Gröbner Bases and Minimum Distane of Ane Varieties Codes C

^{Uma Publiação daSoiedade BrasileiradeMatemátiaApliadae}Computaional.

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 amethodtoestimatetheminimumdistaneofanevari-

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

470416/2011-4,andalsobyFAPEMIGpro. CEX-PPM-00127-12.

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

F q

^{beaniteeldwith}

q

^elements,

n

^{apositiveintegerandfor}

a := (a ¹ , . . . , a n )

^,

b := (b 1 , . . . , b n ) ∈ F ⁿ q

^dene

a ∗ b := (a 1 b 1 , . . . , a n b n )

^{. Let}

C

^{beavetorsubspae}

of

F ⁿ q

^{. Theideaof Andersenand Geilfornding alowerbound fortheminimum}

distaneof

C

^{stemsfromthefatthatif}

c ∈ C

^and

{ b ₁ , . . . , b _n } =: B

isabasisfor

F ⁿ q

^{thenthesubspae}

c ∗ B

generatedby

{ c ∗ b ₁ , . . . , c ∗ b _n }

^{hasdimensionequal}

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 ⊂ F q [X ¹ , . . . , X m ]

^{beanideal,let}

V _F q (I) = {P ¹ , . . . , P n }

^{betheassoiatedvariety}

of

F q

^{-rational points and set}

R := F q [X ¹ , . . . , X m ]/I

^{. Consider the evaluation}

morphism

ϕ : R → F ⁿ q

^{given by}

f + I 7→ (f (P 1 ), . . . , f(P n ))

^{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 , . . . , X m ]

^{is endowed with a monomial order}

4

^{. Thefootprintof}

I

^{(withrespetto}

4

^),denotedby

∆(I)

^{,isthesetofmonomials}

whih are notleadingmonomials ofany polynomialin

I

^.

Let

α = (α ¹ , . . . , α m ) ∈ N ^{m 0}

^(where

N ⁰

^{is the set of} nonnegative integers), we will denote by

M α

^{the monomial}

X 1 ^{α 1} . · · · .X m ^{α m}

^{. Thenthe map}

M α 7→ α

^{gives a}

bijetionbetweenthesetofmonomialsof

F q [X ¹ , . . . , X m ]

^and

N ^{m 0}

^{. Denoteby}

Λ

^the

subsetof

N ^{m 0}

orrespondingtothemonomialswhihare notleadingmonomialsof anypolynomialin

I

^{withrespetto amonomialorder}

4

^{. Then}

∆(I) = {M λ | λ ∈ Λ}

^{is thefootprint of}

I

^{(with respet to}

4

^{). Oneof the main propertiesof}

∆(I)

isthat

{M λ + I | λ ∈ Λ}

^{is abasisfor}

R

^asan

F ^q

^{-vetorspae(see e.g. [5,Prop.}

4,

§

^{3, Ch. 5℄), and we observethat itis abasis whih alreadyarries an order.}

Thus, for eah

λ ∈ Λ

^{weonsider the}

F ^q

^-subspae

L λ ⊂ R

^{whih isgenerated by}

allmonomialsin

∆(I)

^{whiharelessorequalthan}

M λ

^{. Clearly,if}

M σ 4 M λ

^then

L σ ⊆ L λ

^,sothat

C(L σ ) ⊆ C(L λ )

^{. Thenextresultshowsforwhihvaluesof}

λ

^we

get

C(L σ ) $ C(L λ )

^.

Theorem 2.1. Let

I q := I + (X 1 ^q − X ¹ , . . . , X _m ^q − X m )

^{. Then}

dim C(L λ ) >

dim C(L σ )

^(with

M λ ≻ M σ

^{)if andonlyif}

M λ ∈ ∆(I q )

^.

Proof. Observe initially that sine

I ⊂ I q

^then

∆(I q ) ⊂ ∆(I)

^{, so that the laim}

makessense. Denote by

F ^q

^{an algebrailosure of}

F ^q

^{, learly wehave}

V _{F q} (I) = V _{F q} (I q )

^{anddenotingby}

V _{F q} (I q ) ⊂ F q

m

thevarietyof

I q

^asanidealof

F q [X ¹ , . . . , X m ]

wealsoget

V _{F q} (I q ) = V _F

q (I q )

(onsideringthenaturalinlusion

F ^{q m} ⊂ F ^{q m}

^{). From}

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

I q

^{is a radial}

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

R/I q

^isan

F ^q

^{-vetor spaeof nite di-}

mension

#V _F

q (I q )

^{, and sine the lasses of the monomials in}

∆(I q )

^{form a basis}

for

R/I q

^{we get}

#∆(I q ) = n

^{, where}

n = #(V _F q (I))

^{. We will prove now that if}

M λ ∈ ∆(I q )

^then

dim C(L λ ) > dim C(L σ )

^{for any}

M σ ≺ M λ

^{. Assume that it}

is not the ase, sothere exists

σ

^with

M σ ≺ M λ

^{suh that}

C(L λ ) = C(L σ )

^{. In}

partiular,there existsanonzeronitelinearombination

P

M _σ ′ 4M σ a σ ′ M σ ′ ∈ L σ

suh that

( P

M _σ ′ 4M σ a σ ′ M σ ′ )(P i ) = M λ (P i )

^{for all}

i = 1, . . . , n

^{. From Hilbert's}

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

§1

^{,Ch. 4℄) weget that thepolynomial}

M λ − P

M _{σ ′} 4M _σ a σ ^′ M σ ^′

^isin

p

I q = I q

^andafortiori

M λ ∈ / ∆(I q )

^,aontradition. This ompletes the proof of the if assertion, for the only if part observe that the

dimensionofthespaes

dim C(L λ )

^{mayjumpfrom1to}

n

^atmost

n − 1

^times,but

wejust provedthatitwilljump

n − 1

^{times,soeveryjumpmustorrespondtoan}

elementof

∆(I q )

^.

