TEMA (São Carlos) vol.13 número3

(1)

Uma Publiação daSoiedade BrasileiradeMatemátiaApliadaeComputaional.

Gröbner Bases and Minimum Distane of Ane

Varieties Codes

C. CARVALHO 1

, Fauldade de Matemátia, Universidade Federal de Uberlândia,

Av.J.N. Ávila2121,38.408-902Uberlândia,MG,Brazil.

Abstrat. Wepresent amethodtoestimatetheminimumdistaneofane

vari-etiesodes. Ourtehniqueusespropertiesofthefootprintofanidealobtainedby

enlargingthedeningidealofthevariety,andmaybeappliedalsotoodeswhih

donotomefromtheso-alledweightdomains.

Keywords. Anevarietiesodes, Gröbnerbases,footprint ofanideal,minimum

distane.

AMS Classiation. 13P10,13F20,94B27

1. Introdution

Sine the appearane of thegeometri Goppa odes in theeighties,many papers

havedealt with improvementson thelowerbound fortheminimum distane of a

ode. One of themost suessful methods for this improvementwasobtainedby

Feng and Rao (see [6℄ and [7℄). Manyrelated bounds appeared after their work,

oneofthembeingabound derivedbyAndersenandGeilin[1℄. Inthatpaperthe

authorsrstderiveageneralapproahtoobtainaboundfortheminimumdistane

(atually, for the generalized Hamming weights) of a linear ode, and then show

howtoapplytheirmethodtoodesdenedfromweightdomains. Aweightdomain

isan

F

-algebra,where

F

isaeld,whihadmitsafuntionto

N

0

∪ {−∞}

satisfying ertainproperties, whih makesthedomainsuitabletobeusedfordening odes,

when

F

isaniteeld. Theywereintroduedin[12℄byT.Høholdt,J.H.vanLint andR.PellikaaninordertopresentanalternativeonstrutionforgeometriGoppa

odeswith simple toolsfrom ommutativealgebra. In the present work weshow

howtoapplyAndersenandGeil'sgeneralapproahtoanevarietyodes. Similarly

to odesobtainedfrom weightdomains, these areevaluation odes obtainedfrom

theringofregularfuntionsofananevarietybutweightfuntionsplaynorolein

thistheory. Sine thealgebraswhih appearin theweightfuntion theoryarethe

ringofregularfuntions ofertaintypeofvariety(see[11℄)ourresultappliesto a

moreomprehensivelassofrings(seeExample 3.2). Thus, distintlyfrom reent

works (see e.g. [9℄ and [10℄) wedo notneedonepts likewell-behavingbasis or

one-way well behaving basis. An important set of data to obtain a bound for

1

ieroufu.br. The author was supported in part by CNPq grants 302280/2011-1 and

(2)

theminimumdistaneisthesetofindexeswhere thereisadimensionjump in a

sequeneofnestedvetorspaes. Whilein [1℄there areseveralresultsonsuh set

for theaseof odes from weightdomains, and in partiular, one-pointgeometri

Goppaodes,hereweshowthatthissetmaybereaddiretlyfromthefootprintof

anidealobtainedbyenlargingthedening idealoftheurve.

Inthenextsetion we reallAndersen andGeil's approah to obtainabound

for the minimum distane of a linear ode, we introdue the ane variety odes

and reallthedenition and someproperties of thefootprintof anideal, thenwe

proveourmainresult. Followingthat, wepresentsomeexamplesto illustrateour

method, inludingodesobtainedfrom analgebrawhih doesnotadmit aweight

funtion.

2. Main result

Let

Fq

beaniteeldwith

1

b

1

, . . . , anbn

)

. Let

C

beavetorsubspae

of

F

n

q

. Theideaof Andersenand Geilfornding alowerbound fortheminimum

distaneof

C

stemsfromthefatthatif

c

∈

C

and

{

b

1

, . . . ,

b

n

totheweightof

c

. Thuswehavethefollowingresult,whihisnotexpliitlystated

in[1℄but isusedthere.

Lemma2.1. Theminimumdistane

d

(

C

)

isequalto

min

{

dim

c

_∗

B

_;

c

_∈

_C

_{\ {}

₀

_}}

.

