On the small-weight codewords of some Hermitian codes

Contents lists available atScienceDirect

Journal

of

Symbolic

Computation

www.elsevier.com/locate/jsc

On

the

small-weight codewords

of

some

Hermitian

codes

Chiara Marcolla

a

,

Marco Pellegrini

b

, Massimiliano Sala

a a_{DepartmentofMathematics,UniversityofTrento,Italy}

b_{DepartmentofMathematics,UniversityofFirenze,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received28April2014 Accepted19February2015 Availableonline5March2015

Keywords: Affine-varietycode Hammingweight Hermitiancode Hermitiancurve Linearcode Minimum-weightwords

For anyaffine-variety code weshow how to constructan ideal whose solutions correspond to codewords with any assigned weight. We are able to obtain geometric characterizations for small-weightcodewordsforsomefamiliesofHermitiancodesover anyFq2.Fromthesegeometriccharacterizations,weobtainexplicit

formulas.In particular, wedetermine the number of minimum-weight codewords for all Hermitian codes with d≤q and all second-weightcodewordsfordistance-3,4 codes.

©2015ElsevierLtd.All rights reserved.

1. Introduction

Letq be apowerofa prime,thentheHermitiancurve

H

istheplane curvedefinedover

F

_q2 by

theaffineequationxq+1

=

yq

+

y,wherex

,

y

∈ F

_q2.

Thiscurvehasgenusg

=

q(q₂−1) andhasq3

F

q2-rationalaffinepoints,plusonepointatinfinity,so

ithasq3

₊

_{1 rationalpointsover}

_F

q2 andthereforeitisamaximalcurve(RuckandStichtenoth,1994).

Thisisthebestknownexampleofmaximalcurveandthereisavastliteratureonitsproperties,see

Hirschfeld et al. (2008) for a recent survey. Moreover, the Goppa code (Goppa, 1981, 1988) con-structedonthiscurveisbyfarthemoststudied,duetothesimplebasisofitsRiemann–Rochspace (Stichtenoth,1993),whichcanbe writtenexplicitly.TheGoppaconstructionhasbeengeneralized in

E-mailaddresses:chiara.marcolla@unitn.it(C. Marcolla),pellegrin@math.unifi.it(M. Pellegrini),maxsalacodes@gmail.com (M. Sala).

http://dx.doi.org/10.1016/j.jsc.2015.03.003 0747-7171/©2015ElsevierLtd.All rights reserved.

Vl˘adu ¸t andManin (1984)tohigherdimensions.A simplerdescriptioncanbefoundinFitzgeraldand Lax (1998)fortheso-calledaffine-varietycodes.

Inthispaperwe provideanalgebraicandgeometricdescriptionforcodewordsofagivenweight belongingtoanyfixedaffine-varietycode.InAugot (1996)thesolvingofa(multivariate)polynomial equation systemwasproposed forthefirsttime todetermineminimum-weightcodewordsofcyclic codes,whileinSala (2007)amoreefficientsystemwasproposed.Ourproposalcanbeseenasa gen-eralizationtotheaffine-varietycaseofSala (2007).A similarapproachcanbe usedtodecodecodes, asdescribed,forexample, inde BoerandPellikaan (1999) andsurveyedinMoraandOrsini (2009). ThespecializationofourresultstotheHermitiancaseallows ustogiveexplicitformulasforthe num-ber ofsome small-weight codewords.Codesoverthe Hermitian curvehave beenstudiedalong the yearsandinHøholdtet al. (1998),alongwithasurveyofknownresults,a newchallengingapproach asexplicitevaluationcodesisproposed.Weexpandonour2006previousresult,presentedorallyas

SalaandPellegrini (2006),whereweprovedtheintimateconnectionbetweencurveintersectionsand minimum-weightcodewords.

Thepaperisorganizedasfollows:

•

In Section 2 we provide our notation, our first preliminary results on the algebraic charac-terization of fixed-weight codewords of any affine-variety code and some easy results on the intersectionbetweentheHermitiancurveandanyline.

•

InthebeginningofSection 3weprovideadivisionofHermitiancodes infourphases,whichis a slightmodificationofthedivision inHøholdtet al. (1998),andwe give ouralgebraic charac-terizationoffixed-weightcodewordsofsomeHermitiancodes.Westudyindepththefirstphase (that is,d

≤

q) inSection 3.2andwe usetheseresultsto completelyclassify geometricallythe minimum-weightcodewordsforallfirst-phasecodesinSection3.3.InSection3.4wecancount some specialconfigurations ofsecond weight codewordsforanyfirst-phasecodeandfinally in Section 3.5 we can count the exact number of second-weight codewords for the special case when d

=

3

,

4.A result inthis sectionrelies onour results(Marcollaet al.,2014) on intersec-tionpropertiesof

H

withsomespecialconics,firstlypresentedatEffectiveMethodinAlgebraic Geometry,MEGA2013.

•

InSection4wedrawsomeconclusionsandproposesomeopenproblems.

2. Preliminaryresults

2.1. KnownfactsonHermitiancurveandaffine-varietycodes

Fromnowonweconsider

F

qthefinitefieldwithq elements,whereq isapowerofaprimeand

F

q2 thefinitefieldwithq2 elements.Also,

F

_q2 willdenotethealgebraicclosureof

F

qand

F

q2.Let

α

be a fixedprimitive elementof

F

_q2, andwe consider

β

=

α

q+1 asa primitiveelementof

F

q.From nowonq

,

q2

,

α

and

β

areunderstoodasabove.

TheHermitiancurve

H = H

q isdefinedover

F

q2 bytheaffineequation

xq+1

=

yq

+

y where x

,

y

∈ F

_q2

.

(1)

Thiscurvehasgenus g

=

q(q₂−1) andhasn

=

q3_{rationalaffinepoints,denotedby P}

1

,

. . . ,

Pn.Forany

x

∈ F

_q2,Eq.(1)hasexactly q distinctsolutionsin

F

_q2. Thecurve contains alsoone pointatinfinity P_∞,soithasq3

+

1 rationalpointsover

F

q2 (RuckandStichtenoth,1994).