Fromthe(proofofthe) abovetheoremwegetthefollowingresult.

Corollary 2.1.1. Let

∆(I q ) := {M λ ₁ , . . . , M λ _n }

^,then

{ϕ(M λ ₁ ), . . . , ϕ(M λ _n )}

^isa

basis for

F ⁿ q

^,where

n = #(V _{F q} (I))

^.

For simpliity we will denote by

M ¹ , . . . , M n

^{the elements of}

∆(I q )

^{and we}

assumethat

M ¹ < · · · < M n

^{. 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(M i + 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 ∈ F q [X ¹ , . . . , X m ]

^,sothat

c _t ∗ b _i = ϕ(f M i + I)

^.

Nowobservethatif

R ti

^{istheremainderinthedivisionof}

c _t ∗ b _i

byaGröbnerbasis of

I q

^{(withrespetto}

^)then

R ti

^{isalinearombinationof theelementsin}

∆(I q )

and the evaluation of

f M i

^{at the points of}

V _F q (I) = V _F q (I q )

^{oinides with the}

evaluation of

R ti

^{at these points, whih produes thedesired linear} ombination.

Let

Γ ti ⊂ ∆(I q )

^{denote the set of monomials}

M j

^{suh that the oeient of}

b _j

in that linear ombination is not zero. Let

M ti

^{be the greatest element in}

Γ ti

^.

Wewantto ndalowerbound for theweightofwordsof thetype

c = P t i =1 a i c _i

with

a 1 , . . . , a t ∈ F ^q

^and

a t 6= 0

^{. Forthis wedene}

D t := {M t1 }

^{andthen forall}

i = 2, . . . , n

^{we add}

M ti

^{to the set}

D t

^if

M si ≺ M ti

^{for all}

1 ≤ s < t

^{. We laim}

thattheweightof

c

satises

w( c ) ≥ #(D t )

^{. Infat,let}

v = #(D t )

^,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}

M ti ₁ , . . . , M ti _v

^bethedistint

elementsof

D t

^,thenforeah

j ∈ {1, . . . , v}

^wehave

M ti _j = M λ _j ∈ ∆(I q )

^,andfrom

theonstrutionof

D t

^{weget thatin the}

j

^{-thline of}

A

^the

λ j

^{-th entryisnonzero}

and all entries after it are equalto zero. This provesthat

w( c ) ≥ #(D t )

^{, so the}

minimumdistaneof

C

^{islowerboundedby}

min{#(D t ) | 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

∆(I q )

^{. 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}

3a + 4b ≤ 3a ^′ + 4b ^′

^,

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

M i

^the

i

^{-th monomial in}

∆(I q )

^{we get that}

{ b _i := ϕ(M i + I) | i = 1, . . . , 27}

^{isabasis for}

F ^{27 9}

^{. Let}

C

^{be the ode generatedbythe evaluation of(the}

lass of) the rst 5 elements of the basis, namely,

C = h b ₁ , . . . , b ₅ i

^{. We start}

by nding a lower bound for the weight of odewords of the type

c = P ⁵

i =1 a i b _i

, where

a ¹ , . . . , a 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 ∈ ∆(I q )

^{and also}

M ≺ XM ≺ Y M ≺ X ² M ≺ XY M

^{, then we must add}

XY M

^to

D 5

^{for all}

M ∈ M

^{, so that (so far)}

#(D 5 ) = 10

^{. We get ten other}

elements tobeaddedto

D ⁵

^{fromthe tables below}

Remainder inthe division of

c _i ∗ b _j

bythe Gröbnerbasisof

I q

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 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 X ³ Y ³ + Y X ³ Y 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

I q

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

D ⁵

^{;forexample,theremainders}

in the divisionof

c ₁ ∗ b ₂₅

,

c ₂ ∗ b ₂₅

,

c ₃ ∗ b ₂₅

,

c ₄ ∗ b ₂₅

and

c ₅ ∗ 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 ≺ Y ⁸

^{). Likewise, weom-}

pute

#(D ⁴ )

^,

#(D ³ )

^,

#(D ² )

^and

#(D ¹ )

^obtaining respetively 21, 22, 24 and 27.

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(7P ^∞ )

^{,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) = 2P ¹ + 2P ²

^and

div ^∞ (y) = 5P ¹ + 5P 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

2a + 5b ≤ 2a ^′ + 5b ^′

^{,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 +I, X ³ +I, XY +I}

at the rational points in the ane plane (hene, from the above theorem we know

that this ode has dimension 6). Observe that this is the geometri Goppa ode

generated bya baseof

L(7P ¹ + 7P ² )

^{,or inother words, the geometri Goppa ode}

assoiatedwiththedivisors

G = 7P ¹ +7P ²

^and

D

^,where

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}

a j 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 adistâniamí-

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.

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