optimal linear codes with hull dimension one over small finite fields have been focused on. For all positive integers

𝑛 ≥ 3

and

^{𝑞 ∈ {2,3}}

, optimal linear

^{[𝑛, 2, 𝑑]}𝑞

codes with Euclidean hull dimension one are constructed. Moreover, for

^{𝑞 = 2}

, the enumeration of such optimal codes is given up to equivalence. For all lengths

^{𝑛 ≥ 3}

such that

𝑛 ≡ 1, 2, 4 (mod 5)

, optimal quaternary linear

[𝑛, 2]₄

codes with Hermitial hull dimension one are constructed. For positive integers

^{𝑛 ≡}

0, 3 (mod 5)

, good lower and upper bounds on the minimum weight of quaternary

^{[𝑛, 2, 𝑑]}4

codes with Hermitian hull dimension one are given as well as some illustrative examples.

D

Acknowledgements

This thesis has been completed by the involvement of people about whom I would like to mention here.

I would like to express my deep gratitude to my thesis advisor, Associate Professor Dr. Somphong Jitman, for his help and support in all stages of my thesis studies.

I would like to thank to my thesis committees, Assistant Professor Dr. Ratana Srithus and Professor Dr. Patanee Udomkavanich, for comments and suggestions.

I would like to thank all the teachers who have instructed and taught me for valuable knowledge.

I would like to thank the Development and Promotion of Science and Technology Talents Project (DPST) for financial support throughout my undergraduate and graduate studies.

Finally, I would like to thank my family, my friends and those whose names are not mentioned here but have greatly inspired and encouraged me throughout the period of this research.

Todsapol MANKEAN

E

Table of Contents

Page Abstract……… C Acknowledgements………... D Chapter

1. Introduction………... 1

2. Preliminaries………. 3

2.1 Linear Codes over Finite Fields……….. 3

2.2 Hulls of Linear Codes………. 4

2.3 Constructions and Bounds……….. 7

3. Optimal Binary Linear Codes with Euclidean Hull Dimension One……… 10

3.1 Basic Concepts……… 10

3.2

𝑛 ≡ 1 (mod 6)

………. 12

3.3

𝑛 ≡ 3 (mod 6)

……… 15

3.4

𝑛 ≡ 5 (mod 6)

……… 18

3.5

𝑛 ≡ 0,2,4 (mod 6)

………. 20

4. Optimal Ternary Linear Codes with Euclidean Hull Dimension One………. 26

4.1 Basic Concepts……… 26

4.2

𝑛 ≡ 1 (mod 4)

………. 28

4.3 𝑛 ≡ 2 (mod 4)

……… 30

4.4

𝑛 ≡ 3 (mod 4)

……… 32

4.5

𝑛 ≡ 0 (mod 4)

………... 33

5. Quaternary Linear Codes with Hermitian Hull Dimension One……….... 36

5.1 Basic Concepts……… 36

F

5.2 Optimal Quaternary Linear Codes with Hermitian

Hull Dimension One with

𝑛 ≡ 1, 2, 4 (mod 5)

………. 39

5.3 Bounds on

𝐷₄^𝐻(𝑛, 2,1)

with

𝑛 ≡ 0, 3 (mod 5)

………... 46

6. Conclusion ……….. 49

References………. 52

Publications………... 55

Biography………. 56

*?Ti2` R AMi`Q/m+iBQM

*Q/BM; i?2Q`v BMi`Q/m+2/ BM RN93 #v *Hm/2 a?MMQM Bb i?2 bim/v Q7 i?2 T`QT@

2`iB2b Q7 +Q/2b M/ /2Hb rBi? i?2 /2bB;M Q7 2``Q`@+Q``2+iBM; +Q/2b 7Q` i?2 `2HB#H2 i`MbKBbbBQM Q7 BM7Q`KiBQM +`Qbb MQBbv +?MM2HbX GBM2` +Q/2b- bm#bT+2b Q7 i?2 p2+iQ` bT+2 F<sup>n</sup>q- `2 BMi2`2biBM; 2ti2MbBp2Hv bim/B2/ /m2 iQ i?2B` MB+2 H;2#`B+ bi`m+@

im`2b M/ rB/2 TTHB+iBQMbX HBM2` +Q/2 +QMiBM2/ BM Bib /mH Bb +HH2/ b2H7@

Q`i?Q;QMH +Q/2M/ HBM2` +Q/2 K22ib Bib /mH i`BpBHv Bb +HH2/ HBM2` +QKTH2K2M@

i`v /mH UG*.V +Q/2X a2H7@Q`i?Q;QMH M/ HBM2` +QKTH2K2Mi`v /mH +Q/2b 7Q`K BKTQ`iMi +Hbb2b Q7 HBM2` +Q/2b /m2 iQ i?2B` H;2#`B+ bi`m+im`2b- T`+iB+H TTHB@

+iBQMb BM +QKKmMB+iBQMb bvbi2Kb- M/ HBMF2/ rBi? Qi?2` Ki?2KiB+H Q#D2+ib b b?QrM BM (k- j- 8- e- 3- Ry- RR- Rk- R8- ky- kR- k9) M/ `272`2M+2b i?2`2BMX a2H7@Q`i?Q;QMH +Q/2b ?p2 #22M TTHB2/ BM +QMbi`m+iBQMb Q7 [mMimK +Q/2b BM (RR) M/ (ky)X AM (d)- G*. +Q/2b ?p2 #22M b?QrM iQ #2 bvKTiQiB+HHv ;QQ/ M/ K22i i?2 bvKTiQiB+

:BH#2`i@o`b?KQp #QmM/X

h?2 +QM+2Tib Q7 b2H7@Q`i?Q;QMH +Q/2b M/ HBM2` +QKTH2K2Mi`v /mH +Q/2b ?p2

#22M ;2M2`HBx2/ BM i?2 MQiBQM Q7 ?mHHbX h?Bb +QM+2Ti ?b #22M }`bi BMi`Q/m+2/ M/

TTHB2/ BM i?2 +?`+i2`BxiBQM Q7 }MBi2 T`QD2+iBp2 THM2b BM (R)X S`2+Bb2Hv- i?2 ?mHH Q7 HBM2` +Q/2 Bb i?2 BMi2`b2+iBQM Q7 i?2 +Q/2 M/ Bib /mHX Ai Bb 2bBHv b22M i?i b2H7@Q`i?Q;QMH +Q/2b `2 HBM2` +Q/2b rBi? KtBKH ?mHH M/ G*. +Q/2b `2 HBM2`

+Q/2b rBi? KBMBKH Ui`BpBHV ?mHHX Gi2`- Bi ?b #22M b?QrM i?i i?2 +QKTH2tBiv Q7 R

bQK2 H;Q`Bi?Kb BM +Q/BM; i?2Q`v ?b #22M /2i2`KBM2/ #v i?2 ?mHH /BK2MbBQM Q7 +Q/2b (Rj- R9- k8- ke)X S`2+Bb2Hv- KQbi Q7 i?2 H;Q`Bi?Kb rQ`F B7 i?2 ?mHH Q7 +Q/2b Bb bKHHX aBM+2 i?2M- ?mHHb Q7 +Q/2b ?p2 #2+QK2 Q7 BMi2`2bi M/ 2ti2MbBp2Hv bim/B2/X AM (kj)- i?2 MmK#2` Q7 HBM2` +Q/2b rBi? +QKKQM ?mHH /BK2MbBQM M/ i?2 p2`;2 ?mHH /BK2MbBQM Q7 BM2` +Q/2b ?p2 #22M bim/B2/X _2+2MiHv- `B;Q`Qmb i`2iK2Mi Q7 ?mHHb Q7 HBM2` +Q/2b

?p2 #22M ;Bp2M M/ TTHB2/ BM +QMbi`m+iBQMb Q7 ;QQ/ [mMimK 2``Q` +Q``2+iBM; +Q/2b BM (3- Rd- k3)X PTiBKH M/ ;QQ/ G*. +Q/2b ?p2 #22M bim/B2/ BM (j- d- Ry- RN)X "QmM/b M/ i?2 QTiBKHBiv Q7 b2H7@Q`i?Q;QMH +Q/2b ?p2 #22M 2bi#HBb?2/ BM (Rk- ky- kk- kd)X h?2`27Q`2- Bi Bb Q7 BMi2`2bi iQ bim/v +QMbi`m+iBQMb M/ QTiBKHBiv Q7 HBM2` +Q/2b rBi?

T`2b+`B#2/ ?mHH /BK2MbBQM M/ i?2B` TTHB+iBQMbX

AM i?Bb i?2bBb- r2 7Q+mb QM +QMbi`m+iBQMb M/ QTiBKHBiv Q7 HBM2` +Q/2b Qp2` bKHH }MBi2 }2H/b rBi? ?mHH /BK2MbBQM QM2 rBi? `2bT2+i iQ i?2 1m+HB/2M M/ >2`KBiBM BMM2`

T`Q/m+ibX aQK2 #bB+ T`QT2`iB2b Q7 HBM2` +Q/2b- ?mHH Q7 HBM2` +Q/2b- M/ #QmM/b QM i?2 T`K2i2`b Q7 +Q/2b `2 /Bb+mbb2/ BM *?Ti2` kX *QMbi`m+iBQMb Q7 QTiBKH #BM`v [n,2]2 HBM2` +Q/2b rBi? 1m+HB/2M ?mHH /BK2MbBQM QM2 `2 ;Bp2M 7Q` HH H2M;i?b n≥3 BM *?Ti2` j iQ;2i?2` rBi? i?2 2MmK2`iBQM Q7 bm+? +Q/2b mT iQ 2[mBpH2M+2X AM

*?Ti2` 9- +QMbi`m+iBQMb Q7 QTiBKH i2`M`v [n,2]3 HBM2` +Q/2b rBi? 1m+HB/2M ?mHH /BK2MbBQM QM2 `2 T`2b2Mi2/ 7Q` HH H2M;i?b n ≥ 3X PTiBKH [mi2`M`v [n,2]4 +Q/2b rBi? >2`KBiBM ?mHH /BK2MbBQM QM2 `2 2bi#HBb?2/ BM *?Ti2` 8 7Q` HH H2M;i?b n≥3 bm+? i?i n ≡ 1,2,4 (mod 5)X am#b2[m2MiHv- ;QQ/ HQr2` M/ mTT2` #QmM/b QM i?2 KBMBKmK r2B;?i Q7 [mi2`M`v[n,2]4 HBM2` +Q/2b rBi? >2`KBiBM ?mHH /BK2MbBQM QM2