Let t

≥

1. For any ideal I in the polynomial ring

F

q

[

X

]

, where X

= {

x1

,

. . . ,

xt

}

, we denote by

V(

I

)

⊂ (F

q

)

t its variety,thatis,thesetofitscommonroots.Forany Z

⊂ (F

q

)

t wedenoteby

I(

Z

)

⊂

F

q

[

X

]

thevanishingidealofZ ,thatis,

I(

Z

)

= {

f

∈ F

q

[

X

]

|

f

(

Z

)

=

0

}

.

Let g1

,

. . . ,

gs

∈ F

q

[

X

]

,wedenoteby I

= 

g1

,

. . . ,

gs



theidealgeneratedbythe gi’s.Let

{

xq1

−

x1

,

Wehaveanisomorphismof

F

q-vectorspaces(anevaluationmap):

φ

:

R

= F

q

[

x1

, . . . ,

xt

]/

I

−→ (F

q

)

n

f

−→ (

f

(

P1

), . . . ,

f

(

Pn

)).

(2)

LetL

⊆

R bean

F

q-vectorsubspaceofR withdimensionr.

Definition1.The affine-varietycode C

(

I

,

L

)

istheimage

φ (

L

)

andtheaffine-varietycodeC⊥

(

I

,

L

)

is itsdualcode.

Our definition is slightly different from Fitzgerald and Lax (1998) and follows instead that in

Marcollaetal. (2012).LetL belinearlygeneratedbyb1

,

. . . ,

br thenthematrix

H

=

⎛

⎝

b1

(

P1

)

b1

(

P2

)

. . .

b1

(

Pn

)

..

.

..

.

. .

.

..

.

br

(

P1

)

br

(

P2

)

. . .

br

(

Pn

)

⎞

⎠

isageneratormatrixforC

(

I

,

L

)

andaparity-checkmatrixforC⊥

(

I

,

L

)

.

Formorerecentresultsonaffine-varietycodesseeGeil (2008),Marcollaetal. (2012),Lax (2012).

2.2. Firstresultsonwordsofgivenweight

Let0

≤

w

≤

n, C bealinearcode andc

∈

C . Werecallthat theweight of c,denotedby w

(

c

)

,is thenumberofcomponentsofc thataredifferentfromzeroand

Aw

(

C

)

= |{

c

∈

C

|

w

(

c

)

=

w

}|.

Letc

∈ (F

q

)

n,c

= (

c1

,

. . . ,

cn

)

.Then c

∈

C

(

I

,

L

)

⊥

⇐⇒

HcT

=

0

⇐⇒

n



i=1 cibj

(

Pi

)

=

0

,

j

=

1

, . . . ,

r

.

(3)

Proposition1.Let1

≤

w

≤

n.LetI

= 

g1

,

. . . ,

gs



besuchthat

{

xq1

−

x1

,

. . . ,

xtq

−

xt

}

⊂

I.LetL bea

sub-spaceof

F

_q2

[

x1

,

. . . ,

xt

]/

I ofdimensionr.LetL belinearlygeneratedby

{

b1

,

. . . ,

br

}

.Let Jw betheidealin

F

q

[

x1,1

,

. . . ,

x1,t

,

. . . ,

xw,1

,

. . . ,

xw,t

,

z1

,

. . . ,

zw

]

generatedby w



i=1 zibj

(

xi,1

, . . . ,

xi,t

)

for j

=

1

, . . . ,

r (4) gh

(

xi,1

, . . . ,

xi,t

)

for i

=

1

, . . . ,

w and h

=

1

, . . . ,

s (5) zq_i−1

−

1 for i

=

1

, . . . ,

w (6)



1≤l≤t

((

xj,l

−

xi,l

)

q−1

−

1

)

for 1

≤

j

<

i

≤

w

.

(7)

Thenanysolutionof JwcorrespondstoacodewordofC⊥

(

I

,

L

)

withweightw.Moreover,

Aw

(

C⊥

(

I

,

L

))

=

|

V(

Jw

)

|

w

!

.

Proof. Let

σ

beapermutation,

σ

∈

Sw.Itinducesapermutation

σ

ˆ

actingon

{

x1,1

,

. . . ,

x1,t

, . . . ,

xw,1

,

. . . ,

xw,t

,

z1

,

. . . ,

zw

}

as

σ(

ˆ

xi,l

)

=

xσ(i),l and

σ

ˆ

(

zi

)

=

zσ(i).Itiseasytoshowthat Jw isinvariantw.r.t. any

σ

ˆ

,sinceeachof(4),(5),(6)and(7)isso.

Let Q

= (¯

x1,1

,

. . . ,

x

¯

1,t

,

. . . ,

¯

xw,1

,

. . . ,

¯

xw,t

,

¯

z1

,

. . . ,

z

¯

w

)

∈

V(

Jw

)

.We canassociateacodewordto Q inthe following way. Foreach i

=

1

,

. . . ,

w, Pri

= (¯

xi,1

,

. . . ,

x

¯

i,t

)

isin

V(

I

)

, by (5).We can assume

r1

<

r2

< . . . <

rw,viaa permutation

σ

ˆ

ifnecessary.Notethat (7)ensures thatforeach

(

i

,

j

)

,with

i

=

j,we have Pri

=

Prj,since thereisanl such thatxi,l

=

xj,l.Sincez

¯

q−1 i

=

1 (6),

¯

zi

∈ F

q

\ {

0

}

.Let c

∈ (F

q

)

nbe c

= (

0

, . . . ,

0

,

¯

z1 ↑ r1

,

0

, . . . ,

0

,

z

¯

i ↑ ri

,

0

, . . . ,

0

,

¯

zw ↑ rw

,

0

, . . . ,

0

).

Wehavethatc

∈

C⊥

(

I

,

L

)

,since(4)isequivalentto(3).

Reversingthepreviousargument,wecanassociatetoanycodewordasolutionof Jw.Byinvariance of Jw,weactuallyhave w

!

distinctsolutionsforanycodeword.So,togetthenumberofcodewords ofweightw,wedivide

|

V(

Jw

)

|

by w

!

.

2

2.3. IntersectionbetweentheHermitiancurve

H

andaline

Weconsiderthenormandthetrace,thetwofunctionsdefinedasfollows.

Definition2.The norm NFqm

Fq andthe trace Tr

Fqm

Fq aretwofunctionsfrom

F

qm to

F

qsuchthat NF_Fqm

q

(

x

)

=

x

1+q+...+qm−1 _{and Tr}Fqm

Fq

(

x

)

=

x

+

x

q

_{+ . . . +}

_xqm−1

_.