Wewill usethisresulttoestimatetheminimumdistaneoftheso-alledane

variety odes, whih were introdued by J. Fitzgerald and R. F. Lax in [8℄. Let

I

⊂

Fq

[

X

1

, . . . , Xm

]

beanideal,let

V

F

q

(

I

) =

{

P

1

, . . . , Pn

}

betheassoiatedvariety

of

Fq

-rational points and set

R

:=

Fq

[

X

1

, . . . , Xm

]

/I

. Consider the evaluation morphism

ϕ

:

R

→

F

n

q

given by

f

+

I

7→

(

f

(

P

1

)

, . . . , f

(

Pn

))

and let

L

be an

F

q

-vetorsubspaeof

R

.

Denition2.1. The anevarietyode

C

(

L

)

isthe image

ϕ

(

L

)

.

We observe that as an

F

q

-vetor spae

R

may not have nite dimension. A usefulwayofndingabasisfor

R

is bymeansoftheso-alledfootprintanideal.

Denition 2.2. Assume that

F

q

[

X

1

, . . . , Xm

]

is endowed with a monomial order

4

. Thefootprintof

I

(withrespetto

4

),denotedby

∆(

I

)

,isthesetofmonomials

whih are notleadingmonomials ofany polynomialin

I

.

Let

α

= (

α

1

, . . . , αm

)

∈

N

m

0

(where

N

0

is the set of nonnegative integers), we

will denote by

Mα

the monomial

X

α

₁

1

.

· · ·

.X

α

m

. Thenthe map

Mα

7→

α

gives a

bijetionbetweenthesetofmonomialsof

Fq

[

X

1

, . . . , Xm

]

and

N

m

0

. Denoteby

Λ

the

subsetof

N

m

0

orrespondingtothemonomialswhihare notleadingmonomialsof

anypolynomialin

I

withrespetto amonomialorder

4

. Then

∆(

I

) =

{

Mλ

|

λ

∈

Λ

}

is thefootprint of

I

(with respet to

4

). Oneof the main propertiesof

∆(

I

)

(3)

Thus, for eah

λ

∈

Λ

weonsider the

F

q

-subspae

Lλ

⊂

R

whih isgenerated by allmonomialsin

∆(

I

)

whiharelessorequalthan

Mλ

. Clearly,if

Mσ

4 Mλ

then

Lσ

⊆

Lλ

,sothat

C

(

Lσ

)

⊆

C

(

Lλ

)

. Thenextresultshowsforwhihvaluesof

λ

we

get

C

(

Lσ

)

$

C

(

Lλ

)

.

Theorem 2.1. Let

Iq

:=

I

+ (

X

q

1

−

X

1

, . . . , X

_m

q

−

Xm

)

. Then

dim

C

(

Lλ

)

>

dim

C

(

Lσ

)

(with

Mλ

≻

Mσ

)if andonlyif

Mλ

∈

∆(

Iq

)

.

Proof. Observe initially that sine

I

⊂

Iq

then

∆(

Iq

)

⊂

∆(

I

)

, so that the laim

makessense. Denote by

F

q

an algebrailosure of

F

q

, learly wehave

V

F

q

(

I

) =

V

F

q

(

Iq

)

anddenotingby

V

F

q

(

Iq

)

⊂

Fq

m

thevarietyof

Iq

asanidealof

Fq

[

X

1

, . . . , Xm

]

wealsoget

V

F

q

(

Iq

) =

V

_F

_q

(

Iq

)

(onsideringthenaturalinlusion

F

q

m

⊂

F

q

m

). From

Seidenberg's Lemma 92 (see [16℄ or[2, Lemma 8.13℄) weget that

Iq

is a radial

ideal, sofrom [2, Thm. 8.32℄weget that

R/Iq

isan

F

q

-vetor spaeof nite di-mension

#

V

F

q

(

Iq

)

, and sine the lasses of the monomials in

∆(

Iq

)

form a basis

for

R/Iq

we get

#∆(

Iq

) =

n

, where

n

= #(

V

F

q

(

I

))

. We will prove now that if

Mλ

∈

∆(

Iq

)

then

dim

C

(

Lλ

)

>

dim

C

(

Lσ

)

for any

Mσ

≺

Mλ

. Assume that it

is not the ase, sothere exists

σ

with

Mσ

≺

Mλ

suh that

C

(

Lλ

) =

C

(

Lσ

)