`2 ;Bp2M 7Q` TQbBiBp2 BMi2;2`b n≡0,3 (mod 5)X

k

*?Ti2` k

S`2HBKBM`B2b

AM i?Bb b2+iBQM- bQK2 /2}MBiBQMb M/ T`QT2`iB2b Q7 HBM2` +Q/2b M/ ?mHHb Q7 HBM2`

+Q/2b `2 `2+HH2/ b r2HH b i?2 T`QQ7b Q7 T`2HBKBM`v `2bmHib `2[mB`2/ BM i?Bb bim/vX

kXR GBM2` *Q/2b Qp2` 6BMBi2 6B2H/b

G2i Fq /2MQi2 i?2 }MBi2 }2H/ Q7 Q`/2` qX 6Q` TQbBiBp2 BMi2;2` n- HBM2` +Q/2 Q7 H2M;i? n Qp2` F^q Bb /2}M2/ iQ #2 bm#bT+2 Q7 i?2 F^q@p2+iQ` bT+2 F<sup>n</sup>qX 6Q` M 2H2K2Mi w BM Fⁿq- H2i wt(w) /2MQi2 i?2 >KKBM; r2B;?i Q7 wX S`2+Bb2Hv- wt(w) =

|{i ∈ {1,2, . . . , n} | wi ̸= 0}| 7Q` HH w = (w1, w2, . . . , wn) ∈ F<sup>n</sup>qX HBM2` +Q/2 C Q7 H2M;i?n Qp2`F^q Bb +HH2/ M[n, k, d]q +Q/2 B7 i?2 F^q@/BK2MbBQM Bbk M/ i?2 KBMBKmK

>KKBM; r2B;?i Q7 HBM2` +Q/2 C Bb

d= wt(C) = min{wt(c)|wt(c)∈C\ {0}},

Q` M[n, k]q+Q/2 B7 i?2 KBMBKmK >KKBM; r2B;?i Q7CBb MQi bT2+B}2/X h?2 KBMBKmK

>KKBM; r2B;?i Bb F2v iQ /2i2`KBM2 i?2 2``Q`@+Q``2+iBM; +T#BHBiv Q7 +Q/2X G2i t #2 TQbBiBp2 BMi2;2`X +Q/2 C Bb bB/ iQ #2 t@2``Q`@+Q``2+iBM; B7 i?2 KBMBKmK /BbiM+2 /2+Q/BM; Ub22 (Re- .2}MBiBQM kX8XR)V Bb #H2 iQ +Q``2+i t Q` 72r2` 2``Q`bX AM (Re- h?2Q`2K kX8XRy)- Bi Bb T`Qp2/ i?i +Q/2 C Bb t@2``Q`@+Q``2+iBM; B7 M/ QMHv B7 wt(C)≥2t+1X >2M+2-wt(C)/2i2`KBM2b i?2 2{+B2M+v Q7 i?2 +Q/2CX k×nKi`Bt

j

GQp2` F<sup>q</sup> Bb +HH2/ ;2M2`iQ` Ki`Bt7Q` M [n, k, d]q +Q/2C B7 i?2 `Qrb Q7 G7Q`K

#bBb Q7 CX ;2M2`iQ` Ki`Bt Q7 i?2 7Q`K ! Ik|X

"

Bb bB/ iQ #2 BMbiM/`/ 7Q`KX 1tKTH2 kXRXRX G2iC #2 HBM2` +Q/2 Q7 H2M;i? 5Qp2` F² rBi? ;2M2`iQ` Ki`Bt

G=

⎡

⎣1 0 1 0 1 0 1 1 1 1

⎤

⎦.

h?2M C = {0000,10101,01111,11010}X aBM+2 ri(10101) = 3, ri(01111) = 4, M/

ri(11010) = 3- r2 ?p2 wt(C) = 3X h?Bb BKTHB2b i?iC Bb [5,2,3]2 HBM2` +Q/2X GBM2` [n, k]q +Q/2b C1 M/ C2 `2 bB/ iQ #2 UT2`KmiiBQMV 2[mBpH2Mi B7 i?2`2 2tBbib T2`KmiiBQM δ QM {1,2, . . . , n} bm+? i?i C2 = {(cδ(1), cδ(2), , . . . , cδ(n)) | (c₁, c₂, . . . , c_n) ∈ C₁}. 6Q` (Re- h?2Q`2K 9XeXj) 2p2`v HBM2` +Q/2 Bb 2[mBpH2Mi iQ HBM2` +Q/2 i?i ?b ;2M2`iQ` Ki`Bt BM biM/`/ 7Q`KX

1tKTH2 kXRXkX G2i q = 2 M/ n = 4X G2i ρ = (1 4)(3 4) #2 T2`KmiiBQM QM {1,2,3,4}X h?2M

C1 ={0000,0101,0010,0111} Bb 2[mBpH2Mi iQ i?2 +Q/2

C₂ ={0000,1100,0001,1110} T2`Kmi2/ #vρX

6Q` TQbBiBp2 BMi2;2`b m, n, /2MQi2 #v Mm,n(F^q) i?2 b2i Q7 m×n Ki`B+2b r?Qb2 2Mi`B2b `2 BMF<sup>q</sup>.6Q`A = [aij]∈Mm,n(F^q),H2iA^T /2MQi2 i?2 i`MbTQb2 Ki`Bt Q7 X AM //BiBQM- B7 q =r² Bb b[m`2- H2i A^†= [a^r_ji].

kXk >mHHb Q7 GBM2` *Q/2b

6Q`u= (u1, u2, . . . , un)M/v = (v1, v2, . . . , vn)BMF<sup>n</sup>q- irQ BMM2` T`Q/m+ib #2ir22M u M/ v `2 /2}M2/ b 7QHHQrb ,

RX ⟨u,v⟩E:='n

i=1uivi Bb +HH2/ i?2 1m+HB/2M BMM2` T`Q/m+i #2ir22Mu M/ vX 9

kX 6Q`q=r²-⟨u,v⟩^H:='n

i=1uivi =⟨u,v⟩^EBb +HH2/ i?2>2`KBiBM BMM2` T`Q/m+i

#2ir22M u M/ v- r?2`2a=a<sup>r</sup> 7Q` HH a∈F^q

6Q` HBM2` +Q/2 C Q7 H2M;i? n Qp2` Fq- i?2 1m+HB/2M /mH M/ U`2bTX- >2`KBiBM /mHV Q7 HBM2` +Q/2C Bb /2}M2/ iQ #2 i?2 b2i

C^⊥E :={u∈Fⁿq |⟨u,c⟩E= 0 7Q` HH c∈C} U`2bTX-C^⊥H :={u∈Fⁿq |⟨u,c⟩H= 0 7Q` HH c∈C})

M/ i?2 1m+HB/2M U`2bTX- >2`KBiBMV ?mHH Q7 C Bb /2}M2/ iQ #2 Hull(C) = C∩C^⊥E U`2bTX- HullH(C) = C∩C^⊥HVX

M (n−k)×n Ki`Bt H Qp2` F^q Bb +HH2/ T`Biv@+?2+F Ki`Bt Q7 M [n, k, d]q

+Q/2C B7H Bb ;2M2`iQ` Ki`Bt Q7C<sup>⊥E</sup>. T`Biv@+?2+F Ki`Bt Q7 i?2 7Q`K!

Y|In−k

"

Bb bB/ iQ #2 BM biM/`/ 7Q`KUb22 (Re- .2}MBiBQM 9X8Xj)VX

h?2Q`2K kXkXR U(Re- h?2Q`2K 9X8XN)VX A7 G= [Ik|A] Bb i?2 biM/`/ 7Q`K ;2M2`iQ`

Ki`Bt Q7 M [n, k]<sub>q</sub>@+Q/2 C- i?2M i?2 biM/`/ 7Q`K ;2M2`iQ` Ki`Bt Q7 C^⊥E Bb H = [−A^T|In−k]X

1tKTH2 kXkXkX G2iC #2 HBM2` +Q/2 Q7 H2M;i? 5Qp2` F² rBi? ;2M2`iQ` Ki`Bt

G=

⎡

⎣1 0 1 0 1 0 1 1 1 1

⎤

⎦,

h?2M C ={00000,10101,01111,11010} ?b T`K2i2`b [5,2,3]2X "v h?2Q`2K kXkXR- r2 ?p2 i?i

H =

⎡

⎢⎢

⎢⎣

1 1 1 0 0 0 1 0 1 0 1 1 0 0 1

⎤

⎥⎥

⎥⎦

Bb T`Biv@+?2+F Ki`Bt 7Q` C M/ ?2M+2-

C^⊥E ={00000,10011,11001,01010,01111,11100,10110,00101}.

"v i?2 /2}MBiBQM Q7 i?2 ?mHH- r2 ?p2 >mHH(C) = {00000,01111}X 8

1tKTH2 kXkXjX G2i F⁴ ={0,1,ω,ω² =ω+ 1} M/ H2iC #2 HBM2` +Q/2 Q7 H2M;i?

5 Qp2`F<sup>4</sup> rBi? ;2M2`iQ` Ki`Bt G=

⎡

⎣1 0 1 ω ω² 0 1 0 1 ω

⎤

⎦.

h?2M C ?b T`K2i2`b [5,2,4]₄."v h?2Q`2K kXkXR- C<sup>⊥H</sup> ?b ;2M2`iQ` Ki`Bt

H =

⎡

⎢⎢

⎢⎣

1 0 1 0 0 ω 1 0 1 0 ω² ω 0 0 1

⎤

⎥⎥

⎥⎦.

lbBM; /B`2+i +H+mHiBQM- r2 ?p2 HullH(C) = {00000,101ωω<sup>2</sup>,ω0ωω<sup>2</sup>1,ω<sup>2</sup>0ω<sup>2</sup>1ω}X 6Q`n∈{1,2}- Bi Bb 2bv iQ b22 i?i i?2`2 `2 MQ[n,2]2 +Q/2b rBi? ?mHH /BK2MbBQM QM2X h?`Qm;?Qmi- r2 bbmK2 i?i i?2 H2M;i? n Q7 i?2 +Q/2b Bb ;`2i2` i?M 2X

h?2 ?mHH /BK2MbBQM Q7 HBM2` +Q/2 +M #2 /2i2`KBM2/ mbBM; Bib ;2M2`iQ` Ki`Bt BM (3) b 7QHHQrbX

S`QTQbBiBQM kXkX9 U(3- S`QTQbBiBQMb jXR)VX G2i C #2 HBM2` [n, k, d]q +Q/2 rBi?