We denote by N and Tr, respectively, the norm and the trace from

F

q2 to

F

q. It is clear that

H = {

N

(

x

)

=

Tr

(

y

)

|

x

,

y

∈ F

q2

}

.

Lemma1.Foranyt

∈ F

q,theequationTr

(

y

)

=

yq

+

y

=

t hasexactlyq distinctsolutionsin

F

q2.Theequation

N

(

x

)

=

xq+1

=

t hasexactlyq

+

1 distinctsolutions,ift

=

0,otherwiseithasjustonesolution.

Proof. The trace isa linear surjectivefunctionbetween two

F

q-vector spacesofdimension, respectively, 2 and 1. Thus,dim

(

ker

(

Tr

))

=

1,andthismeansthatforanyt

∈ F

q thesetofsolutionsoftheequation Tr

(

y

)

=

yq

₊

_y

₌

_{t isnon-emptyandthenithasthesamecardinalityof}

_F

q,thatis,q.

Theequationxq+1

=

0 hasobviouslyonlythesolutionx

=

0.Ift

=

0,sincet

∈ F

q,wecanwritet

= β

i,so

thatx

=

α

i+j(q−1)_{areallsolutions.Wecanassignj}

₌

₀

_,

_{. . . ,}

_{q,andsowehaveq}

₊

_{1 distinctsolutions.}

₂

Lemma2.Let

H

betheHermitiancurve.

(i) Everyline

L

of

P

2

₍

_F

q2

)

eitherintersects

H

inq

+

1 distinctpoints,oritistangentto

H

atapoint P (with contactorderq

+

1).Inthelattercase,

L

doesnotintersect

H

inanyotherpointdifferentfrom P .

(ii) Througheachpointof

H

,thereisonetangentandq2linesof

P

2

(

F

q2

)

thatintersect

H

inq

+

1 points.

Proof. SeeHirschfeld (1998),Lemma 7.3.2atp. 247.

2

Lemma3.Let

L

beanyverticalline

{

x

=

t

}

,witht

∈ F

_q2.Then

L

intersects

H

inq affinepoints.

Proof. For anyt

∈ F

_q2,tq+1

∈ F

q,andso theequation yq

+

y

=

tq+1 hasexactlyq distinctsolutions byapplyingLemma 1.

2

Lemma4.Let

L

beanyhorizontalline

{

y

=

b

}

,withb

∈ F

_q2.ThenifTr

(

b

)

=

0,

L

intersects

H

inoneaffine point,otherwise,ifTr

(

b

)

=

0,

L

intersects

H

inq

+

1 affinepoints.

Proof. ByLemma 1,foranyb

∈ F

_q2,theequation xq+1

=

bq

+

b hasexactlyq

+

1 distinctsolutionsif

Lemma5.Intheaffineplane

A

2

(

F

_q2

)

,thetotalnumberofnon-verticallinesisq4.Ofthese,q4

−

q3intersect

H

inq

+

1 affinepoints,whiletheremainingq3_{linesaretangentto}

_H

_{andtheyintersect}

_H

_{inonlyoneaffine} point.

Proof. Let

L

beanynon-verticallinein

A

2

₍

_F

q2

)

,then

L

= {

y

=

ax

+

b

}

,witha

,

b

∈ F

_q2.Wehaveq2

choicesforbotha andb,sothetotalnumberis q4.

By(ii) ofLemma 2wehavethatthrougheachpointof

H

thereisonetangent.SincetheHermitian curvehasq3 affinepoints, thenthereexistq3 tangent linesto

H

(thatmeet

H

atasingle pointof order q

+

1).

The linescontaining the point atinfinity are onlythe verticallines. ByLemma 3, their number is q2 _{(sincetheyare}

_L

_{= {}

_x

₌

_t

_}

_wheret

_{∈ F}

q2).

Theremaininglinesareq4

−

q3.By(i) ofLemma 2theymeetthecurveatq

+

1 affinepoints.

2

3. Small-weightcodewordsofHermitiancodes

Werecallthatanaffine-varietycodeisIm

(φ (

L

))

,where

φ

isas(2).Weconsideraspecialcaseof affine-varietycode,whichistheHermitiancode.

LetI

= 

xq+1

₋

_yq

₋

_y

_,

_xq2

−

x

,

yq2

−

y



⊂ F

_q2

[

x

,

y

]

andletR

= F

_q2

[

x

,

y

]/

I.WetakeL

⊆

R generated

by

B

m,q

= {

xrys

+

I

|

qr

+ (

q

+

1

)

s

≤

m

,

0

≤

s

≤

q

−

1

,

0

≤

r

≤

q2

−

1

},

wherem isanintegersuchthat0

≤

m

≤

q3

+

q2

−

q

−

2.Forsimplicity,wealsowritexrysforxrys

+

I.

We have the following affine-variety codes: C

(

I

,

L

)

=

Span_F

q2

φ(

B

m,q

)



where

φ

is the evaluation

map(2)andwedenotebyC

(

m

,

q

)

= (

C

(

I

,

L

))

⊥itsdual.Thentheaffine-varietycodeC

(

m

,

q

)

iscalled

theHermitiancode withparity-checkmatrix H :

H

=

⎛

⎝

f1

(

P1

)

. . .

f1

(

Pn

)

..

.

. .

.

..

.

fi

(

P1

)

. . .

fi

(

Pn

)

⎞

⎠

where fj

∈

B

m,q

.

(8)

Remark1.We recallthatthe Riemann–Rochspace associatedtoa divisor E ofa curve

χ

isavector space

L(

E

)

over

F

q

(χ)

definedas

L(

E

)

= {

f

∈ F

q

(

χ

)

| (

f

)

+

E

≥

0

} ∪ {

0

},

where

F

q

(χ

)

isa rationalfunctionfield on

χ

. In particular, ifwe consider the Hermitian curve

H

, we knowthat it hasq3

+

1 points ofdegree one, namelya pole Q_∞ andq3 distinct affinepoints

Pγ,δ

= (

γ

,

δ)

suchthat

γ

q+1

= δ

q

+ δ

.LetD bethedivisorD

=



γq+1_=δq_+δPγ,δonthecurve

H

.For m

∈ Z

the affine-varietycodeC

(

m

,

q

)

⊥ isthesamecode astheGoppacode C

(

D

,

m Q_∞

)

associated withthedivisors D andm Q_∞as

C

(

D

,

m Q_∞

)

= {(

f

(

P1

), . . . ,

f

(

Pn

))

|

f

∈

L(

m Q∞

)

} ⊂ (F

q2

)

n

,

where

L(

m Q_∞

)

isavectorspaceover

F

_q2

(

H)

.