. In

partiular,there existsanonzeronitelinearombination

P

M

_σ

′

4 M

σ

aσ

′

Mσ

′

∈

Lσ

suh that

(

P

M

_σ

′

4 M

σ

aσ

′

Mσ

′

)(

Pi

) =

Mλ

(

Pi

)

for all

i

= 1

, . . . , n

. From Hilbert's

Nullstellensatz (see e.g. [5, Thm. 2,

§

1

,Ch. 4℄) weget that thepolynomial

Mλ

−

P

M

_σ

′

4 M

σ

aσ

′

Mσ

′

isin

p

Iq

=

Iq

andafortiori

Mλ

∈

/

∆(

Iq

)

,aontradition. This

ompletes the proof of the if assertion, for the only if part observe that the

dimensionofthespaes

dim

C

(

Lλ

)

mayjumpfrom1to

n

atmost

n

−

1

times,but

wejust provedthatitwilljump

n

−

1

times,soeveryjumpmustorrespondtoan

elementof

∆(

Iq

)

.

Fromthe(proofofthe) abovetheoremwegetthefollowingresult.

Corollary 2.1.1. Let

∆(

Iq

) :=

{

Mλ

1

, . . . , Mλ

n

}

,then

{

ϕ

(

Mλ

1

)

, . . . , ϕ

(

Mλ

n

)

}

isa

basis for

F

n

q

,where

n

= #(

V

F

q

(

I

))

.

For simpliity we will denote by

M

1

, . . . , Mn

the elements of

∆(

Iq

)

and we

assumethat

M

1

<

· · ·

< Mn

. Wenowshowhow to usethe aboveresultsto nd

a lowerbound for the minimum distane of an ane variety ode

C

⊂

F

n

q

. Let

{

c

1

, . . . ,

c

ℓ

}

beabasisfor

C

, anddenoteby

B

:=

{

b

1

, . . . ,

b

n

}

the(ordered)basis

for

F

n

q

where

b

i

=

ϕ

(

Mi

+

I

)

,

i

= 1

, . . . , n

. Forall

t

∈ {

1 , . . . , ℓ

}

and

i

∈ {

1 , . . . , n

}

write

c

t

∗

b

i

as alinear ombinationof theelementsin

B

. Oneway to dothis is

towrite

c

t

=

ϕ

(

f

+

I

)

forsome

f

∈

Fq

[

X

1

, . . . , Xm

]

,sothat

c

t

∗

b

i

=

ϕ

(

f Mi

+

I

)

. Nowobservethatif

Rti

istheremainderinthedivisionof

c

t

∗

b

i

byaGröbnerbasis

of

Iq

(withrespetto

)then

Rti

isalinearombinationof theelementsin

∆(

Iq

)

and the evaluation of

f Mi

at the points of

V

F

q

(

I

) =

V

F

q

(

Iq

)

oinides with the

evaluation of

Rti

at these points, whih produes thedesired linear ombination.

Let

Γ

ti

⊂

∆(

Iq

)

denote the set of monomials

Mj

suh that the oeient of

b

_j

in that linear ombination is not zero. Let

Mti

be the greatest element in

Γ

ti

. Wewantto ndalowerbound for theweightofwordsof thetype

c

=

Pt

i

=1

ai

c

i

(4)

i

= 2

, . . . , n

we add

Mti

to the set

Dt

if

Msi

≺

Mti

for all

1 ≤

s < t

. We laim

thattheweightof

c

satises

w

(

c

)

≥

#(

Dt

)

. Infat,let

v

= #(

Dt

)

,thenthe

v

×

n

matrix

A

whoselines are the oordinates of thevetors

c

∗

b

i

1

, . . . ,

c

∗

b

i

v

in the

base

B

hasa

v

×

v

invertibleminor. Toseethis, let

Mti

1

, . . . , Mti

v

bethedistint

elementsof

Dt

,thenforeah

j

∈ {

1 , . . . , v

}

wehave

Mti

j

=

Mλ

j

∈

∆(

Iq

)

,andfrom

theonstrutionof

Dt

weget thatin the

j

-thline of

A

the

λj

-th entryisnonzero

and all entries after it are equalto zero. This provesthat

w

(

c

₎

_≥

_#(

_Dt

₎

, so the

minimumdistaneof

C

islowerboundedby

min

{

#(

Dt

)

|

t

= 1

, . . . ,

dim(

C

)

}

.