;2M2`iQ` Ki`Bt GX h?2M rank(GG^T) Bb BM/2T2M/2Mi Q7 G M/

rank(GG^T) =k−dim(Hull(C)) =k−dim(Hull(C^⊥E)).

AM //BiBQM- B7 q Bb b[m`2- i?2M

rank(GG^†) =k−dim(HullH(C)) = k−dim(HullH(C^⊥H)).

1tKTH2 kXkX8X 6`QK 1tKTH2 kXkXk- r2 ?p2 k = 2, G=

⎡

⎣1 0 1 0 1 0 1 1 1 1

⎤

⎦ M/

GG^T =

⎡

⎣1 0 0 0

⎤

⎦.

h?Bb BKTHB2b i?i rank(GG^T) = 1X "v S`QTQbBiBQM kXkX9- Bi 7QHHQrb i?i dim(Hull(C)) = k−rank(GG^T)

= 2−1 = 1.

>2M+2- dim(Hull(C)) = 1X

e

1tKTH2 kXkXeX 6`QK 1tKTH2 kXkXj- r2 ?p2 k = 2- G=

⎡

⎣1 0 1 ω ω² 0 1 0 1 ω

⎤

⎦ M/

GG^†=

⎡

⎣0 0 0 1

⎤

⎦.

h?Bb BKTHB2b i?i rank(GG^†) = 1X "v S`QTQbBiBQM kXkX9- Bi +M #2 /2/m+2/ i?i

dim(HullH(C)) = k−rank(GG^†)

= 2−1 = 1.

h?2`27Q`2-dim(HullH(C)) = 1X

kXj *QMbi`m+iBQMb M/ "QmM/b

lbBM; i?2 MHvbBb QM ;2M2`iQ` Ki`Bt Q7 HBM2` +Q/2b BM (RN)- r2 ?p2 i?2 7QH@

HQrBM; `2bmHibX 6Q` M [n,2]q +Q/2 rBi? ;2M2`iQ` Ki`Bt

G=

⎡

⎣a11 a12 · · · a1n

a₂₁ a₂₂ · · · a_2n

⎤

⎦,

r2 r`Bi2

G=

⎡

⎣α₁ α2

⎤

⎦, UkXRV

r?2`2 α1 =!

a11 a12 · · · a1n

"

M/ α2 =!

a21 a22 · · · a2n

"

X Hi2`MiBp2Hv- #v b2iiBM; βl=

⎡

⎣a1l

a_2l

⎤

⎦ 7Q` 1≤l ≤n-G +M #2 pB2r2/ b

G=!

β1 β2 · · · βn

"

. UkXkV

6Q`i, j ∈F^q- H2i

Sij :=|{l∈{1,2, . . . , n} |βl =

⎡

⎣i j

⎤

⎦}|. UkXjV

d

1tKTH2 kXjXRX G2iC M/ C^′ #2 i2`M`v HBM2` +Q/2b Q7 H2M;i? 7;2M2`i2/ #v G=

⎡

⎣1 1 0 1 1 2 2 0 0 1 1 1 1 1

⎤

⎦

M/

G^′ =

⎡

⎣1 1 0 1 1 2 2 1 1 1 0 0 1 1

⎤

⎦.

q2 b22 i?i G M/ G^′ `2 /2i2`KBM2/ #v S10 = 2, S01 = 1, S11 = 2, S21 = 2, M/

S_ij = 0Qi?2`rBb2X Ai Bb 2bBHv b22M i?i C M/ C<sup>′</sup> `2 2[mBpH2Mi #v i?2 T2`KmiiBQM (1 4)(2 5)X

_2K`F kXjXkX h?2 ;2M2`iQ` Ki`B2b /2i2`KBM2/ #v Sij :=|{l ∈{1,2, . . . , n} |βl=

⎡

⎣i j

⎤

⎦}|

;2M2`i2 2[mBpH2Mi HBM2` +Q/2bX h?2 2[mBpH2M+2b `2 BM/m+2/ #v T2`KmiiBQM QM {1,2, . . . , n}.

h?2 :`B2bK2` "QmM/ BM (N) Bb TTHB2/ BM i?Bb rQ`FX

h?2Q`2K kXjXj U:`B2bK2` "QmM/- (N- h?2Q`2K kXdX9)VX G2i q #2 T`BK2 TQr2` M/

H2i n- k M/ d #2 TQbBiBp2 BMi2;2`bX A7 i?2`2 2tBbib M [n, k, d]_q +Q/2- i?2M n ≥

*k−1 i=0

+d qⁱ

, .

h?2 KBMBKmK >KKBM; r2B;?i Q7 HBM2` +Q/2b Bb BKTQ`iMi iQ /2i2`KBM2 i?2 2``Q`@+Q``2+iBM; +T#BHBiv Q7 +Q/2bX AM i?Bb i?2bBb- r2 7Q+mb QM i?2 KtBKmK Q7 i?2 KBMBKmK >KKBM; r2B;?i Q7 HBM2` +Q/2bX 6Q` T`BK2 TQr2` q- TQbBiBp2 BMi2;2`b n, k M/ MQM@M2;iBp2 BMi2;2` ℓ- H2i

Dq(n, k,ℓ) := max{d|∃[n, k, d]q +Q/2 rBi? dim(Hull(C)) =ℓ} M/

D^H_q(n, k,ℓ) := max{d|∃[n, k, d]q +Q/2 rBi? dim(HullH(C)) = ℓ}.

"b2/ QM i?2 :`B2bK2` "QmM/- i?2 7QHHQrBM; mTT2` #QmM/b QMDq(n, k,ℓ)M/D<sup>H</sup><sub>q</sub>(n, k,ℓ) +M #2 /2`Bp2/ BM i?2 M2ti H2KKX

3

G2KK kXjX9X G2i q #2 T`BK2 TQr2` M/ H2i n, k M/ ℓ #2 BM2;2`b bm+? i?i 1≤k≤n M/ 0≤ℓX h?2M

D_q(n, k,ℓ)≤

-(q−1)q^k−1n (q^k−1)

.

M/

D_q^H(n, k,ℓ)≤

-(q−1)q^k−1n (q^k−1)

. B7 q Bb b[m`2.

S`QQ7X 6`QK i?2 :`B2bK2` "QmM/ BM h?2Q`2K kXjXj- 7Q` Mv HBM2` [n, k, d]_q +Q/2- r2

?p2

n ≥

*k−1 i=0

+d qⁱ

,

≥d

*k−1 i=0

1

qⁱ = d(q^k−1) (q−1)q^k−1. Ai 7QHHQrb i?i

d≤

-(q−1)q^k−1n (q^k−1)

. .

>2M+2- Dq(n, k,ℓ)≤/

(q−1)q^k−1n (q^k−1)

0 b /2bB`2/X

aBM+2 i?2 #Qp2 T`QQ7 Bb BM/2T2M/2Mi Q7 i?2 BMM2` T`Q/m+i-D_q^H(n, k,ℓ)≤/

(q−1)q^k−1n (q^k−1)

0

7QHHQrb bBKBH`HvX

AM i?Bb i?2bBb- r2 7Q+mb QM Dq(n, k,1) M/ D_q^H(n, k,1)X h?2 /2iBHb `2 /Bb+mbb2/

BM i?2 7QHHQrBM; +?Ti2`bX

N

*?Ti2` j

PTiBKH "BM`v GBM2` *Q/2b rBi?

1m+HB/2M >mHH .BK2MbBQM PM2

AM i?Bb +?Ti2`- r2 7Q+mb QM +QMbi`m+iBQM Q7 #BM`v HBM2` +Q/2b rBi? ?mHH /BK2MbBQM QM2X 6Q` 2+? BMi2;2` n ≥ 3- M QTiBKH [n,2]2 +Q/2 rBi? ?mHH /BK2MbBQM QM2 Bb +QM@

bi`m+i2/X JQ`2Qp2`- i?2 2MmK2`iBQM Q7 bm+? QTiBKH +Q/2b Bb ;Bp2M mT iQ 2[mBpH2M+2X

jXR "bB+ *QM+2Tib

6`QK i?2 b2imT BM am#b2+iBQM kXj U+X7X (RN)V- HBM2` [n,2]₂ +Q/2 C rBi? ;2M2`iQ`

Ki`Bt

G=

⎡

⎣α1

α2

⎤

⎦ UjXRV

rBi? α1 = %

a11 a12 · · · a1n

&

M/ α2 = %

a21 a22 · · · a2n

&

+M #2 pB2r2/ Q7 i?2 7Q`K C = {0,α1,α2,α1 +α2}. 6`QK UkXjV- r2 ?p2 α1 = %

a₁₁ a₁₂ · · · a_1n

&

M/

α2 =%

a21 a22 · · · a2n

&

M/ Bi Bb MQi /B{+mHi iQ b22 i?i

ri(α1) =S10+S11,ri(α2) =S01+S11,ri(α1+α2) =S10+S01.