Notethat

B

m,q isamonomialbasisfor

L(

m Q∞

)

.

The Hermitian codes can be divided in four phases (Høholdt et al., 1998), any of them having specificexplicitformulaslinkingtheirdimensionandtheirdistance(Marcolla,2013),asinTable 1.

Intheremainder ofthispaperwe focusonthefirstphase.Thiscasecanbecharacterized bythe conditiond

≤

q.

Table 1

Thefour“phases”ofHermitiancodes(Marcolla,2013).

Phase m Distance d Dimension k

1 0≤m≤q2₋₂ m=aq+b 0≤b≤a≤q−1 b=q−1 a+1 a>b a+2 a=b ⇐⇒d≤q q 3₋a(a+1) 2 − (b+1) 2 q2_{−1≤m≤4g−3} m=2q2_{−q−aq−b−3} 1≤a≤q−2 0≤b≤q−2 (q−a)q−b−1 a≤b (q−a)q a>b n−g−q 2_+aq+b+2 3 4g−2≤m≤n−2 m−2g+2 n−m+g−1 4 n−1≤m≤n+2g−2 m=n+2g−2−aq−b 0≤b≤a≤q−2, n−aq−b a(a+₂1)+b+1

3.1. Cornercodesandedgecodes

In Section 5.3 of Høholdt et al. (1998), the first phase denotes case (3) at p. 933, where it is characterized by l

<

g, which is equivalent to consider the first g

−

1 nongaps in the numerical semigroup

= 

q

,

q

+

1



, that is all nongaps up to the conductor c

=

2g

=

q2

₋

_{q. With respect} tothisdescription,weareabletoextendthisphasetoincludealsonongaps

{

q2

−

q

+

1

,

. . . ,

q2

−

2

}

, asfollows.

By analyzingprecisely the monomialsinvolved,we are able topartition this phasein two sets:

the edgecodes andthecornercodes. Whenconsidering nongapsup totheconductor,there are

non-gaps immediatelyprecedinga gap.Thesecorrespond tocornercodes.The othersare edgecodes. For example,if

= 

5

,

6



then g

=

10 andtheconductorc

=

20.Thenon-gaps upto c are

{

0

,

5

,

6

,

10

,

11

,

12

,

15

,

16

,

17

,

18

,

20

}

.Obviously,

{

0

,

6

,

12

,

18

}

arefollowedbythegaps,respectively,

{

1

,

7

,

13

,

19

}

. So

{

0

,

6

,

12

,

18

}

correspondtocornercodes,while

{

5

,

10

,

11

,

15

,

16

,

17

,

20

,

21

,

22

,

23

}

correspondto edgecodes(notethatwehaveincludedouraddition

{

21

,

22

,

23

}

,withq2

−

2

=

23).

WeobservethatcornercodesarecodesoftheformC

(

m

,

q

)

,withm

= (

q

+

1

)

s and0

≤

s

≤

q

−

2, whileedgecodesarecodesoftheformC

(

m

,

q

)

,withm

=

qr

+ (

q

+

1

)

s,1

≤

r

≤

q

−

1 andr

+

s

≤

q

−

1.

Weprovideaformaldefinitionintermsofmonomials.

Definition3.Let2

≤

d

≤

q andlet1

≤

j

≤

d

−

1. LetL0 d

= {

1

,

x

,

. . . ,

xd−2

},

Ld1

= {

y

,

xy

,

. . . ,

xd−3y

},

. . . ,

L d−2 d

= {

yd−2

}

. Letl1 d

=

xd−1

,

. . . ,

l j d

=

xd−jyj−1.

•

If

B

m,q

=

L_d0

∪ . . . ∪

Ld_d−2,thenwesaythatC

(

m

,

q

)

isa cornercode andwedenoteitbyH0_d.

•

If

B

m,q

=

L0d

∪ . . . ∪

L d−2

d

∪ {

l1d

,

. . . ,

l j

d

}

,thenwesaythatC

(

m

,

q

)

isan edgecode andwedenoteit byH_dj.

FromtheformulasinTable 1wehavethefollowingtheorem.

Theorem4.Let2

≤

d

≤

q,1

≤

j

≤

d

−

1.Then d

(

H0_d

)

=

d

(

H_dj

)

=

d

,

dim_F q2

(

H 0 d

)

=

n

−

d

(

d

−

1

)

2

,

dimFq2

(

H j d

)

=

n

−

d

(

d

−

1

)

2

−

j

In other words, all

φ (

xrys

)

are linearlyindependent (i.e. H has maximal rank) andforany dis-tance d thereareexactlyd Hermitiancodes(onecornercodeandd

−

1 edgecodes).Wecanrepresent theabovecodesasinthefollowingpicture,whereweconsiderthefivesmallestnon-trivialcodes(for anyq

≥

3).

H0₂ isan

[

n

,

n

−

1

,

2

]

code.

B

m,q

=

L02

= {

1

}

, so theparity-check matrix of H0 2is

(

1

,

. . . ,

1

)

. H1₂ isan

[

n

,

n

−

2

,

2

]

code.

B

m,q

=

L02

∪ {

l12

}

= {

1

,

x

}

H0₃ isan

[

n

,

n

−

3

,

3

]

code.

B

m,q

=

L03

∪

L13

= {

1

,

x

,

y

}

H1₃ isan

[

n

,

n

−

4

,

3

]

code.

B

m,q

=

L03

∪

L13

∪ {

l13

}

= {

1

,

x

,

y

,

x2

}

H2₃ isan

[

n

,

n

−

5

,

3

]

code.

B

m,q

=

L03

∪

L13

∪ {

l13

,

l23

}

= {

1

,

x

,

y

,

x2

,

xy

}

3.2. Firstresultsforthefirstphase

Ideal Jw ofProposition 1forC

(

m

,

q

)

is

Jw

=



wi=1zixr_iys_i



xr_ys_∈B m,q

,

xq_i+1

−

yq_i

−

yi



i=1,...,w

,

zq_i2−1

−

1



i=1,...,w

,

xq_i2

−

xi



i=1,...,w

,

yq_i2

−

yi



i=1,...,w

,

((

xi

−

xj

)

q 2₋₁

−

1

)((

yi

−

yj

)

q 2₋₁

−

1

)



1≤i<j≤w



.