Theabovemethod appliesto any anevarietyode in

F

n

q

but tosimplify the

alulationsin thefollowingexamples we useodeswhih are generated by some

vetorsofthebase

B

induedby

∆(

Iq

)

. Theseexamplesalsoshowthatthemethod

an yield sharp bounds, and may be applied to odes whih do not ome from

evaluationorderdomains.

3. Examples

Example 3.1. For the rst example we take the hermitian urve given by

Y

3

+

Y

−

X

4

= 0

, dened over

F

9

. Codes over this urve has been studied extensively,

and the minimum distane of one point geometri Goppa odes has been

deter-minedbyStihtenoth([17 ℄)andYangandKumar([18 ℄). Buildingontheexperiene

of those who have dealt with these odeswe hoose a weighted lexiographi order

for

F

9

[

X, Y

]

by stating that

X

a

_Y

b

₄

_X

a

′

Y

b

′

if and only if

3 a

+ 4

b

≤

3 a

′

+ 4

b

′

,

and if equality holds then

X

a

_Y

b

_<

lex

X

a

′

Y

b

′

(with

Y <

lex

X

). These weights

ome from the pole orders of the rational funtions

x

=

X/Z

and

y

=

Y /Z

at

the point at innity

P

∞

:= (0 : 1 : 0)

, whih is their only pole. Using CoCoA

([4 ℄) or Maaulay2 ([13 ℄) we may alulate a Gröbner basis for the ideal

I

9

:=

(

Y

3

+

Y

−

X

4

, Y

9

−

Y, X

9

−

X

)

with respet to

4

,whih is

{

Y

3

+

Y

−

X

4

, Y

9

−

Y, XY

6

−

XY

4

+

XY

2

−

X

}

and from that we get that the footprint of

I

9

(w.r.t.

4

) is

∆(

I

9

) =

{

X

a

_Y

b

_|

₀

_≤

_a

_≤

₃

_,

₀

_≤

_b

_≤

₅

_{} ∪ {}

_Y

6

, Y

7

, Y

8

}

. The urve has

27 rational points (in the ane plane), whih isthe same number of elements in

the footprint, asexpetedfrom theproof oftheorem2.1. Whenweorderthe

mono-mials in

∆(

I

9

)

we get

{

1 , X, Y, X

2

, XY, Y

2

, X

3

, X

2

Y, XY

2

, Y

3

, X

3

Y, X

2

Y

2

, XY

3

,

Y

4

, X

3

Y

2

, X

2

Y

3

, XY

4

, Y

5

, X

3

Y

3

, X

2

Y

4

, XY

5

, Y

6

, X

3

Y

4

, X

2

Y

5

, Y

7

, X

3

Y

5

, Y

8

}

so

denoting by

Mi

the

i

-th monomial in

∆(

Iq

)

we get that

{

b

i

:=

ϕ

(

Mi

+

I

)

|

i

=

1 , . . . ,

27 }

isabasis for

5

∈

F

9

and

a

5

6 = 0

. Following the proedure (and the notation) de-sribedatthe endof thelastsetion,sine

b

₁

₌

_ϕ

_{(1 +}

_I

₎

wemustset

D

5

=

{

XY

}

to start. Then, from

c

₅

_∗

b

₂

₌

_ϕ

₍

_XY

₊

_I

₎

_∗

_ϕ

₍

_X

₊

_I

_{) =}

_ϕ

₍

_X

2

Y

+

I

) =

b

₈

,

c

₄

_∗

b

₂

₌

_ϕ

₍

_X

3

+

I

) =

b

₇

,

c

₃

_∗

b

₂

₌

b

₅

,

c

₂

_∗

b

₂

₌

b

₄

and

c

₁

_∗

b

₂

₌

b

₂

we must add

X

2

Y

to

D

5

obtaining

D

5

=

{

XY, X

2

Y

}

. Atually, sine for any

N

∈ {

1 , X, Y, X

2

, XY

}

and

M

∈ M

:=

{

X

a

_Y

b

_|

₀

_≤

_a

_≤

₁

_,

₀

_≤

_b

_≤

₄

_}

we have

N M

∈

∆(

Iq

)

and also

M

≺

XM

≺

Y M

≺

X

2

M

≺

XY M

, then we must add

(5)