Ry

>2M+2-

wt(C) = min{ri(α1),ri(α2),ri(α1+α2)}

= min{S10+S11, S01+S11, S10+S01}. UjXkV 6`QK UjXRV- r2 ?p2

GG^T =

⎡

⎣S10+S11 S11

S11 S01+S11

⎤

⎦(mod 2). UjXjV

1tKTH2 jXRXRX G2i C #2 i?2 #BM`v HBM2` +Q/2 Q7 H2M;i? 5 rBi? ;2M2`iQ` Ki`Bt

G=

⎡

⎣1 0 0 1 1 0 1 1 0 1

⎤

⎦

h?2M S00 = 0, S10 = 2, S01 = 2 M/ S11 = 1 r?B+? BKTHB2b i?i S10 +S11 = 3 M/

S01+S11= 3 "v UjXjV- Bi 7QHHQrb i?i

GG^T =

⎡

⎣1 1 1 1

⎤

⎦

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

_2K`F jXRXkX h?2 #BM`v +Q/2b /2i2`KBM2/ #v

(S01, S10, S11)∈{(a, b, c),(b, a, c),(c, b, a),(a, c, b),(b, c, a),(c, a, b)}

`2 2[mBpH2MiX h?Bb 7QHHQrb 7`QK i?2 7+i i?i

⎡

⎣α₁ α2

⎤

⎦-

⎡

⎣α₂ α1

⎤

⎦-

⎡

⎣α₁+α₂ α2

⎤

⎦-

⎡

⎣ α₁ α1+α2

⎤

⎦-

⎡

⎣α₁+α₂ α1

⎤

⎦- M/

⎡

⎣ α₂ α1+α2

⎤

⎦

;2M2`i2 i?2 bK2 +Q/2X

1tKTH2 jXRXjX "v _2K`F jXRXk- H2i a= 2, b = 0 M/ c= 3X h?2M

(S01, S10, S11)∈{(2,0,3),(0,2,3),(3,0,2),(2,3,0),(0,3,2),(3,2,0)}.

RR

6Q` (S01, S10, S11) = (2,0,3)- r2 ?p2 i?i α1 = 00111 M/ α2 = 11111X G2i C #2 HBM2` +Q/2 rBi? ;2M2`iQ` Ki`Bt

G=

⎡

⎣α1

α2

⎤

⎦=

⎡

⎣0 0 1 1 1 1 1 1 1 1

⎤

⎦.

h?2M C ={00000,00111,11111,11000}X "v _2K`F jXRXk- r2 ?p2 i?i

⎡

⎣α2

α1

⎤

⎦=

⎡

⎣0 0 1 1 1 1 1 0 0 0

⎤

⎦/2i2`KBM2/ #v (S₀₁, S₁₀, S₁₁) = (0,2,3)

⎡

⎣α1+α2

α2

⎤

⎦=

⎡

⎣1 1 0 0 0 1 1 1 1 1

⎤

⎦/2i2`KBM2/ #v (S01, S10, S11) = (3,0,2)

⎡

⎣ α1

α1+α2

⎤

⎦=

⎡

⎣0 0 1 1 1 1 1 0 0 0

⎤

⎦/2i2`KBM2/ #v (S01, S10, S11) = (2,3,0)

⎡

⎣α1+α2

α1

⎤

⎦=

⎡

⎣1 1 0 0 0 0 0 1 1 1

⎤

⎦/2i2`KBM2/ #v (S₀₁, S₁₀, S₁₁) = (0,3,2),M/

⎡

⎣ α2

α1+α2

⎤

⎦=

⎡

⎣1 1 1 1 1 1 1 0 0 0

⎤

⎦/2i2`KBM2/ #v (S01, S10, S11) = (3,2,0)

;2M2`i2 HBM2` +Q/2 *X

6Q` #BM`v HBM2` +Q/2b rBi? /BK2MbBQM irQ M/ ?mHH /BK2MbBQM QM2- i?2 mTT2`

#QmM/ BM G2KK kXjX9 +M #2 bBKTHB}2/ b D2(n,2,1)≤

-2n 3

.

. UjX9V

lbBM; i?2 MQiiBQMb #Qp2- i?2 2t+i pHm2 Q7D₂(n,2,1)+M #2 /2`Bp2/ iQ;2i?2` rBi?

i?2 2MmK2`iBQM Q7 QTiBKH [n,2]2 +Q/2b b 7QHHQrbX

jXk n ≡ 1 (mod 6)

AM i?2 7QHHQrBM; i?2Q`2K- r2 ;Bp2 +QMbi`m+iBQMb Q7 QTiBKH [n,2,1_2n

3

2]2 +Q/2b Q7 bm+? H2M;i?b rBi? ?mHH /BK2MbBQM QM2 7Q` HH n ≡1 (mod 6)X

Rk

h?2Q`2K jXkXRX G2i n ≥3 #2 M BMi2;2`X A7 n ≡1 (mod 6)- i?2M D2(n,2,1) =

-2n 3

.

M/ i?2`2 2tBbib mMB[m2 UmT iQ 2[mBpH2M+2V QTiBKH 3

n,2,1_2n

3

24

2 +Q/2 rBi? ?mHH /BK2MbBQM QM2X

S`QQ7X bbmK2 i?i n ≡1 (mod 6)X h?2M n = 6t+ 1 7Q` bQK2 TQbBiBp2 BMi2;2` t M/

1_2n

3

2 =/

2(6t+1) 3

0= 4tX

6Q` i?2 2tBbi2M+2- #v UjX9V- Bi bm{+2b iQ b?Qr i?i i?2`2 2tBbib M [n,2,4t]₂ +Q/2 rBi? ?mHH /BK2MbBQM QM2X G2i C #2 HBM2` +Q/2 rBi? ;2M2`iQ` Ki`Bt G /2}M2/ BM UjXRV bm+? i?i (S01, S10, S11) = (2t,2t,2t+ 1)X h?2MC Bb M [n,2, d]2 +Q/2- r?2`2

d= min{S01+S11, S10+S11, S10+S01}

= min{4t+ 1,4t+ 1,4t}

= 4t #v UjXkV.

6`QK UjXjV- r2 ?p2

GG^T =

⎡

⎣1 1 1 1

⎤

⎦

r?B+? BKTHB2b i?i rank(GG^T) = 1X q2 i?2`27Q`2 ?p2 dim(Hull(C)) = 1 #v S`QTQ@

bBiBQM kXkX9X

6Q` i?2 mMB[m2M2bb- r2 }`bi b?Qr i?i S01+S10+S11 = nX amTTQb2 i?i i?2`2 2tBbib M [n,2,4t]2 +Q/2 rBi? ?mHH /BK2MbBQM QM2 M/S00>0X h?2M

S₀₁+S₁₀+S₁₁ ≤n−1 = 6t.

aBM+2min{S10+S11, S01+S11, S10+S01}= 4t Bb `2[mB`2/- i?2 QMHv TQbbB#H2 +?QB+2 Q7 (S01, S10, S11) Bb (2t,2t,2t)X AM i?Bb +b2- r2 ?p2

GG^T =

⎡

⎣0 0 0 0

⎤

⎦

Rj

#v UjXjVX "v S`QTQbBiBQM kXkX9- r2 i?2`27Q`2 ?p2dim(Hull(C)) = 2̸= 1- +QMi`/B+@

iBQMX >2M+2- n=S01+S10+S11X

Ai `2KBMb iQ 7Q+mb QM (S01, S10, S11) biBb7vBM; S01+S10+S11 =n = 6t+ 1 M/

min{S10+S11, S01+S11, S10+S01}= 4tX Ai 7QHHQrb i?i

(S01, S10, S11)∈{(2t,2t,2t+ 1),(2t,2t+ 1,2t),(2t+ 1,2t,2t), (2t−1,2t+ 1,2t+ 1),(2t+ 1,2t−1,2t+ 1), (2t+ 1,2t+ 1,2t−1)}.

q2 +QMbB/2` i?2 7QHHQrBM; 2 +b2bX

*b2 RX (S₀₁, S₁₀, S₁₁)∈{(2t,2t,2t+ 1),(2t,2t+ 1,2t),(2t+ 1,2t,2t)}."v UjXjV- i?2 +Q``2bTQM/BM; Ki`B+2b GG<sup>T</sup> `2 Q7 i?2 7Q`Kb

⎡

⎣1 1 1 1

⎤

⎦-

⎡

⎣1 0 0 0

⎤

⎦- M/

⎡

⎣0 0 0 1

⎤

⎦-

`2bT2+iBp2HvX AM i?2b2 +b2b- r2 ?p2 rank(GG<sup>T</sup>) = 1 r?B+? BKTHB2b i?i i?2`2 ?mHH /BK2MbBQMb `2 QM2 #v S`QTQbBiBQM kXkX9X 6`QK _2K`F jXRXk- i?2b2 +Q/2b `2 2[mBp@

H2MiX

*b2 kX (S01, S10, S11)∈{(2t−1,2t+ 1,2t+ 1),(2t+ 1,2t−1,2t+ 1),(2t+ 1,2t+ 1,2t−1)}X 6`QK UjXjV- Bi Bb 2bBHv b22M i?i

GG^T =

⎡

⎣0 1 1 0

⎤

⎦.

>2M+2- rank(GG^T) = 2 r?B+? BKTHB2b i?i i?2 +Q/2b `2 G*. #v S`QTQbBiBQM kXkX9X

1tKTH2 jXkXkX G2i t = 2X "v h?2Q`2K 3.2.1- r2 ?p2 i?i (S01, S10, S11) = (2t,2t,2t+ 1) = (4,4,5)X G2i C #2 i?2 #BM`v HBM2` +Q/2 Q7 H2M;i?13/2i2`KBM2/ #v i?2 i`BTH2 (4,4,5) rBi? ;2M2`iQ` Ki`Bt

G=

⎡

⎣0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1

⎤

⎦.

R9

"v UjXjV- Bi 7QHHQrb i?i

GG^T =

⎡

⎣1 1 1 1

⎤

⎦

r?B+? BKTHB2b i?i rank(GG^T) = 1 M/ dim(Hull(C)) = 2− rank(GG^T) = 1 #v S`QTQbBiBQM kXkX9X h?2M C Bb M QTiBKH[13,2,8]2 +Q/2 rBi? ?mHH /BK2MbBQM QM2X

jXj n ≡ 3 (mod 6)

L2ti- r2 b?Qr i?i i?2`2 /Q2b MQi 2tBbi [n,2,1_2n

3

2]2 +Q/2b rBi? ?mHH /BK2MbBQM QM2 7Q` HH n≡3 (mod 6)X Gi2`- r2 ;Bp2 +QMbi`m+iBQMb Q7 QTiBKH [n,2,1_2n

3

2−1]2 +Q/2b Q7 bm+? H2M;i?b rBi? ?mHH /BK2MbBQM QM2X

h?2Q`2K jXjXRX G2i n ≥3 #2 M BMi2;2`X A7 n ≡3(mod 6)- i?2M D2(n,2,1) =

-2n 3

.

−1.