(9)

Letw

≥

v

≥

1.Let Q

= (¯

x1

,

. . . ,

x

¯

w

,

¯

y1

,

. . . ,

y

¯

w

,

z

¯

1

,

. . . ,

z

¯

w

)

∈

V(

Jw

)

.Weknow(seeProposition 1) that Q correspondtoacodewordc.Wesaythattheset

{(¯

x1

,

y

¯

1

),

. . . ,

(

¯

xw

,

¯

yw

)

}

isthe support ofc.

Wesay that Q isin v-blockposition if wecan partition

{

1

,

. . . ,

w

}

in v blocks I1

,

. . . ,

Iv such that

¯

xi

= ¯

xj

⇐⇒ ∃

1

≤

h

≤

v such that i

,

j

∈

Ih

.

Thismeans that, inthe support ofc,we have

|

I1

|

points ona vertical line,

|

I2

|

points on another verticalline,andsoon.

W.l.o.g.wecanassume

|

I1

|

≤ . . . ≤ |

Iv

|

andI1

= {

1

,

. . . ,

u

}

. Weneedthefollowingtechnicallemmas.

Lemma6.Wealwayshaveu

+

v

≤

w

+

1.Ifu

≥

2 andv

≥

2,thenv

≤ 

w₂



andu

+

v

≤ 

w₂



+

2.

Proof. Notethat uv

≤

w since

|

I1

|

+ . . . + |

Iv

|

=

w and

|

I1

|

≤ . . . ≤ |

Iv

|

.Sothe worstcaseis when

w

=

uv. Thisisequivalent to thecasewhen allblocks havethesamesize. Let w

:=

uv,note that wealways have w

≤

w andhence anylower bound for w impliesalower bound for w.If u

=

1, thenwehaveu

+

v

=

1

+

v

=

1

+

w.Ifv

=

1,then u

=

w

=

w andsou

+

v

=

w

+

1.Ifu

≥

2 and

v

≥

2,wehaveu

≤ 

w₂



andv

≤ 

w₂



,andthenu

+

v

≤ 

w₂



+ 

w₂



≤

w

<

w

+

1.Sowealwayshave

u

+

v

≤

w

+

1.

Toprove that u

+

v

≤ 

w₂



+

2, we studythe real-valued function f

(

v

)

=

u

+

v

=

w_v

+

v, with 2

≤

v

≤

w₂ (w

≥

4).Wehave f

(

2

)

=

f

(

w₂

)

=

w₂

+

2 where v

=

√

w istheminimumpoint.Infact, the derivativeof f (inthevariable v)is f

(

v

)

= (

v2

₋

_w

_)/

_v2_{,itszerois v}

₌

√

_{w, f} _{ispositive inthe} wholeintervalunderconsiderationandwehave2

≤

√

w

≤

w₂.Thus,thefunctiontakesitsmaximum valueattheendpointsoftheinterval.Thenwe haveu

+

v

≤

w₂

+

2.Since u and v areintegers,we actuallyhavethatu

+

v

≤ 

w₂



+

2.

2

Lemma7.LetusconsidertheedgecodeH_djwith1

≤

j

≤

d

−

1

≤

q

−

1 and3

≤

d

≤

w

≤

2d

−

3.LetQ

=

(

x

¯

1

,

. . . ,

x

¯

w

,

¯

y1

,

. . . ,

¯

yw

,

¯

z1

,

. . . ,

z

¯

w

)

beasolutionofJwinv-blockposition,thenexactlyoneofthefollowing

(a) u

=

1,v

≥

d

+

1 andw

≥

d

+

1,

or

(b) v

=

1,thatis,x

¯

1

= . . . = ¯

xw.

Proof. Wedefine,forallh suchthat1

≤

h

≤

v: Xh

= ¯

xi if i

∈

Ih

,

Zh

=



i∈Ih

¯

zi

.

(a) u

=

1.Wewanttoprove,bycontradiction,thatv

≥

d

+

1.

Letv

≤

d.Since Q

∈

V(

Jw

)

,forany f

∈

L0_d

,

{

l1_d

}

wehave f

(

Q

)

=

0 andsowehavethefollowing equations 0

=

w



i=1

¯

xr_i

¯

zi

=

v



h=1



i∈Ih X_hr

¯

zi

=

v



h=1 Xr_hZh 0

≤

r

≤

d

−

1

.

(10)

Since v

≤

d,onecanrestricttothefirstv equationsof(10)togetav

×

v system,thatis,

⎛

⎜

⎜

⎝

1

. . .

1 X1

. . .

Xv

..

.

. .

.

..

.

X₁v−1

. . .

X_vv−1

⎞

⎟

⎟

⎠

⎛

⎝

Z1

..

.

Zv

⎞

⎠ =

0 (11)

TheabovematrixisaVandermondematrixandthe Xi’sarepairwisedistinct,soithasmaximal rank v.Therefore,thesolutionof(11)is

(

Z1

,

. . . ,

Zv

)

= (

0

,

. . . ,

0

)

.Sinceu

=

1,then Z1

= ¯

z1

=

0, whichcontradictsz

¯

i

∈ F

q2

\ {

0

}

.Thus,v

≥

d

+

1;wehavew

+

1

≥

u

+

v

=

v

+

1

>

d

+

1 andhence

w

≥

d

+

1.

(b) u

≥

2.Wesupposebycontradictionthat v

≥

2. Weneedtodefine: Yh,s

=



i∈Ih

¯

ys_i

¯

zi with 1

≤

s

≤

u

−

1

WeconsiderProposition 1.A subsetofequationsofcondition(4)isthefollowingsystem:

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩



w i=1x

¯

ri

¯

zi

=

0



w i=1x

¯

ri

¯

yiz

¯

i

=

0

..

.



w i=1x

¯

ri

¯

y u−1 i

¯

zi

=

0

⇐⇒

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩



v h=1XhrZh

=

0



v h=1XhrYh,1

=

0

..

.



v h=1XhrYh,u−1

=

0 0

≤

r

≤

v

−

1

.

