Some remarks on the asymptotic behavior
of cyclic AG-codes
R. Toledano1
Facultad de Ingenierı́a Quı́mica - Univ. Nacional del Litoral
CLA 2016
1
Joint work with M. Chara and R. Podestá
Linear codes
A linear code C (over the alphabet Fq ) of length n is a
linear subspace of Fnq . The elements of C are usually called
codewords.
If k = dim C is the dimension of the code (as a vector space
over Fq ) and
d = mı́n{d(c, c0 ) : c, c0 ∈ C, c 6= c0 }
is the minimum distance of C, where d is the Hamming
distance in Fnq , we shall say that C is an [n, k, d]−code.
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There is a natural action of the permutation group Sn on
Fnq given by
π · (v1 , . . . , vn ) = (vπ(1) , . . . , vπ(n) )
where π ∈ Sn and (v1 , . . . , vn ) ∈ Fnq .
The set of all π ∈ Sn such that π · c ∈ C for all codewords c
of C forms a subgroup Perm(C) of Sn which is called the
permutation group of C:
Perm(C) = {π ∈ Sn : π(C) = C}.
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Transitive and cyclic codes
Let c = (c1 , . . . , cn−1 , cn ) a codeword in C.
C is transitive if Perm(C) acts transitively on C, i.e. if for
any 1 ≤ i < j ≤ n there is a π ∈ Sn such that π(i) = j.
C is cyclic if it is invariant under the action of the cyclic shift
s ∈ Sn defined as s(1) = n and s(i) = i − 1 for i = 2, . . . , n,
i.e. if
s · c = (cn , c1 , . . . , cn−1 ) ∈ C
for every c ∈ C.
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Transitive and cyclic codes
Let c = (c1 , . . . , cn−1 , cn ) a codeword in C.
C is transitive if Perm(C) acts transitively on C, i.e. if for
any 1 ≤ i < j ≤ n there is a π ∈ Sn such that π(i) = j.
C is cyclic if it is invariant under the action of the cyclic shift
s ∈ Sn defined as s(1) = n and s(i) = i − 1 for i = 2, . . . , n,
i.e. if
s · c = (cn , c1 , . . . , cn−1 ) ∈ C
for every c ∈ C.
Cycic codes are transitive codes
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AG-codes
Let F be an algebraic function field over Fq , i.e. a finite
extension of a rational function field Fq (x).
Let D and G be disjoint divisors of F , with D = P1 +· · ·+Pn
where P1 , . . . , Pn are different rational places.
Let L(G) be the Riemann-Roch space associated to G
L(G) = {x ∈ F : (x) ≥ −G} ∪ {0}
where (x) denotes the principal divisor associated to x ∈ F.
The AG-code defined by F , D and G is
C(D, G) = {(x(P1 ), . . . , x(Pn )) ∈ Fnq : x ∈ L(G)}
where x(Pi ) stands for the residue class of x modulo Pi .
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Cyclic AG-codes
An AG-code CL (D, G) with D = P1 + · · · + Pn is cyclic if for any
codeword
(x(P1 ), . . . x(Pn )) ∈ CL (D, G),
where x ∈ L(G), we have that
(x(Pn ), x(P1 ), . . . x(Pn−1 )) ∈ CL (D, G).
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Cyclic AG-codes
An AG-code CL (D, G) with D = P1 + · · · + Pn is cyclic if for any
codeword
(x(P1 ), . . . x(Pn )) ∈ CL (D, G),
where x ∈ L(G), we have that
(x(Pn ), x(P1 ), . . . x(Pn−1 )) ∈ CL (D, G).
This happens if and only if there exists z ∈ L(G) such that
z(P1 ) = x(Pn ) ,
z(P2 ) = x(P1 ) ,
..
.
(1)
z(Pn ) = x(Pn−1 ) .
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Cyclic AG-codes
An AG-code CL (D, G) with D = P1 + · · · + Pn is cyclic if for any
codeword
(x(P1 ), . . . x(Pn )) ∈ CL (D, G),
where x ∈ L(G), we have that
(x(Pn ), x(P1 ), . . . x(Pn−1 )) ∈ CL (D, G).
This happens if and only if there exists z ∈ L(G) such that
z(P1 ) = x(Pn ) ,
z(P2 ) = x(P1 ) ,
..
.
(1)
z(Pn ) = x(Pn−1 ) .
The existence of such an element z ∈ L(G) is a crucial question
to answer in the theory of cyclic AG-codes.
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Let E/F a finite cyclic extension of the function field Fq and C
an AG-code such that
every place in Sup(D) = {P1 , . . . , Pn } ⊂ P(E) lies over a
unique place P of F ;
there is an element σ ∈ Aut(E/Fq (x)) such that
σ(G) = G
and σ(Pi ) = Pi−1 mod n ,
for i = 1, . . . , n.