AM i?Bb +b2- i?2`2 2tBbib mMB[m2 UmT iQ 2[mBpH2M+2V QTiBKH [3,2,1]<sub>2</sub> +Q/2 rBi? ?mHH /BK2MbBQM QM2 M/ i?2`2 `2 irQ UmT iQ 2[mBpH2M+2V QTiBKH 3

n,2,1_2n

3

2−14

2 +Q/2b rBi? ?mHH /BK2MbBQM QM2 7Q` HH n >3X

S`QQ7X bbmK2 i?in ≡3(mod 6)X h?2Mn = 6t+ 3 7Q` bQK2 BMi2;2`t≥0X "v UjX9V- r2 ?p2

D2(n,2,1)≤ -2n

3 .

=

-2(6t+ 3) 3

.

= 4t+ 2.

amTTQb2 i?2`2 2tBbib M3

n,2,1_2n

3

2= 4t+ 24

2 +Q/2 rBi? ?mHH /BK2MbBQM QM2X h?2M-

#v UjXkV- r2 ?p2 wt(C) = min{S01+S11, S10+S11, S10+S01} = 4t+ 2 M/ S01+ S10+S11≤n = 6t+ 3X >2M+2- i?2 QMHv TQbbB#H2 +?QB+2 Q7(S01, S10, S11)Bb(2t+ 1,2t+ 1,2t+ 1)X "v UjXjV- r2 ?p2

GG^T =

⎡

⎣0 1 1 0

⎤

⎦.

"v S`QTQbBiBQM kXkX9- Bi 7QHHQrb i?i dim(Hull(C)) = 0 ̸= 1- +QMi`/B+iBQMX >2M+2- i?2`2 /Q2b MQi 2tBbi M 3

n,2,1_2n

3

24

2 +Q/2X R8

6Q` i?2 2tBbi2M+2 Q7 M[n,2,4t+ 1]<sub>2</sub>+Q/2 rBi? ?mHH /BK2MbBQM QM2- H2iC#2 HBM2`

+Q/2 rBi? ;2M2`iQ` Ki`Bt G pB2r2/ BM i?2 7Q`K Q7 UjXRV bm+? i?i (S01, S10, S11) = (2t,2t+ 2,2t+ 1)X h?2MC Bb M [n,2, d]2 +Q/2- r?2`2

d= min{S₀₁+S₁₁, S₁₀+S₁₁, S₁₀+S₀₁}

= min{4t+ 1,4t+ 3,4t+ 2}

= 4t+ 1 #v UjXkV.

6`QK UjXjV- r2 ?p2

GG^T =

⎡

⎣1 1 1 1

⎤

⎦

r?B+? BKTHB2b i?i rank(GG^T) = 1X b /2bB`2/- r2 ?p2 dim(Hull(C)) = 1 #v S`QTQbBiBQM kXkX9X

hQ /2i2`KBM2 i?2 MmK#2` Q7[n,2,4t+ 1]₂ +Q/2b rBi? ?mHH /BK2MbBQM QM2- r2 +HBK i?i S₀₁+S₁₀+S₁₁=nX amTTQb2 i?i S₀₀>0X h?2M

S01+S10+S11≤n−1 = 6t+ 2.

aBM+2min{S10+S11, S01+S11, S10+S01}= 4t+1- i?2 TQbbB#H2 +?QB+2b Q7(S01, S10, S11)

`2 (2t,2t+ 1,2t+ 1)- (2t+ 1,2t+ 1,2t)- M/(2t+ 1,2t,2t+ 1)X "v UjXjV- i?2 Ki`Bt GG^T Bb Q7 i?2 7Q`K

⎡

⎣0 1 1 1

⎤

⎦-

⎡

⎣1 0 0 1

⎤

⎦- M/

⎡

⎣1 1 1 0

⎤

⎦-

`2bT2+iBp2HvX >2M+2- rank(GG<sup>T</sup>) = 2 r?B+? BKTHB2b i?i dim(Hull(C)) = 0 ̸= 1 #v S`QTQbBiBQM kXkX9X

LQr- Bi `2KBMb iQ 7Q+mb QM i?2 i`BTH2b (S01, S10, S11)biBb7vBM; S01+S10+S11= n= 6t+ 3 M/ min{S10+S11, S01+S11, S10+S01}= 4t+ 1X "v BMbT2+iBQM- r2 ?p2 (S₀₁, S₁₀, S₁₁)∈T₁∪T₂∪T₃- r?2`2

T1 ={(2t,2t+ 2,2t+ 1),(2t+ 2,2t,2t+ 1),(2t+ 2,2t+ 2,2t−1)}, T₂ ={(2t−1,2t+ 2,2t+ 2),(2t+ 1,2t,2t+ 2),(2t+ 1,2t+ 2,2t)}, T3 ={(2t,2t+ 1,2t+ 2),(2t+ 2,2t−1,2t+ 2),(2t+ 2,2t+ 1,2t)}.

Re

"v UjXjV- i?2 Ki`B+2b GG<sup>T</sup> `2 Q7 i?2 7Q`Kb

⎡

⎣1 1 1 1

⎤

⎦-

⎡

⎣0 0 0 1

⎤

⎦- M/

⎡

⎣1 0 0 0

⎤

⎦

7Q` (S01, S10, S11) BMT1- T2- M/ T3- `2bT2+iBp2HvX Ai 7QHHQrb i?i rank(GG^T) = 1 M/

i?2 +Q/2b ?p2 ?mHH /BK2MbBQM QM2 #v S`QTQbBiBQM kXkX9X 6`QK _2K`F jXRXk- i?2 +Q/2b /2i2`KBM2/ #v

(S01, S10, S11)∈{(2t,2t+ 2,2t+ 1),(2t+ 2,2t,2t+ 1),(2t+ 2,2t+ 1,2t), (2t,2t+ 1,2t+ 2),(2t+ 1,2t,2t+ 2),(2t+ 1,2t+ 2,2t)}

`2 2[mBpH2Mi- M/ i?2 +Q/2b /2i2`KBM2/ #v

(S01, S10, S11)∈{(2t+ 2,2t+ 2,2t−1),(2t−1,2t+ 2,2t+ 2), (2t+ 2,2t−1,2t+ 2)}

`2 2[mBpH2MiX aBM+2 +Q/2 BM i?2 }`bi 7KBHv +QMiBMb +Q/2rQ`/ Q7 r2B;?i 4t+ 3

#mi i?2 Hii2` /Q2b MQi- irQ +Q/2b 7`QK /Bz2`2Mi 7KBHB2b `2 MQi 2[mBpH2MiX >2M+2- i?2`2 `2 irQ UmT iQ 2[mBpH2M+2V QTiBKH3

n,2,1_2n

3

2−14

2 +Q/2b rBi? ?mHH /BK2MbBQM QM2 7Q` HH n > 3X 6Q` n = 3- r2 ?p2 t = 0 M/ ?2M+2- i?2 b2+QM/ 7KBHv /Q2b MQi 2tBbi r?B+? BKTHB2b i?i i?2`2 2tBbib mMB[m2 UmT iQ 2[mBpH2M+2V QTiBKH [3,2,1]₂ +Q/2 rBi? ?mHH /BK2MbBQM QM2X

1tKTH2 jXjXkX G2it= 2X "v h?2Q`2K3.3.1- r2 ?p2 i?i(S01, S10, S11) = (2t,2t+

2,2t+ 1) = (4,6,5) M/ (S01, S10, S11) = (2t+ 2,2t+ 2,2t−1) = (6,6,3) BM/22/ k BM2[mBpH2Mi +Q/2bX

RX G2i C1 #2 i?2 #BM`v HBM2` +Q/2 Q7 H2M;i? 15 /2i2`KBM2/ #v i?2 i`BTH2(4,6,5) rBi? ;2M2`iQ` Ki`Bt

G1 =

⎡

⎣0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1

⎤

⎦.

"v UjXjV- Bi 7QHHQrb i?i

G1G^T₁ =

⎡

⎣1 1 1 1

⎤

⎦ Rd

r?B+? BKTHB2b i?irank(G1G1T) = 1M/dim(Hull(C)) = 2−rank(G1G1T) = 1

#v S`QTQbBiBQM kXkX9X h?2MC1 Bb M QTiBKH[15,2,9]2 +Q/2 rBi? ?mHH /BK2MbBQM QM2X

kX G2i C2 #2 i?2 #BM`v HBM2` +Q/2 Q7 H2M;i? 15 /2i2`KBM2/ #v i?2 i`BTH2(6,6,3) rBi? ;2M2`iQ` Ki`Bt

G2 =

⎡

⎣0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1

⎤

⎦.

"v UjXjV- Bi 7QHHQrb i?i

G2G^T₂ =

⎡

⎣1 0 0 0

⎤

⎦

r?B+? BKTHB2b i?irank(G2G2T) = 1M/dim(Hull(C)) = 2−rank(G2G2T) = 1

#v S`QTQbBiBQM kXkX9X h?2MC2 Bb M QTiBKH[15,2,9]2 +Q/2 rBi? ?mHH /BK2MbBQM QM2X

"v h?2Q`2K jXkXR- r2 ?p2 i?i C1 M/ C2 `2 MQi 2[mBpH2MiX

jX9 n ≡ 5 (mod 6)

AM i?2 7QHHQrBM; i?2Q`2K- r2 ;Bp2 +QMbi`m+iBQMb Q7 M QTiBKH[n,2,1_2n

3

2]2 +Q/2 rBi?

/BK2bBQM 2 M/ 1m+HB/2M ?mHH /BK2MbBQM QM2 Bb ;Bp2M 7Q` HH H2M;i?bn ≡5 (mod 6)X h?2Q`2K jX9XRX G2i n ≥3 #2 M BMi2;2`X A7 n ≡5(mod 6)- i?2M

D2(n,2,1) = -2n

3 .

M/ i?2`2 2tBbib mMB[m2 UmT iQ 2[mBpH2M+2V QTiBKH 3

n,2,1_2n

3

24

2 +Q/2 rBi? ?mHH /BK2MbBQM QM2X

S`QQ7X bbmK2 i?i n ≡5(mod 6)X h?2M n = 6t+ 5 7Q` bQK2 TQbBiBp2 BMi2;2` tX "v UjX9V- r2 ?p2

D2(n,2,1)≤ -2n

3 .

=

-2(6t+ 5) 3

.