(12)

Infact, system(12)isa subsetof(4)ifandonly ifdeg

(

x

¯

_iv−1y

¯

u_i−1

)

≤

d

−

2 foranyi

=

1

,

. . . ,

w.

Thatis,

(

v

−

1

)

+ (

u

−

1

)

≤

d

−

2

⇐⇒

v

+

u

≤

d.

To verifythis,since v

≥

2, itis sufficientto applyLemma 6 andwe obtainu

+

v

≤ 

w₂



+

2

≤



2d−3

2



+

2

=

d.

System (12)can be decomposedin u systems, all withthe sameVandermonde matrix,having rank v:

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩



v h=1Zh

=

0



v h=1XhZh

=

0

..

.



v h=1Xhv−1Zh

=

0

,

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩



v h=1Yh,1

=

0



v h=1XhYh,1

=

0

..

.



v h=1Xhv−1Yh,1

=

0

, . . . ,

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩



v h=1Yh,u−1

=

0



v h=1XhYh,u−1

=

0

..

.



v h=1Xhv−1Yh,u−1

=

0 (13)

Therefore,thesolutionsofthesesystemsarezero-solutions.So,inparticular,wehaveZ1

=

Y1,1

=

. . .

=

Y1,u−1

=

0,thatis

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩



u i=1

¯

zi

=

0



u i=1

¯

yiz

¯

i

=

0

..

.



u i=1

¯

yui−1

¯

zi

=

0

⇐⇒

⎛

⎜

⎜

⎝

1

. . .

1

¯

y1

. . .

¯

yu

..

.

. .

.

..

.

¯

yu₁−1

. . .

¯

yu_u−1

⎞

⎟

⎟

⎠

⎛

⎝

¯

z1

..

.

¯

zu

⎞

⎠ =

0

.

Sincethex

¯

i’sinI1 areallequal,thenthe y

¯

i’sarealldistinct.ThenthelastVandermondematrix hasranku,andsoz

¯

1

= . . . = ¯

zu

=

0,butthisisimpossiblebecauseeveryz

¯

i

∈ F

q2

\ {

0

}

.Therefore v

=

1 andu

=

w.

2

3.3. Minimum-weightcodewords

Corollary1.LetusconsidertheedgecodeH_djwith1

≤

j

≤

d

−

1

≤

q

−

1.

IfQ

= (¯

x1

,

. . . ,

x

¯

d

,

y

¯

1

,

. . . ,

y

¯

d

,

z

¯

1

,

. . . ,

z

¯

d

)

∈

V(

Jd

)

,thenx

¯

1

= . . . = ¯

xd.Inotherwords,thesupportofa

minimum-weightwordliesintheintersectionoftheHermitiancurve

H

andaverticalline.

Whereasifd

≥

4 andQ

= (¯

x1

,

. . . ,

x

¯

d+1

,

¯

y1

,

. . . ,

¯

yd+1

,

¯

z1

,

. . . ,

¯

zd+1

)

∈

V(

Jd+1

)

,thenoneofthefollowing casesholds:

(a) x

¯

i

= ¯

xjfori

=

j,1

≤

i

,

j

≤

d

+

1,

or

(b) x

¯

1

= . . . = ¯

xd+1.

Proof. WeareinthehypothesesofLemma 7.Ifw

=

d,thenu

>

1,hencev

=

1.Whereas,ifw

=

d

+

1, we can apply Lemma 7 onlyifd

≥

4. We havetwo possibilities:in case (a) ofLemma 7, we have

v

=

d

+

1,thenall

¯

xi’saredifferent,otherwiseweareincase (b).

2

Nowwecanprovethefollowingtheoremforedgecodes.

Theorem5.Let2

≤

d

≤

q andlet1

≤

j

≤

d

−

1,thenthenumberofminimumweightwordsofanedgecode

H_djis Ad

=

q2

(

q2

−

1

)



q d



.

Proof. ByProposition 1weknowthat Jd representsall thewordsofminimumweight.Thefirstset ofidealbasis (9)hasexactly d(d₂−1)

+

j equations,where1

≤

j

≤

d

−

1.So,if j

=

1,thissetimplies thefollowingsystem:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

¯

z1

+ . . . + ¯

zd

=

0

¯

x1

¯

z1

+ . . . + ¯

xd

¯

zd

=

0

¯

y1z

¯

1

+ . . . + ¯

yd

¯

zd

=

0

..

.

¯

yd₁−2

¯

z1

+ . . . + ¯

yd_d−2z

¯

d

=

0

¯

xd₁−1

¯

z1

+ . . . + ¯

xd_d−1z

¯

d

=

0 (14)

Whereas,if j

>

1,thenwehavetoaddthefirst j

−

1 ofthefollowingequations:

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

¯

xd₁−2y

¯

1z

¯

1

+ . . . + ¯

xd_d−2y

¯

dz

¯

d

=

0

¯

xd₁−3y

¯

2₁z

¯

1

+ . . . + ¯

xd_d−3y

¯

_d2z

¯

d

=

0

..

.

¯

x1

¯

yd₁−2z

¯

1

+ . . . + ¯

xd

¯

yd_d−2z

¯

d

=

0 (15)

Butx

¯

1

= . . . = ¯

xd,sinceweareinthehypothesesofCorollary 1.So,forany j,thesystembecomes:

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

¯

z1

+ . . . + ¯

zd

=

0

¯

y1

¯

z1

+ . . . + ¯

ydz

¯

d

=

0

..

.

¯

yd₁−2z

¯

1

+ . . . + ¯

yd_d−2

¯

zd

=

0 (16)

Wehaveq2 _{choicesforthecommonvalueofthe}

_¯

_x

i’sand,byLemma 3,wehave



q d



d

!

different

¯

yi’s, sinceforanychoiceof

¯

xi thereareexactlyq possiblevaluesforthe y

¯

i’s,butweneedjustd ofthem, andanypermutationofthesewillbeagainasolution.Nowwehavetocomputethesolutionsforthe

¯

zi’s.

The matrix of thesystem (16)is a Vandermonde matrix,with rankd

−

1. Thismeans that the solutionspacehaslineardimension1.Sothesolutionsare

(

a1

α,

a2

α,

. . . ,

ad−1

α)

with

α

∈ F

∗_q2,where aj are fixed since they dependon the y

¯

i’s.So the numberofthe

¯

zi’s is

|F

_q∗2

|

=

q2

−

1, then Ad

=

1 d!



q2



q_d



d

!(

q2

−

1

)



.