elements tobeaddedto

D

5

fromthe tables below

Remainder inthe division of

c

i

∗

b

j

bythe Gröbnerbasisof

Iq

b

₄

₌

_X

2

_b

7

=

X

3

_b

8

=

X

2

Y

b

₁₁

₌

_X

3

Y

b

₁₂

₌

_X

2

Y

2

XY

X

3

Y

4

+

Y

2

X

3

Y

2

Y

5

+

Y

3

X

3

Y

3

X

2

Y

3

+

Y

XY

3

+

XY

Y

4

+

Y

2

XY

4

+

XY

2

Y

5

+

Y

3

Y

X

2

Y

X

3

Y

X

2

Y

2

X

3

Y

2

X

2

Y

3

X

3

Y

3

+

Y

X

3

Y

4

+

Y

2

X

3

Y

2

1 X

2

X

3

X

2

Y

X

3

Y

X

2

Y

2

Remainderin the divisionof

c

i

∗

b

j

bythe Gröbner basis of

Iq

b

15

=

X

3

Y

2

b

16

=

X

2

Y

3

b

19

=

X

3

Y

3

b

20

=

X

2

Y

4

b

23

=

X

3

Y

4

XY

Y

6

+

Y

4

X

3

Y

4

Y

7

+

Y

5

X

3

Y

5

Y

8

+

Y

6

X

2

XY

5

+

XY

3

Y

6

+

Y

4

−

XY

4

−

XY

2

+

X

Y

7

+

Y

5

−

XY

5

−

XY

3

+

XY

Y

X

3

Y

3

X

2

Y

4

X

3

Y

4

X

2

Y

4

X

3

Y

4

X

Y

5

+

Y

3

X

3

Y

3

Y

6

+

Y

4

X

3

Y

4

Y

7

+

Y

5

1 X

3

Y

2

X

2

Y

3

X

3

Y

3

X

2

Y

4

X

3

Y

4

whihare

{

X

3

Y, Y

4

, X

3

Y

2

, Y

5

, X

3

Y

3

, Y

6

, X

3

Y

4

, Y

7

, X

3

Y

5

, Y

8

}

,sothatnow

#(

D

5

) =

20

. The produt of the elements in the basis of

C

with

b

18

,

b

21

,

b

22

,

b

24

,

b

25

,

b

26

and

b

27

willnotyieldanymonomialtobeaddedto

∗

b

25

by the Gröbner

basis of

I

9

are respetively

Y

7

, XY

5

−

XY

3

+

XY, Y

8

, X

2

Y

5

−

X

2

Y

3

+

X

2

Y

and

X

,sowe annotadd

X

to

D

5

(beause,for instane,

X

Thus our bound for the minimum distane is 20 butthis is the atual value of the

minimum distane, whih we may hek by observing that the ode orrespondsto

the geometri Goppaode generatedbyabasis of

L

(7

P

∞

)

,see[17 ℄.

Example 3.2. In this example we deal with the ane urve dened over

F

9

by the equation

X

6

Y

4

+

X

8

+ 1 = 0

. Observe that the losureof the urve in

P

2

(

F

9

)

has two nonsingular points, namely,

P

1

:= (0 : 1 : 0)

and

P

2

:= (1 : 0 : 0)

so

R

=

F

9

[

X, Y

]

/

(

X

6

Y

4

+

X

8

+ 1)

isnot aweight domain (see [14 ℄, see also [3℄for

results on odes dened by means of near weight domains, whih inlude the ring

R

). The poledivisor ofthe funtions

x

and

y

in thefuntion eldof the urveare

respetively,

div

∞

(

x

) = 2

P

1

+ 2

P

2

and

div

∞

(

y

) = 5

P

1

+ 5

P

2

(alulationsdonewith

KASH/KANT - [15 ℄) so wehoosea weightedlexiographi order for

F

9

[

X, Y

]

by statingthat

X

a

_Y

b

₄

_X

a

′

Y

b

′

ifandonlyif

2 a

+ 5

b

≤

2 a

′

+ 5

b

′

,andifequalityholds

then

X

a

_Y

b

_<

lex

X

a

′

Y

b

′

(with

Y <

lex

X

). Using CoCoA ([4 ℄) orMaaulay2 ([13 ℄)

wemayalulateaGröbnerbasisfor

I

9

= (

X

6

Y

4

+

X

8

+1

, Y

9

−

Y, X

9

−

X

)