= 4t+ 3.

R3

6Q` i?2 2tBbi2M+2 Q7 M[n,2,4t+ 3]<sub>2</sub>+Q/2 rBi? ?mHH /BK2MbBQM QM2- H2iC#2 HBM2`

+Q/2 rBi? ;2M2`iQ` Ki`Bt G pB2r2/ BM i?2 7Q`K Q7 UjXRV bm+? i?i (S01, S10, S11) = (2t+ 2,2t+ 1,2t+ 2)X h?2M C Bb M [n,2, d]2 +Q/2- r?2`2

d= min{S01+S11, S10+S11, S10+S01}

= min{4t+ 3,4t+ 3,4t+ 3}

= 4t+ 3 #v UjXkV.

6`QK UjXjV- r2 ?p2

GG^T =

⎡

⎣1 0 0 0

⎤

⎦

r?B+? BKTHB2b i?i rank(GG^T) = 1X >2M+2- r2 ?p2 dim(Hull(C)) = 1 #v S`QTQbB@

iBQM kXkX9X

hQ /2i2`KBM2 i?2 MmK#2` Q7[n,2,4t+ 3]₂ +Q/2b rBi? ?mHH /BK2MbBQM QM2- r2 +HBK i?i S₀₁+S₁₀+S₁₁=nX amTTQb2 i?i S₀₀>0X h?2M

S01+S10+S11≤n−1 = 6t+ 4.

aBM+2 min{S10+S11, S01+S11, S10+S01} = 4t+ 3- i?2`2 `2 MQ TQbbB#H2 +?QB+2b Q7 (S₀₁, S₁₀, S₁₁)- +QMi`/B+iBQMX >2M+2- S₀₁+S₁₀+S₁₁=nX

Ai Bb MQi /B{+mHi iQ b22 i?i i?2 i`BTH2b(S01, S10, S11)biBb7vBM; S01+S10+S11= n= 6t+ 5 M/ min{S10+S11, S01+S11, S10+S01}= 4t+ 3 `2 BM i?2 b2i

{(2t+ 1,2t+ 2,2t+ 2),(2t+ 2,2t+ 1,2t+ 2),(2t+ 2,2t+ 2,2t+ 1)}.

"v UjXjV- i?2 +Q``2bTQM/BM; Ki`B+2b GG<sup>T</sup> `2 Q7 i?2 7Q`Kb

⎡

⎣0 0 0 1

⎤

⎦-

⎡

⎣1 0 0 0

⎤

⎦- M/

⎡

⎣1 1 1 1

⎤

⎦-

`2bT2+iBp2HvX q2 ?p2 rank(GG<sup>T</sup>) = 1 M/ i?2B` ?mHH /BK2MbBQMb `2 QM2 #v S`QTQbB@

iBQM kXkX9X h?2b2 +Q/2b `2 2[mBpH2Mi #v _2K`F jXRXkX

RN

1tKTH2 jX9XkX G2i t = 2X "v h?2Q`2K 3.4.1- r2 ?p2 i?i (S01, S10, S11) = (2t+ 2,2t+ 2,2t+ 1) = (6,6,5)X G2i C #2 i?2 #BM`v HBM2` +Q/2 Q7 H2M;i? 17/2i2`KBM2/

#v i?2 i`BTH2(6,6,5)rBi? ;2M2`iQ` Ki`Bt

G=

⎡

⎣0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1

⎤

⎦.

"v UjXjV- Bi 7QHHQrb i?i

GG^T =

⎡

⎣1 1 1 1

⎤

⎦

r?B+? BKTHB2b i?i rank(GG^T) = 1 M/ dim(Hull(C)) = 2− rank(GG^T) = 1 #v S`QTQbBiBQM kXkX9X h?2M C Bb M QTiBKH[13,2,11]2 +Q/2 rBi? ?mHH /BK2MbBQM QM2X

jX8 n ≡ 0, 2, 4 (mod 6)

L2ti- r2 b?Qr i?i i?2`2 /Q2b MQi 2tBbi[n,2,1_2n

3

2]2 +Q/2b rBi? ?mHH /BK2MbBQM QM2 7Q` HH n ≡ 0,2,4 (mod 6)X Gi2`- r2 ;Bp2 +QMbi`m+iBQMb Q7 QTiBKH [n,2,1_2n

3

2−1]₂

+Q/2b Q7 bm+? H2M;i?b rBi? ?mHH /BK2MbBQM QM2X

G2KK jX8XRX G2i n ≥ 3 #2 M BMi2;2`X A7 n ≡ 0,2,4 (mod 6)- i?2M i?2`2 `2 MQ [n,2,1_2n

3

2]2 +Q/2b rBi? ?mHH /BK2MbBQM QM2X

S`QQ7X bbmK2 i?i n ≡ 0,2,4 (mod 6)X amTTQb2 i?i i?2`2 2tBbib M [n,2,1_2n

3

2]2

+Q/2 C rBi? ?mHH /BK2MbBQM QM2X

6B`bi- r2 T`Qp2 i?i S10+S01+S11 =nX amTTQb2 i?i S00 >0X

*b2 AXn ≡0,2 (mod 6)X A7 7QHHQrb i?i/

2(n−1) 3

0 < ²ⁿ₃ =1_2n

3

27Q` HHn ≡0 (mod 6)- M/ /

2(n−1) 3

0

< ²ⁿ₃⁻¹ = 1_2n

3

2 7Q` HH n ≡ 2 (mod 6)X aBM+2 S00 > 0- i?2`2 2tBbib M [n−1,2,1_2n

3

2]2 +Q/2 C^′ #v TmM+im`BM; i QM2 Q7 i?2 x2`Q +QHmKMbX 6`QK UjX9V- r2

?p2 -

2n 3

.

= wt(C^′)≤

-2(n−1) 3

.

<

-2n 3

. , +QMi`/B+iBQMX

ky

*b2 AAX n ≡ 4 (mod 6)X aBM+2 S00 >0- i?2`2 2tBbib M [n−1,2,1_2n

3

2]2 +Q/2 C^′ #v TmM+im`BM; i QM2 Q7 i?2 x2`Q +QHmKMbX aBM+2 n−1≡3 (mod 6)- r2 ?p2

-2n 3

.

= wt(C^′)≤D2(n−1,2,1) =

-2(n−1) 3

.

−1 = -2n

3 .

−1<

-2n 3

.

#v h?2Q`2K jXjXRX h?Bb Bb +QMi`/B+iBQMX

6`QK i?2 /Bb+mbbBQM #Qp2- r2 ?p2S<sub>10</sub>+S<sub>01</sub>+S<sub>11</sub>=nX L2ti- r2 +QMbB/2` i?2 7QHHQrBM;2 +b2bX

*b2 RX S11 Bb Q//X aBM+2 n Bb 2p2M- Bi 7QHHQrb i?i 2Bi?2` S10 Q` S01 Bb Q//X 6`QK UjXjV- r2 ?p2

GG^T =

⎡

⎣0 1 1 1

⎤

⎦ Q` GG^T =

⎡

⎣1 1 1 0

⎤

⎦.

AM #Qi? +b2b- rank(GG^T) = 2. "v S`QTQbBiBQM kXkX9- r2 ?p2 dim(Hull(C)) = 2− rank(GG^T) = 0X

*b2 kX S11 Bb 2p2MX h?2MS10 M/ S01 ?p2 i?2 bK2 T`BivX A7 S10M/ S01`2 2p2M- i?2M

GG^T =

⎡

⎣0 0 0 0

⎤

⎦

#v UjXjVX A7 S₁₀ M/ S₀₁ `2 Q//- i?2M

GG^T =

⎡

⎣1 0 0 1

⎤

⎦

#v UjXjVX AM #Qi? +b2b- dim(Hull(C)) = 2−rank(GG^T)̸= 1 #v S`QTQbBiBQM kXkX9X 6`QK i?2 irQ +b2b- r2 ?p2 dim(Hull(C))̸= 1- +QMi`/B+iBQMX >2M+2- i?2`2 `2 MQ[n,2,1_2n

3

2]2 +Q/2b rBi? ?mHH /BK2MbBQM QM2X

h?2Q`2K jX8XkX G2i n ≥3 #2 M BMi2;2`X A7 n ≡0,2,4(mod 6)- i?2M D2(n,2,1) =

-2n 3

.

−1.

JQ`2Qp2`- i?2 7QHHQrBM; bii2K2Mib ?QH/X

RX h?2`2 2tBbib mMB[m2 UmT iQ 2[mBpH2M+2V QTiBKH 3

n,2,1_2n

3

2−14

2 +Q/2 rBi?

?mHH /BK2MbBQM QM2 7Q` HH n= 4 Q` n≡0,2(mod 6)X kR

kX h?2`2 `2 irQ UmT iQ 2[mBpH2M+2V QTiBKH 3

n,2,1_2n

3

2−14

2 +Q/2b rBi? ?mHH /B@

K2MbBQM QM2 7Q` HH n >4 bm+? i?i n≡4(mod 6)X

S`QQ7X bbmK2 i?i n ≡0,2,4(mod 6)X "v G2KK jX8XR- r2 ?p2 D2(n,2,1)≤

-2n 3

.

−1.

6B`bi- r2 Q#b2`p2 i?i i?2`2 `2 MQ i`BTH2b(S01, S10, S11)biBb7vBM;S01+S10+S11≤ n−2M/min{S₁₀+S₁₁, S₀₁+S₁₁, S₁₀+S₀₁}=1_2n

3

2−1X >2M+2- S₀₁+S₁₀+S₁₁ ≥n−1X q2 +QMbB/2` i?2 7QHHQrBM; irQ +b2bX

*b2 RX n =S01+S10+S11X q2 b?Qr i?i i?2`2 `2 MQ [n,2,1_2n

3

2−1]2 +Q/2b rBi?