2

Weconsidernowcornercodes.Wehavethefollowinggeometriccharacterization.

Proposition2.LetusconsiderthecornercodeH_d0and2

≤

d

≤

q.Thenthepoints

(

x

¯

1

,

y

¯

1

),

. . . ,

(

x

¯

d

,

y

¯

d

)

cor-respondingtominimum-weightwordslieonasameline.

Proof. The minimum-weight words of a corner code have to verify the first condition set of Jw, whichhas d(d₂−1) equations.Thatis,

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

¯

z1

+ . . . + ¯

zd

=

0

¯

x1

¯

z1

+ . . . + ¯

xdz

¯

d

=

0

¯

y1

¯

z1

+ . . . + ¯

ydz

¯

d

=

0

..

.

¯

yd₁−2z

¯

1

+ . . . + ¯

yd_d−2

¯

zd

=

0 (17)

Thissystemisthesameas(14),butwithamissingequation.Thismeansthat (17)hasall the solu-tionsofsystem(14)andothersolutions.

We claimthatthe

¯

zi’sareall non-zeroonlyifeitherall x

¯

i’saredistinct, orallareequal.Infact, suppose that we havesome (butnot all) x

¯

i’s equal, then we havethat the point Q

∈

V(

Jw

)

, that correspondstoacodeword c,isinv-blockpositionwith1

<

v

<

d.Wecanrepeatthesameargument oftheproofofLemma 7.Inparticular,wehavetwodifferentcases:

u

=

1 wecan restricttothefirst v ofequationsofa subsetof(17),whichhasjustthevariables x’s

andz’s.Inthiswaywe obtaina v

×

v system as(11).As before,

¯

z1

=

0 sinceu

=

1, soitis impossible.

u

≥

2 andv

≥

2.Wehaveu systems,allwiththesameVandermondematrix,havingrank v as(13). Asinthepoint(b)oftheproofofLemma 7,weobtainthat

¯

z1

= . . . = ¯

zu

=

0.

Therefore,we haveonly two possibilities for the

¯

xi’s: eitherall are differentor they coincide.The sameconsiderationistrue forthe y

¯

i’s,becausewhenwe consider(17)andwe exchangex with y, weobtainagain(17).

Sowehavetwoalternatives:

•

The

¯

xi’sareallequalorthe y

¯

i’sareallequal,soourpropositionistrue.

•

The x

¯

i’s andthe

¯

yi’s are all distinct. We willprove that they lie on anon-horizontal linethat intersectstheHermitiancurve.

Let y

= β

x

+ λ

beanon-verticallinepassing throughtwopoints inaminimumweight configu-ration.Wecandoanaffinetransformationofthistype:



x

=

x

y

=

y

+

ax

,

a

∈ F

_q2

such that atleast two ofthe y’s are equal.Substituting theabove transformation in (17)and applyingsomeoperationsbetweentheequations,weobtainasystemthatisequivalent to(17). Butthisnewsystemhasall y’sequaloralldistinct, sothe y’shavetobe allequal.Hencewe canconcludethatthepointslieonasameline.

2

Wefinallyprovethefollowingtheorem:

Theorem6.Let2

≤

d

≤

q,thenthenumberofwordshavingweightd ofacornercodeH0_dis Ad

=

q2

(

q2

−

1

)

q3

₋

_d

₊

₁ d



q d

−

1



.

Proof. Again, the points corresponding to minimum-weight words of a cornercode have to verify

(17).ByProposition 2,weknowthatthesepointslieintheintersectionsofanylineandtheHermitian curve

H

.

LetQ

= (¯

x1

,

. . . ,

x

¯

d

,

y

¯

1

,

. . . ,

y

¯

d

,

¯

z1

,

. . . ,

¯

zd

)

∈

V(

Jd

)

suchthatx

¯

1

= . . . = ¯

xd,thatis,thepoints

(

¯

xi

,

y

¯

i

)

lieonaverticalline.Weknowthatthenumberofsuch Q ’sis

q2

(

q2

−

1

)



q d



d

! .

Nowwehavetocomputethenumberofsolutions Q

∈

V(

Jd

)

suchthat

(

x

¯

i

,

¯

yi

)

lieonanon-vertical line.

ByLemma 5wehavethatthenumberofd-tuplesofpointsis

(

q4

−

q3

)



q

+

1 d



d

!,

becausewehaveq4

₋

_q3 _{non-verticallinesthat intersect}

_H

_inq

₊

_{1 points, andforanychoice ofa} lineweneedjustd ofthesepoints(andthesystemisinvariant).Asregardsthenumberofthe

¯

zi’s, wehavetocomputethenumberofsolutionsofsystem(17).

Weapply anaffinetransformationtothesystem(17)toobtainahorizontalline, thatis,tohave allthex

¯

i’sdifferentandallthey

¯

i’sequal,soweobtainasystemequivalenttosystem(16).Therefore wehaveaVandermondematrix,hencethenumberofthez

¯

i’sisq2

−

1.So

Ad

=

d1!



q2

(

q2

−

1

)



q_d



d

! + (

q4

−

q3

)(

q2

−

1

)



q+_d1



d

!



=

q2

₍

_q2

₋

₁

₎



q d



+ (

q2

₋

_q

₎



q+1 d



=

=

q2

(

q2

−

1

)



(q−d+1)q! d(d−1)!(q−d+1)!

+ (q

2

−

q

)

d(d−(1)q+!(1)q−qd!+1)!



=

q2

(

q2

−

1

)



_d−q₁



q−d_d+1

+

(q2−q_d)(q+1)



=

q2

(

q2

−

1

)

q3−_dd+1



_d−q₁



.

2

3.4. Second-weightcodewords

In thissectionwe statemore theoremsforedgeandcornercodes.We studythecasewhenthe

¯

xi’scoincideorwhenthe y

¯

i’scoincide.

Theorem7.Let2

≤

d

≤

q,thenthenumberofwordsofweightd

+

1 withy

¯

1

= . . . = ¯

yd+1ofacornercode H0_dis:

(

q2

−

q

)(

q4

− (

d

+

1

)

q2

+

d

)



q

+

1 d

+

1



.

WhereasforanedgecodeH_djwith1

≤

j

≤

d

−

1 thisnumbersis:

(

q2

−

q

)(

q2

−

1

)



q

+

1 d

+

1



.

