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)spool exampleagcode.output
)set message auto off
)clear all
-- This example is to show how to compute the generator matrix
-- of an algebraic geometric code (AG-codes, or geometric Goppa code).
-- Here we use the same curve has the example in the example1.input file,
-- that is the curve defined by
--
-- 5 2 3 4
-- X + Y Z + Y Z = 0
--
--
-- NOTE THAT we will use the package PAFFFF instead of PAFF.
--
-- With PAFFFF the computation are done using dynamic extension
-- over the FINITE ground field that is given to PAFFFF.
--
-- For example,
--
-- PAFFFF(K,[X,Y,Z])
--
-- will do computation in any finite extension of K when NEEDED ONLY,
-- while
--
-- PAFF(K,[X,Y,Z])
--
-- only do computation over the field K and, when
-- some extension is needed, PAFF cannot go further and stop.
--
--
-- In fact the difference between PAFF(K,[X,Y,Z]) and PAFFFF(K,[X,Y,Z])
-- is that PAFFFF(K,[X,Y,Z]) will do the computation using the domain
--
-- PseudoAlgebraicClosureOfFiniteField (abbreviation PACOFF)
--
-- (see pseudoAlgCls.spad in the */spad directory).
--
-- Note that it is almost right to say that PAFFFF(K,[X,Y,Z]) is the same as
-- PAFF(PACOFF(K),[X,Y,Z]) but PAFFFF(K,[X,Y,Z]) is easier to use.
--
--
--
-- We want here to construct an AG-code over the field GF(2^4)
-- and also, we want the code to be of length equal to the number of places
-- of degree 1 of the function field of the curve. To do this we consider
-- the following
--
-- Let
--
-- F be the function field of the curve with constant field GF(2) and
-- F4 be the function field of the curve with constant field GF(2^4)
-- (F4 is the field obtained from F by
-- taking a constant field extension)
--
-- It is clear that F is a subfield of F4, but since GF(2^3)
-- is not a subfield of GF(2^4),
-- any place of F of degre 3 will not split in F4 and in particular,
-- any place of F of degre 3 is dominated by a unique place of F4 of degre 3.
--
-- Let P be a place of degree 3 of F and Q be the unique place
-- of F4 above P (i.e. Q | P).
-- It is well known that a basis of L(nP) is a basis of L(nQ)
-- ( n being an integer ).
-- Using this fact, we can contruct an AG-code of length
-- equal to the number of places
-- of degree 1 of the function field of the curve.
--
--
-- First, let us find all the places of degree 3 of F.
-- The results from PAFFFF will be something like this:
--
-- 2 3 2 3
-- [[%D4:%D4 + 1:1] ,[%D4:%D4 :1] ]
--
-- What you have here is 2 places of degree 3
-- (the degree is given by the exponant).
-- Also, they correspond to simple points of the curve
-- defined in an extension of degree 3.
--
--S 1 of 19
K1:= PF(2)
--R
--R
--R (1) PrimeField(2)
--R Type: Domain
--E 1
--S 2 of 19
R1:= DMP([X,Y,Z],K1)
--R
--R
--R (2) DistributedMultivariatePolynomial([X,Y,Z],PrimeField(2))
--R Type: Domain
--E 2
--S 3 of 19
P1:= PAFFFF(K1,[X,Y,Z],BLQT)
--R
--R
--R (3)
--R PackageForAlgebraicFunctionFieldOverFiniteField(PrimeField(2),[X,Y,Z],BlowUpW
--R ithQuadTrans)
--R Type: Domain
--E 3
--S 4 of 19
C1:R1:=X**5 + Y**2*Z**3+Y*Z**4
--R
--R
--R 5 2 3 4
--R (4) X + Y Z + Y Z
--R Type: DistributedMultivariatePolynomial([X,Y,Z],PrimeField(2))
--E 4
--S 5 of 19
setCurve(C1)$P1
--R
--R
--R 5 2 3 4
--R (5) X + Y Z + Y Z
--R Type: DistributedMultivariatePolynomial([X,Y,Z],PrimeField(2))
--E 5
--S 6 of 19
plc3:= placesOfDegree(3)$P1
--R
--R
--R 2 3 2 3
--R (6) [[%D5:%D5 :1] ,[%D5:%D5 + 1:1] ]
--R Type: List(PlacesOverPseudoAlgebraicClosureOfFiniteField(PrimeField(2)))
--E 6
-- %D4 is an elemenent created by the domain PACOFF: it is a root of the
-- irreducible polynomial of degree 3 that is used to defined the
-- extension of degree 3 of GF(2).
-- To see the irreducible polynomial you can issue the following that will
-- retreive the first coordinate of the simple point
-- corresponding to the first place
-- of the list of places of degree 3.