?mHH /BK2MbBQM QM2 b 7QHHQrbX

*b2 RXRX n ≡ 0(mod 6)X h?2M n = 6t 7Q` bQK2 TQbBiBp2 BMi2;2` tX h?2 i`BTH2b (S01, S10, S11) biBb7vBM; S01+S10+S11 =n = 6t M/ min{S10+S11, S01+S11, S10+ S₀₁}=1_2n

3

2−1 = 4t−1`2 BM i?2 b2i

{(2t,2t+ 1,2t−1),(2t,2t−1,2t+ 1),(2t+ 1,2t,2t−1),

(2t+ 1,2t+ 1,2t−2),(2t+ 1,2t−2,2t+ 1),(2t−2,2t+ 1,2t+ 1), (2t+ 1,2t−1,2t),(2t−1,2t,2t+ 1),(2t−1,2t+ 1,2t)}.

h?2 +Q``2bTQM/BM; ;2M2`iQ` Ki`B+2b G ?p2 i?2 T`QT2`iv i?i rank(GG<sup>T</sup>) = 2 r?B+? BKTHB2b i?i dim(Hull(C)) = 0̸= 1 #v S`QTQbBiBQM kXkX9X

*b2 RXkX n ≡ 2(mod 6)X h?2M n = 6t+ 2 7Q` bQK2 TQbBiBp2 BMi2;2` tX h?2 i`BTH2b (S01, S10, S11)biBb7vBM;S01+S10+S11=n= 6t+2M/min{S10+S11, S01+S11, S10+ S01}=1_2n

3

2−1 = 4t `2 BM i?2 b2i

{(2t,2t,2t+ 2),(2t+ 2,2t,2t),(2t,2t+ 2,2t),(2t−2,2t+ 2,2t+ 2), (2t+ 2,2t−2,2t+ 2),(2t+ 2,2t+ 2,2t−2),(2t+ 1,2t−1,2t+ 2), (2t+ 2,2t+ 1,2t−1),(2t+ 1,2t+ 2,2t−1),(2t−1,2t+ 1,2t+ 2), (2t+ 2,2t−1,2t+ 1),(2t−1,2t+ 2,2t+ 1)}.

kk

h?2 +Q``2bTQM/BM; ;2M2`iQ` Ki`Bt G Q7 (S01, S10, S11) BM

{(2t,2t,2t+ 2),(2t+ 2,2t,2t),(2t,2t+ 2,2t),(2t−2,2t+ 2,2t+ 2), (2t+ 2,2t−2,2t+ 2),(2t+ 2,2t+ 2,2t−2)}

?b i?2 T`QT2`iv i?i rank(GG^T) = 0 r?B+? BKTHB2b i?i dim(Hull(C)) = 2 ̸= 1 #v S`QTQbBiBQM kXkX9X h?2 +Q``2bTQM/BM; ;2M2`iQ` Ki`Bt G Q7 (S01, S10, S11) BM

{(2t+ 1,2t−1,2t+ 2),(2t+ 2,2t+ 1,2t−1),(2t+ 1,2t+ 2,2t−1), (2t−1,2t+ 1,2t+ 2),(2t+ 2,2t−1,2t+ 1),(2t−1,2t+ 2,2t+ 1)}

?b i?2 T`QT2`iv i?i rank(GG^T) = 2 r?B+? BKTHB2b i?i dim(Hull(C)) = 0 ̸= 1 #v S`QTQbBiBQM kXkX9X

*b2 RXjX n ≡ 4(mod 6)X h?2M n = 6t+ 4 7Q` bQK2 TQbBiBp2 BMi2;2` tX h?2 i`BTH2b (S01, S10, S11)biBb7vBM;S01+S10+S11=n= 6t+4M/min{S10+S11, S01+S11, S10+ S01}=1_2n

3

2−1 = 4t+ 1 `2 BM i?2 b2i

{(2t,2t+ 1,2t+ 3),(2t−1,2t+ 2,2t+ 3),(2t+ 3,2t−1,2t+ 2), (2t+ 3,2t+ 2,2t−1),(2t−1,2t+ 3,2t+ 2),(2t+ 2,2t+ 3,2t−1), (2t+ 2,2t−1,2t+ 3),(2t−1,2t+ 2,2t+ 3),(2t+ 3,2t−1,2t+ 2), (2t+ 3,2t+ 2,2t−1),(2t−1,2t+ 3,2t+ 2),(2t+ 2,2t+ 3,2t−1), (2t+ 3,2t+ 3,2t−2),(2t−2,2t+ 3,2t+ 3),(2t−2,2t+ 3,2t+ 3)}.

h?2 +Q``2bTQM/BM; ;2M2`iQ` Ki`B+2b G ?p2 i?2 T`QT2`iv i?i rank(GG<sup>T</sup>) = 2 r?B+? BKTHB2b i?i dim(Hull(C)) = 0̸= 1 #v S`QTQbBiBQM kXkX9X

*b2 kX n−1 =S₀₁+S₁₀+S₁₁X h?2 2tBbi2M+2 Q7 M [n,2,1_2n

3

2−1]₂ +Q/2 rBi? ?mHH /BK2MbBQM QM2 Bb ;Bp2M b 7QHHQrbX

*b2 kXRX n ≡ 0(mod 6)X h?2M n−1 ≡5(mod 6)X "v h?2Q`2K jX9XR- i?2`2 2tBbib M [n−1,2,/

2(n−1) 3

0]₂ +Q/2 rBi? ;2M2`iQ` Ki`Bt G M/ ?mHH /BK2MbBQM QM2X G2i G<sup>′</sup> = [0 G]X Ai Bb MQi /B{+mHi iQ b22 i?i G<sup>′</sup> ;2M2`i2b M [n,2,/

2(n−1) 3

0]2 +Q/2DrBi?

?mHH /BK2MbBQM QM2X aBM+2 /

2(n−1) 3

0=1_2n

3

2−1- D ?b T`K2i2`b [n,2,1_2n

3

2−1]2X

kj

*b2 kXkX n ≡ 2(mod 6)X h?2M n−1 ≡1(mod 6)X "v h?2Q`2K jXkXR- i?2`2 2tBbib M M[n−1,2,/

2(n−1) 3

0]2 +Q/2 rBi? ;2M2`iQ` Ki`Bt GM/ ?mHH /BK2MbBQM QM2X G2i G<sup>′</sup> = [0 G]X Ai Bb MQi /B{+mHi iQ b22 i?i G<sup>′</sup> ;2M2`i2b M [n,2,/

2(n−1) 3

0

]2 +Q/2DrBi?

?mHH /BK2MbBQM QM2X aBM+2 /

2(n−1) 3

0=1_2n

3

2−1- D ?b T`K2i2`b [n,2,1_2n

3

2−1]2X

*b2 kXjX n ≡ 4(mod 6)X h?2M n−1 ≡3(mod 6)X "v h?2Q`2K jXjXR- i?2`2 2tBbib M M [n−1,2,/

2(n−1) 3

0−1]2 +Q/2 rBi? ;2M2`iQ` Ki`Bt GM/ ?mHH /BK2MbBQM QM2X G2i G<sup>′</sup> = [0G]X Ai Bb MQi /B{+mHi iQ b22 i?i G<sup>′</sup> ;2M2`i2b M [n,2,/

2(n−1) 3

0−1]2

+Q/2 D rBi? ?mHH /BK2MbBQM QM2X aBM+2 /

2(n−1) 3

0−1 = 1_2n

3

2−1- D ?b T`K2i2`b [n,2,1_2n

3

2−1]2X

h?2 2MmK2`iBQM 7Q` 2+? +b2 7QHHQrb 7`QK h?2Q`2K jX9XR- h?2Q`2K jXkXR- M/

h?2Q`2K jXjXR- `2bT2+iBp2HvX

1tKTH2 jX8XjX "v 1tKTH2 jX9Xk- r2 ?p2 i?iC Bb HBM2` +Q/2 rBi? T`K2i2`b [17,2,8]2 ;2M2`i2/ #v

G=

⎡

⎣0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1

⎤

⎦.

G2i G^′ = [0 G]X "v h?2Q`2K jX8Xk- r2 ?p2 i?i G<sup>′</sup> ;2M2`i2b M QTiBKH [18,2,7]2

+Q/2 rBi? ?mHH /BK2MbBQM QM2X

1tKTH2 jX8X9X "v 1tKTH2 jXkXk- r2 ?p2 i?iC Bb HBM2` +Q/2 rBi? T`K2i2`b [13,2,8]2 ;2M2`i2/ #v

G=

⎡

⎣0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1

⎤

⎦.

G2i G^′ = [0 G]X "v h?2Q`2K jX8Xk- r2 ?p2 i?i G<sup>′</sup> ;2M2`i2b M QTiBKH [14,2,7]2

+Q/2 rBi? ?mHH /BK2MbBQM QM2X

1tKTH2 jX8X8X "v 1tKTH2 jXjXk- r2 ?p2 i?iC1 Bb HBM2` +Q/2 rBi? T`K2i2`b [15,2,9]2 ;2M2`i2/ #v

G1 =

⎡

⎣0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1

⎤

⎦.

k9

G2iG^′₁ = [0 G1]X "v h?2Q`2K jX8Xk- r2 ?p2 i?i G<sup>′</sup><sub>1</sub> ;2M2`i2b M QTiBKH[16,2,8]2

+Q/2 rBi? ?mHH /BK2MbBQM QM2X

1tKTH2 jX8XeX "v 1tKTH2 jXjXk- r2 ?p2 i?iC₂ Bb HBM2` +Q/2 rBi? T`K2i2`b [15,2,9]2 ;2M2`i2/ #v

G2 =

⎡

⎣0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1

⎤

⎦.