Proof. Wehaveq2

₋

_{q choicesforthey}

_¯

i’sand,byLemma 4,wehave



q+1 d+1



(

d

+

1

)

!

differentx

¯

i’s,since foranychoice ofthe y

¯

i’sthereareexactlyq

+

1 possiblevaluesforthex

¯

i’s,butweneedjustd

+

1 ofthemandanypermutationofthesewillbeagainasolution.

Nowwehavetocomputethesolutionsforthez

¯

i’s,inthetwodistinctcases.

∗

Case H0

d.ByProposition 1 we knowthat Jd represents all thewords ofminimum weight.The firstsetofidealbasis(9)hasexactlyd(d₂−1) equations,whichissystem(17)withmorevariables.1 Since y

¯

1

= . . . = ¯

yd+1,system(17)becomes

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

¯

z1

+ . . . + ¯

zd+1

=

0

¯

x1

¯

z1

+ . . . + ¯

xd+1

¯

zd+1

=

0

..

.

¯

x₁d−2z

¯

1

+ . . . + ¯

x_dd₊−2₁

¯

zd+1

=

0 (18)

The matrix of this system is a Vandermonde matrix of rank d

−

1. This means that the so-lution space has linear dimension 2, hence the number of the admissible z’s is q4

_{− |{¯}

_z

i

=

0 for at least one i

}|

.Nowwecomputethenumberofsolutionssuchthatz

¯

i

=

0 foratleastone i. Ifwesetone

¯

zi

=

0,wehavealinear(solution)spaceofdimension1,thatcontainsq2solutions, corresponding tothe zero solutionand q2

₋

_{1 codewords ofweight d.We have d}

₊

_{1 ofsuch} subspaces.

Moreover,theintersectionofanytwoofthemisonlythezerosolution,becauseifweset

¯

zi

=

0 fortwo

¯

zi’s,we havea linearspaceofdimension 0.The numberofadmissible z’sisq4

− (

d

+

1

)

q2

+

d,obtainedbycountingtheelementsofd

+

1 subspaces andremoving thezerosolution countedd extratimes.Thus,thenumberofwordsofweightd

+

1 with y

¯

1

= . . . = ¯

yd+1 ofH_d0is:

(

q2

−

q

)(

q4

− (

d

+

1

)

q2

+

d

)



q

+

1 d

+

1



.

∗

Case H_dj.Inthiscasethefirstsetofidealbasis(9)contains exactly d(d₂−1)

+

j equations,where 1

≤

j

≤

d

−

1.So,if j

=

1,thissetimpliesthesystem(14)withmorevariables.Whereas,if j

>

1, thenwehavetoaddthefirst j

−

1 ofEqs.(15)withmorevariables.

1 _Wehave¯x

Since

¯

y1

= . . . = ¯

yd+1,thesystembecomes

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

¯

z1

+ . . . + ¯

zd+1

=

0

¯

x1

¯

z1

+ . . . + ¯

xd+1z

¯

d+1

=

0

..

.

¯

x₁d−1

¯

z1

+ . . . + ¯

x_dd₊−1₁

¯

zd+1

=

0 (19)

Thismeansthat thesolutionspacehaslineardimension 1.Sothe numberofthe z’sis

|F

∗_q2

|

=

q2

−

1,thenthenumberofwordsofweightd

+

1 with y

¯

1

= . . . = ¯

yd+1 ofH_djis

(

q2

−

q

)(

q2

−

1

)



q

+

1 d

+

1



.

2

Theorem8.Let2

≤

d

≤

q

−

1,thenthenumberofwordsofweightd

+

1 withx

¯

1

= . . . = ¯

xd+1ofacorner codeH0_dandofanedgecodeH_djis:

q2

(

q4

− (d

+

1

)

q2

+

d

)



q d

+

1



.

Proof. ByProposition 1we knowthat Jd representsallthewordsofminimumweight.Foran edge codethefirstsetofidealbasis(9)implies,if j

=

1,thesystem(14)withmorevariablesand,if j

>

1, wehavetoadd thefirst j

−

1 ofEqs.(15)withmorevariables.Whereas,foracornercode,thefirst setofidealbasis(9)impliesthesystem(17)withmorevariables.

Butx

¯

1

= . . . = ¯

xd+1,soinbothcasesthesystembecomes:

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

¯

z1

+ . . . + ¯

zd+1

=

0

¯

y1z

¯

1

+ . . . + ¯

yd+1

¯

zd+1

=

0

..

.

¯

y₁d−2

¯

z1

+ . . . + ¯

y_dd₊−2₁z

¯

d+1

=

0 (20)

Wehaveq2 _{choicesforthe}

_¯

_x

i’sand,byLemma 3,wehave



q d+1



(

d

+

1

)

!

different

¯

yi’s,since forany choiceofthe

¯

xi’sthereareexactlyq possiblevaluesforthe y

¯

i’s,butweneedjustd

+

1 ofthemand anypermutationofthesewillbeagainasolution.Moreover,wehave

(

q4

_{− (}

_d

₊

₁

₎

_q2

₊

_d

₎

_{possiblez’s,} becausesystem(20)isanalogoustothesystem(18).

2

Theorem9.Let2

≤

d

≤

q,thenthenumberofwordsofweightd

+

1 ofacornercodeH0_dwith

(

x

¯

i

,

¯

yi

)

lying

onanon-verticallineis:

(

q4

−

q3

)(

q4

− (

d

+

1

)

q2

+

d

)



q

+

1 d

+

1



.

WhereasforanedgecodeH_djwith1

≤

j

≤

d

−

1 thisnumbersis:

(

q4

−

q3

)(

q2

−

1

)



q

+

1 d

+

1



.

Proof(sketched). We haveq4

₋

_q3 _{non-vertical lines, intersecting}

_H

_{in a set ofq}

₊

_{1 points. We} choosealineandd

+

1 pointsonit.Byanaffinetransformation,thesystemcanalwaysbe reduced tosystem(18)forcornercodes,orto system(19)foredge codes.Forcornercodeswe geta linear spaceofdimension2,whereasforedgecodeswegetalinearspaceofdimension1.

2

Remark2.Non-verticallinesincludehorizontallinesofTheorem 7,sothatTheorem 9canbe consid-eredasageneralizationofTheorem 7.