-- Then we look at the defining polynomial of the element:
--S 7 of 19
a:= elt( first % , 1 )
--R
--R
--R (7) %D5
--R Type: PseudoAlgebraicClosureOfFiniteField(PrimeField(2))
--E 7
--S 8 of 19
definingPolynomial(a)
--R
--R
--R 3 2
--R (8) ? + ? + 1
--RType: SparseUnivariatePolynomial(PseudoAlgebraicClosureOfFiniteField(PrimeField(2)))
--E 8
--S 9 of 19
a**3 + a**2 + 1
--R
--R
--R (9) 0
--R Type: PseudoAlgebraicClosureOfFiniteField(PrimeField(2))
--E 9
-- As you can see, %D4 is the root of an irreducible poynomial of degree 3.
--
-- Now we construct a divisor using the places of degree 3.
-- It will be 2 times the sum of the 2 places of degree 3.
--
--S 10 of 19
D:= 2 * reduce(+,(plc3 :: List DIV PLACESPS PF 2))
--R
--R
--R 2 3 2 3
--R (10) 2 [%D5:%D5 :1] + 2 [%D5:%D5 + 1:1]
--R Type: Divisor(PlacesOverPseudoAlgebraicClosureOfFiniteField(PrimeField(2)))
--E 10
-- Now we compute a basis of L(D)
--S 11 of 19
lB1:= lBasis(D)$P1
--R
--R Trying to interpolate with forms of degree:
--R 4
--R Denominator found
--R Intersection Divisor of Denominator found
--R
--R (11)
--R 4 3 2 2 3 2 2 2 2 2 3 3 4
--R [num= [Z ,Y Z ,Y Z ,X Z ,X Y Z ,X Y Z,X Z ,X Y Z,X Z,X Y,X ],
--R 4 2 2 4
--R den= X + X Y Z + X Y Z + Z ]
--IType: Record(num: List(DistributedMultivariatePolynomial([X,Y,Z],...
--E 11
-- Since we want to construct a code over GF(2^4), we defined the
-- package PAFFFF over GF(2^4) to compute all the places of degree 1
--S 12 of 19
K4:= FFCG(2,4)
--R
--R
--R (12) FiniteFieldCyclicGroup(2,4)
--R Type: Domain
--E 12
--S 13 of 19
R4:= DMP([X,Y,Z],K4)
--R
--R
--R (13)
--R DistributedMultivariatePolynomial([X,Y,Z],FiniteFieldCyclicGroup(2,4))
--R Type: Domain
--E 13
--S 14 of 19
P4:= PAFFFF(K4,[X,Y,Z],BLQT)
--R
--R
--R (14)
--R PackageForAlgebraicFunctionFieldOverFiniteField(FiniteFieldCyclicGroup(2,4),[
--R X,Y,Z],BlowUpWithQuadTrans)
--R Type: Domain
--E 14
--S 15 of 19
C4:R4:=C1
--R
--R
--R 5 2 3 4
--R (15) X + Y Z + Y Z
--R Type: DistributedMultivariatePolynomial([X,Y,Z],FiniteFieldCyclicGroup(2,4))
--E 15
--S 16 of 19
setCurve(C4)$P4
--R
--R
--R 5 2 3 4
--R (16) X + Y Z + Y Z
--R Type: DistributedMultivariatePolynomial([X,Y,Z],FiniteFieldCyclicGroup(2,4))
--E 16
--S 17 of 19
plc1 := placesOfDegree(1)$P4
--R
--R
--R (17)
--R 10 1 5 1 1 4 1 1 1 1 2 8 1
--R [[1:%A :1] , [1:%A :1] , [%A :%A :1] , [%A :%A :1] , [%A :%A :1] ,
--R 2 2 1 3 10 1 3 5 1 4 4 1 4 1 1
--R [%A :%A :1] , [%A :%A :1] , [%A :%A :1] , [%A :%A :1] , [%A :%A :1] ,
--R 5 8 1 5 2 1 6 10 1 6 5 1 7 4 1
--R [%A :%A :1] , [%A :%A :1] , [%A :%A :1] , [%A :%A :1] , [%A :%A :1] ,
--R 7 1 1 8 8 1 8 2 1 9 10 1 9 5 1
--R [%A :%A :1] , [%A :%A :1] , [%A :%A :1] , [%A :%A :1] , [%A :%A :1] ,
--R 10 4 1 10 1 1 11 8 1 11 2 1 12 10 1
--R [%A :%A :1] , [%A :%A :1] , [%A :%A :1] , [%A :%A :1] , [%A :%A :1] ,
--R 12 5 1 13 4 1 13 1 1 14 8 1 14 2 1
--R [%A :%A :1] , [%A :%A :1] , [%A :%A :1] , [%A :%A :1] , [%A :%A :1] ,
--R 1 1 1
--R [0:0:1] , [0:1:1] , %I3 ]
--IType: List(PlacesOverPseudoAlgebraicClosureOfFiniteField(...
--E 17
-- Now, we can construct the matrix of the AG-code, which code-words consist
-- of the evaluation of function in L(D) at each places of F4 of degree 1.
-- Note that we call the function eval of the package P4: this function
-- evaluate function at a place by taking as arguments the
-- numerator and the denominator of a function and a place.