G2iG^′₂ = [0 G2]X "v h?2Q`2K jX8Xk- r2 ?p2 i?i G<sup>′</sup><sub>2</sub> ;2M2`i2b M QTiBKH[16,2,8]2

+Q/2 rBi? ?mHH /BK2MbBQM QM2X "v h?2Q`2K jX8Xk- r2 ?p2 i?i G<sup>′</sup><sub>1</sub> M/ G<sup>′</sup><sub>2</sub> ;2M2`i2 irQ BM2[mBpH2Mi +Q/2bX

h?2 #Qp2 `2bmHib +M #2 bmKK`Bx2/ b 7QHHQrbX h?2Q`2K jX8XdX G2i n ≥3 #2 M BMi2;2`X h?2M

D₂(n,2,1) =

⎧⎪

⎪⎨

⎪⎪

⎩ 1_2n

3

2 B7 n ≡1,5 (mod 6), 1_2n

3

2−1 B7 n ≡0,2,3,4 (mod 6).

k8

*?Ti2` 9

PTiBKH h2`M`v GBM2` *Q/2b rBi? 1m+HB/2M >mHH .BK2MbBQM PM2

AM i?Bb +?Ti2`- r2 7Q+mb QM +QMbi`m+iBQMb Q7 [n,2]3 +Q/2b rBi? ?mHH /BK2MbBQM QM2X 6Q` 2+? BMi2;2`n ≥3- M QTiBKH[n,2]₃+Q/2 rBi? ?mHH /BK2MbBQM QM2 Bb +QMbi`m+i2/X

*QMb2[m2MiHv- i?2 2t+i pHm2 Q7 D3(n,2,1)Bb T`2b2Mi2/ 7Q` HH BMi2;2`b n≥3X

9XR "bB+ *QM+2Tib

6`QK i?2 b2imT BM am#b2+iBQM kXj Q` BM (RN)- HBM2` [n,2]3 +Q/2 C rBi? ;2M2`iQ`

Ki`Bt

G=

⎡

⎣α1

α₂

⎤

⎦ U9XRV

+M #2 pB2r2/ Q7 i?2 7Q`K C={0,α1,α2,2α1,2α2,α1+α2,2α1+α2,α1+ 2α2,2α1+ 2α2}. 6`QK UkXjV- r2 ?p2 α1 = %

a11 a12 · · · a1n

&

M/ α2 = %

a21 a22 · · · a2n

&

ke

M/ Bi Bb MQi /B{+mHi iQ b22 i?i ri(α1) = ri(2α1) =

*2 i=1

*2 j=0

S_ij, U9XkV

ri(α2) = ri(2α2) =

*2 i=0

*2 j=1

Sij, U9XjV

ri(α1+α₂) =ri(2α1+ 2α₂) =S₀₁+S₀₂+S₁₀+S₁₁+S₂₀+S₂₂, U9X9V ri(2α1+α2) = ri(α1+ 2α2) =S01+S02+S10+S12+S20+S21. U9X8V

>2M+2-

wtH(C) = min{ri(α1),ri(2α1),ri(α2),ri(2α2),ri(α1+α2), ri(2α1+ 2α2),ri(2α1+α2),ri(α1+ 2α2)}

= min{

*2 i=1

*2 j=0

Sij,

*2 i=0

*2 j=1

Sij, S01+S02+S10+S11+S20+S22,

S01+S02+S10+S12+S20+S21}. U9XeV 6`QK U9XRV- r2 ?p2

GG^T =

⎡

⎢⎢

⎣ '2 i=1

'2 j=0

Sij

'2 i=1

'2 j=1

ijSij

'2 i=1

'2 j=1

ijSij

'2 i=0

'2 j=1

Sij

⎤

⎥⎥

⎦(mod 3). U9XdV

1tKTH2 9XRXRX G2i C #2 i?2 i2`M`v HBM2` +Q/2 Q7 H2M;i? 5rBi? ;2M2`iQ` Ki`Bt G=

⎡

⎣1 0 1 0 2 0 1 1 2 1

⎤

⎦

h?2M S10 = S01 = S11 = S02 = S21 = 1 M/ S00 = S20 = S12 = S22 = 0. AM F³, Bi 7QHHQrb i?i

'2 i=1

'2 j=0

Sij = 0,'²

i=1

'2 j=1

ijSij = 0,'²

i=1

'2 j=1

ijSij = 0 M/ '²

i=0

'2 j=1

Sij = 1.

"v U9XdV- Bi 7QHHQrb i?i

G=

⎡

⎣0 0 0 1

⎤

⎦

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

kd

6`QK G2KK kXjX9- Bi +M #2 +QM+Hm/2/ i?i D3(n,2,1)≤

-3n 4

.

. U9X3V

lbBM; i?2 MQiiBQMb #Qp2- i?2 2t+i pHm2 7Q` D<sub>3</sub>(n,2,1) +M #2 /2`Bp2/ BM i?2 7QHHQrBM; b2+iBQMbX

9Xk n ≡ 1 (mod 4)

AM i?2 7QHHQrBM; i?2Q`2K- r2 ;Bp2 +QMbi`m+iBQMb Q7 M QTiBKH[n,2,1_3n

4

2]3+Q/2 rBi?

/BK2bBQM 2 M/ 1m+HB/2M ?mHH /BK2MbBQM QM2 Bb ;Bp2M 7Q` HH H2M;i?bn ≡1 (mod 4)X h?2Q`2K 9XkXRX G2i n ≥3 #2 M BMi2;2`X A7 n ≡1 (mod 4)- i?2M

D3(n,2,1) = -3n

4 .

.

S`QQ7X bbmK2 i?i n ≡1 (mod 4)X h?2M n = 4t+ 1 7Q` bQK2 TQbBiBp2 BMi2;2` t M/

1_3n

4

2 =/

3(4t+1) 4

0= 3tX

6`QK UjX9V- Bi bm{+2b iQ b?Qr i?2 2tBbi2M+2 Q7 M [n,2,3t]₃ +Q/2 rBi? ?mHH /BK2M@

bBQM QM2X

*b2 RX t Bb 2p2MX G2i C #2 HBM2` +Q/2 rBi? ;2M2`iQ` Ki`Bt G Q7 i?2 7Q`K U9XRV /2i2`KBM2/ #v

S00 = 0, S01=S10 =S02 =S20 =S12 =S11=S22= t

2, M/ S21= t 2+ 1.

h?2M C Bb M [n,2, d]3 +Q/2- r?2`2

d= min{

*2 i=1

*2 j=0

Sij,

*2 i=0

*2 j=1

Sij, S01+S02+S10+S11+S20+S22,

S01+S02+S10+S12+S20+S21}

= min{3t+ 1,3t+ 1,3t,3t+ 1} #v U9XeV

= 3t.

k3

6`QK U9XdV- r2 ?p2

GG^T =

⎡

⎣1 2 2 1

⎤

⎦

r?B+? BKTHB2b i?i rank(GG^T) = 1X b /2bB`2/- r2 ?p2 dim(Hull(C)) = 1 #v S`QTQbBiBQM kXkX9X

*b2 kX t Bb Q//X G2i C #2 HBM2` +Q/2 rBi? ;2M2`iQ` Ki`Bt G Q7 i?2 7Q`K U9XRV /2i2`KBM2/ #v

S00 = 0, S02=S20=S12=S21 =S11 = t+ 1

2 M/ S01 =S10=S22= t−1 2 . h?2M C Bb M [n,2, d]3 +Q/2- r?2`2

d= min{

*2 i=1

*2 j=0

Sij,

*2 i=0

*2 j=1

Sij, S01+S02+S10+S11+S20+S22,

S01+S02+S10+S12+S20+S21}

= min{3t+ 1,3t+ 1,3t,3t+ 1} #v U9XeV

= 3t.

6`QK U9XdV- r2 ?p2

GG^T =

⎡

⎣1 2 2 1

⎤

⎦

r?B+? BKTHB2b i?irank(GG^T) = 1X "v S`QTQbBiBQM kXkX9- r2 i?2`27Q` ?p2dim(Hull(C)) = 1 b /2bB`2/X

1tKTH2 9XkXkX G2i t = 2X "v h?2Q`2K 9XkXR M/ t Bb 2p2M- r2 ?p2 i?i S00 = 0, S<sub>01</sub> =S<sub>10</sub> = S<sub>02</sub> =S<sub>20</sub> =S<sub>12</sub> =S<sub>11</sub> =S<sub>22</sub> = 1 M/ S<sub>21</sub> = 2X G2i C #2 i?2 i2`M`v HBM2` +Q/2 Q7 H2M;i? 9 rBi? ;2M2`iQ` Ki`Bt

G=

⎡

⎣1 0 0 2 1 1 2 2 2 0 1 2 0 2 1 2 1 1

⎤

⎦.

AM F³, r2 ?p2

kN

'2 i=1

'2 j=0

Sij = 1, '2 i=1

'2 j=1

ijSij = 2, '2 i=1

'2 j=1

ijSij = 2 M/ '²

i=0

'2 j=1

Sij = 1.

"v U9XdV- Bi 7QHHQrb i?i

G=

⎡

⎣1 2 2 1

⎤

⎦

r?B+? BKTHB2b i?i rank(GG^T) = 1 M/ dim(Hull(C)) = 2− rank(GG^T) = 1 #v S`QTQbBiBQM kXkX9X h?2M C Bb M QTiBKH [9,2,6]3 +Q/2 rBi? ?mHH /BK2MbBQM QM2 #v U9X3VX

9Xj n ≡ 2 (mod 4)

AM i?2 7QHHQrBM; i?2Q`2K- r2 ;Bp2 +QMbi`m+iBQMb Q7 QTiBKH [n,2,1_3n

4

2]3 +Q/2b Q7 bm+? H2M;i?b rBi? ?mHH /BK2MbBQM QM2 r?2`2 n≡2 (mod 4)X

h?2Q`2K 9XjXRX G2i n ≥3 #2 M BMi2;2`X A7 n ≡2 (mod 4)- i?2M D3(n,2,1) =

-3n 4

. .

S`QQ7X bbmK2 i?i n ≡2 (mod 4)X h?2M n = 4t+ 2 7Q` bQK2 TQbBiBp2 BMi2;2` t M/

1_3n

4

2 =/

3(4t+2) 4

0= 3t+ 1X

6`QK i?2 #QmM/ BM U9X3V- Bi bm{+2b iQ b?Qr i?2 2tBbi2M+2 Q7 M [n,2,3t+ 1]₃ +Q/2 rBi? ?mHH /BK2MbBQM QM2X

*b2 RX t Bb 2p2MX G2i C #2 HBM2` +Q/2 rBi? ;2M2`iQ` Ki`Bt G Q7 i?2 7Q`K U9XRV /2i2`KBM2/ #v

S00= 0, S01 =S10=S11= t

2+ 1, S02 =S20 =S21=S22= t

2 M/ S12 = t 2 −1.

h?2M C Bb M [n,2, d]3 +Q/2- r?2`2 d= min{

*2 i=1

*2 j=0

Sij,

*2 i=0

*2 j=1

Sij, S01+S02+S10+S11+S20+S22,

S01+S02+S10+S12+S20+S21}

= min{3t+ 1,3t+ 1,3t+ 3,3t+ 1} #v U9XeV

= 3t+ 1.

jy