obtaining

{

X

4

−

1 , Y

4

−

X

6

}

sothatthe footprint is

∆(

I

9

) =

{

X

a

_Y

b

_|

₀

_≤

_a

_≤

₃

_,

₀

_≤

_b

_≤

₃

_}

,

andwe onlude thatthere are16 rational pointsinthe ane urve. Let

C

be the odegeneratedbyevaluatingthelasses

{

1 +

I, X

+

I, X

2

+

I, Y

+

D

isthesumofallrational

points. Let

f

=

a

1

+

a

2

X

+

a

3

X

2

+

a

4

Y

+

a

5

X

3

+

a

6

XY

,with

a

1

, . . . , a

6

∈

F

9

and

let

j

∈ {

1 , . . . ,

6 }

be greatest index for whih

aj

6 = 0

. Denoting by

B

the (ordered)

basis

{

ϕ

(

X

a

_Y

b

₊

_I

₎

_|

₀

_≤

_a

_≤

₃

_,

₀

_≤

_b

_≤

₃

_}

of

F

16

9

andproeeding asintheexample

above wesee that

ϕ

(

f

+

I

)

∗

B

hasdimension atleast 9(respetively, 4,12, 8,12,

16) if

j

= 6

(respetively, 5,4, 3, 2,1), soour boundfor the minimum distaneis

4, andone may hek that thisis the atual bound. Wealso see that ifwe disard

X

3

and onsider the ode generated by evaluating the lasses

{

1 +

I, X

+

I, X

2

+

I, Y

+

I, XY

+

I

}

attherational points,then thebound forthe minimum distane

isnow8, andagainone mayhekthat thisisthe atualbound.

Aknowledment. Theauthorwouldliketoexpresshiswarmthankstothereferees

fortheirarefulreadingandmanysuggestionswhihhaveimprovedverymuhthe

readabilityofthistext.

Resumo. Nesse trabalhoapresentamos ummétodoparaestimar adistânia

mí-nimadeódigosdevariedadesans.Nossaténiausapropriedadesdapegadade

umidealobtidoatravésdoaumentodoidealdedeniçãodavariedadeemquestão,

e tambémpode ser apliada a ódigos de que não são produzidos utilizando-se

domínios-pesos.

Palavras-have. Códigos de variedade am, bases de Gröbner, pegada de um

ideal,distâniamínima.

Referenes

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FieldsAppl.,14,(1)(2008),92-123.

[2℄ T. Beker, V. Weispfenning, Gröbner Bases- A omputational approah to

ommutativealgebra,SpringerVerlag,Berlin,1993.

[3℄ C.Carvalho,E. Silva,On algebrasadmitting aomplete set of near weights,

evaluation odes, and Goppa odes, Des.Codes Cryptography 53(2) (2009),

99-110.

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Algebra.Available athttp://ooa.dima.unige.it.

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Springer,NewYork,2007.

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algebrai-geometri odes from ane plane urves, IEEE Trans. In-form. Theory, 40

(7)

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IEEETrans.Inform. Theory, 41),(1995),1678-1693.

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Publ.,Hakensak,NJ,2008,Eds.: E.Martinez-Moro,C.Munuera,D.Ruano,

153-180

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W. C.Human (Eds.), Handbook ofCoding Theory,Elsevier, 1998,pp.

871-961.

[13℄ D. Grayson,M. Stillman, Maaulay2,a softwaresystem forresearh in

alge-braigeometry,availableat http://www.math.uiu.edu/Maaulay2/.

[14℄ R.Matsumoto,Miuras'sgeneralizationofone-pointAGodesisequivalentto

Høholdt,vanLintandPellikaan'sgeneralization,IEICETrans.Fundamentals,

E82-A(1999),665-670.

[15℄ M.E.Pohst,et al.,TheComputerAlgebraSystemKASH/KANT,availableat

http://www.math.tu-berlin.de/

∼

kant.

[16℄ A.Seidenberg,Construtionsin algebra,Transations of the Amerian

Math-ematial Soiety,197(1974),273-313.

[17℄ H.Stihtenoth,AnoteonHermitianodesover

GF

(

q

2

)

,IEEETrans.Inform.

Theory,34(1988),1345-1348.

[18℄ K. Yang, P.V. Kumar, On the true minimum distane of Hermitian odes,