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Let E/F a finite cyclic extension of the function field Fq and C
an AG-code such that
every place in Sup(D) = {P1 , . . . , Pn } ⊂ P(E) lies over a
unique place P of F ;
there is an element σ ∈ Aut(E/Fq (x)) such that
σ(G) = G
and σ(Pi ) = Pi−1 mod n ,
for i = 1, . . . , n.
C is a cyclic code
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Let E/F a finite cyclic extension of the function field Fq and C
an AG-code such that
every place in Sup(D) = {P1 , . . . , Pn } ⊂ P(E) lies over a
unique place P of F ;
there is an element σ ∈ Aut(E/Fq (x)) such that
σ(G) = G
and σ(Pi ) = Pi−1 mod n ,
for i = 1, . . . , n.
C is a cyclic code
Take z = σ −1 (x).
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Cyclic AG-codes and cyclic extension
The previous situation can happen only in the presence of cyclic
extensions:
Proposition
Let F be a function field over Fq and let P1 , . . . , Pn be n different places of F . Suppose there exists σ ∈ Aut(F/Fq (x)) such
that σ(P1 ) = Pn , σ(P2 ) = P1 , . . . , σ(Pn ) = Pn−1 . Then there
exist an intermediate field Fq (x) ⊂ E ⊂ F and a place P of E
such that F/E is a cyclic extension of degree m divisible by n,
and P decomposes exactly in F into the places P1 , . . . , Pn with
e(Pi |P )f (Pi |P ) = m
n for i = 1, . . . , n.
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Cyclic AG-codes and cyclic extension
The previous situation can happen only in the presence of cyclic
extensions:
Proposition
Let F be a function field over Fq and let P1 , . . . , Pn be n different places of F . Suppose there exists σ ∈ Aut(F/Fq (x)) such
that σ(P1 ) = Pn , σ(P2 ) = P1 , . . . , σ(Pn ) = Pn−1 . Then there
exist an intermediate field Fq (x) ⊂ E ⊂ F and a place P of E
such that F/E is a cyclic extension of degree m divisible by n,
and P decomposes exactly in F into the places P1 , . . . , Pn with
e(Pi |P )f (Pi |P ) = m
n for i = 1, . . . , n.
Conversely, let F/E be a cyclic extension of function fields over
Fq of degree m. Let P be a place of E and let P1 , . . . , Pn be all
the places of F lying over P . Then, m is divisible by n and we
have that σ(P1 ) = Pn , σ(P2 ) = P1 , . . . , σ(Pn ) = Pn−1 for any
generator σ of Gal(F/E).
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Towers of function fields
A sequence F = {Fi }∞
i=0 of function fields over Fq is called a
tower if
F0
F1
···
Fi
··· ;
Fi+1 /Fi finite and separable, i ≥ 0;
Fq is the full constant field of Fi , i ≥ 0.
g(Fi ) → ∞ for i → ∞
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Towers of function fields
A sequence F = {Fi }∞
i=0 of function fields over Fq is called a
tower if
F0
F1
···
Fi
··· ;
Fi+1 /Fi finite and separable, i ≥ 0;
Fq is the full constant field of Fi , i ≥ 0.
g(Fi ) → ∞ for i → ∞
The limit of the tower is defined by
N (Fi )
≥ 0.
i→∞ g(Fi )
λ(F) := lı́m
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Towers of function fields
A sequence F = {Fi }∞
i=0 of function fields over Fq is called a
tower if
F0
F1
···
Fi
··· ;
Fi+1 /Fi finite and separable, i ≥ 0;
Fq is the full constant field of Fi , i ≥ 0.
g(Fi ) → ∞ for i → ∞
The limit of the tower is defined by
N (Fi )
≥ 0.
i→∞ g(Fi )
λ(F) := lı́m
The tower is asymptotically good if λ(F) > 0.
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Theorem
Let F = {Fi }∞
i=0 be an asymptotically good tower of function
fields over Fq where F0 = Fq (x) is a rational function field. For
each i ∈ N, let ni be a positive integer and let P1 , . . . , Pni be
different rational places of Fi . Suppose that for each i ∈ N there
is an element σi ∈ Aut(Fi /F0 ) such that
σ(P1 ) = Pni , σ(P2 ) = P1 , . . . , σ(Pni ) = Pni −1 .
Then ni < [Fi : F0 ] and there exists a place P ∈ Ram(F) such
that P1 , . . . , Pni are all the places of Fi lying over P .
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Theorem
Let F = {Fi }∞
i=0 be an asymptotically good tower of function
fields over Fq where F0 = Fq (x) is a rational function field. For
each i ∈ N, let ni be a positive integer and let P1 , . . . , Pni be
different rational places of Fi . Suppose that for each i ∈ N there
is an element σi ∈ Aut(Fi /F0 ) such that
σ(P1 ) = Pni , σ(P2 ) = P1 , . . . , σ(Pni ) = Pni −1 .
Then ni < [Fi : F0 ] and there exists a place P ∈ Ram(F) such
that P1 , . . . , Pni are all the places of Fi lying over P .
This theorem shows that the divisor D has to be defined with all
the rational places lying over a place in the ramification locus of
the tower.
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Thanks!