--S 18 of 19
mG:= matrix [ [ eval( f, lB1.den, pl )$P4 for pl in plc1 ] for f in lB1.num ]
--R
--R
--R (18)
--R [
--R 4 4 8 8 10 10 1 1 10 10 5 5
--R [1, 1, %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A ,
--R 11 11 2 2 5 5 5 5 13 13 10 10 14
--R %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A ,
--R 14 7 7
--R %A , %A , %A , 1, 1, 0]
--R ,
--R
--R 10 5 8 5 1 10 5 5 2 3 12 10
--R [%A , %A , %A , %A , %A , %A , %A , 1, %A , %A , %A , %A , 1, %A , 1,
--R 12 10 4 10 9 6 6 5 3 9
--R %A , %A , %A , 1, %A , %A , %A , %A , 1, %A , 1, %A , 1, 1, %A , 0, 1,
--R 0]
--R ,
--R
--R 5 10 12 6 9 12 5 9 3 11 14 10
--R [%A , %A , %A , %A , %A , %A , 1, %A , %A , %A , %A , %A , %A , 1,
--R 4 13 3 6 10 13 7 14 2 5 7 1
--R %A , %A , %A , %A , %A , 1, %A , %A , %A , %A , 1, %A , %A , %A ,
--R 8 11
--R %A , %A , 0, 1, 0]
--R ,
--R
--R 5 5 10 10 13 13 5 5 11 11 3
--R [1, 1, %A , %A , %A , %A , %A , %A , %A , %A , 1, 1, %A , %A , %A ,
--R 3 10 10 14 14 9 9 7 7 12 12 6
--R %A , %A , %A , %A , %A , 1, 1, %A , %A , %A , %A , %A , %A , %A ,
--R 6
--R %A , 0, 0, 0]
--R ,
--R
--R 10 5 9 6 3 12 8 3 9 6 8 2 6 1
--R [%A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A ,
--R 7 4 3 12 9 4 4 1 2 11 2 12 1 13
--R %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A ,
--R 14 8
--R %A , %A , 0, 0, 0]
--R ,
--R
--R 5 10 13 7 11 14 3 8 13 7 1 4 1
--R [%A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A ,
--R 6 11 5 11 14 4 9 8 2 10 13 12 2
--R %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A ,
--R 5 14 7 10
--R %A , %A , %A , %A , 0, 0, 1]
--R ,
--R
--R 6 6 12 12 1 1 9 9 5 5 2 2 10
--R [1, 1, %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A ,
--R 10 3 3 8 8 10 10 5 5 4 4 10 10
--R %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A ,
--R 5 5
--R %A , %A , 0, 0, 0]
--R ,
--R
--R 10 5 10 7 5 14 11 6 13 10 13 7 12
--R [%A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A ,
--R 7 14 11 11 5 3 13 14 11 13 7 14 9
--R %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A ,
--R 14 11 13 7
--R %A , %A , %A , %A , 0, 0, 0]
--R ,
--R
--R 7 7 14 14 4 4 13 13 10 10 8 8
--R [1, 1, %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A ,
--R 2 2 11 11 2 2 5 5 1 1 1 1 8 8
--R %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A ,
--R 4 4
--R %A , %A , 0, 0, 0]
--R ,
--R
--R 10 5 11 8 7 1 14 9 2 14 3 12 3
--R [%A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A ,
--R 13 6 3 4 13 12 7 9 6 9 3 11 6
--R %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A , %A ,
--R 12 9 12 6
--R %A , %A , %A , %A , 0, 0, 0]
--R ,
--R
--R 8 8 1 1 7 7 2 2 14 14 9 9
--R [1, 1, %A , %A , %A , %A , %A , %A , %A , %A , 1, 1, %A , %A , %A , %A ,
--R 4 4 11 11 12 12 13 13 6 6 3 3
--R %A , %A , %A , %A , 1, 1, %A , %A , %A , %A , %A , %A , %A , %A ,
--R 0, 0, 0]
--R ]
--R Type: Matrix(FiniteFieldCyclicGroup(2,4))
--E 18
-- The preceding matrix is the generator matrix of a [n,k,d]-code over GF(2^4)
-- where
--
-- n = nb. of places of degree 1 = 33
--
-- k = dim L(D) = deg D - g + 1 = 11 ( since deg D >= 2g -1)
--
-- d >= 33 - degree D = 33 - 12 = 21
--
--
-- In fact, if one look at the row echelon form of the matrix, he will
-- find out that there is a code word of weight 21, so that code has a
-- minimum distance d = 21. (OF COURSE this doesn't work all the time,
-- that we don't always find a word of minimal weight
-- in the row echelon form of the generator matrix).
--S 19 of 19
reduce(min, [33 - count(zero?,l) for l in listOfLists rowEchelon mG])
--R
--R
--R (19) 21
--R Type: PositiveInteger
--E 19
)spool
)lisp (bye)
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