  
  [1X2. Coding theory functions in GAP[0X
  
  This  chapter  will  recall  from the GAP4.4.5 manual some of the GAP coding
  theory  and  finite  field functions useful for coding theory. Some of these
  functions are partially written in C for speed. The main functions are
  
  --    [10XAClosestVectorCombinationsMatFFEVecFFE[0X,
  
  --    [10XAClosestVectorCombinationsMatFFEVecFFECoords[0X,
  
  --    [10XCosetLeadersMatFFE[0X,
  
  --    [10XDistancesDistributionMatFFEVecFFE[0X,
  
  --    [10XDistancesDistributionVecFFEsVecFFE[0X,
  
  --    [10XDistanceVecFFE[0X and [10XWeightVecFFE[0X,
  
  --    [10XConwayPolynomial[0X and [10XIsCheapConwayPolynomial[0X,
  
  --    [10XIsPrimitivePolynomial[0X, and [10XRandomPrimitivePolynomial[0X.
  
  However,  the  GAP  command [10XPrimitivePolynomial[0X returns an integer primitive
  polynomial not the finite field kind.
  
  
  [1X2.1 Distance functions[0X
  
  [1X2.1-1 AClosestVectorCombinationsMatFFEVecFFE[0X
  
  [2X> AClosestVectorCombinationsMatFFEVecFFE( [0X[3Xmat, F, vec, r, st[0X[2X ) _____[0Xfunction
  
  This  command  runs  through the [3XF[0X-linear combinations of the vectors in the
  rows of the matrix [3Xmat[0X that can be written as linear combinations of exactly
  [3Xr[0X  rows  (that  is without using zero as a coefficient) and returns a vector
  from  these that is closest to the vector [3Xvec[0X. The length of the rows of [3Xmat[0X
  and  the  length  of  [3Xvec[0X must be equal, and all elements must lie in [3XF[0X. The
  rows  of  [3Xmat[0X must be linearly independent. If it finds a vector of distance
  at  most  [3Xst[0X, which must be a nonnegative integer, then it stops immediately
  and returns this vector.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> F:=GF(3);;[0X
    [4Xgap> x:= Indeterminate( F );; pol:= x^2+1;[0X
    [4Xx_1^2+Z(3)^0[0X
    [4Xgap> C := GeneratorPolCode(pol,8,F);[0X
    [4Xa cyclic [8,6,1..2]1..2 code defined by generator polynomial over GF(3)[0X
    [4Xgap> v:=Codeword("12101111");[0X
    [4X[ 1 2 1 0 1 1 1 1 ][0X
    [4Xgap> v:=VectorCodeword(v);[0X
    [4X[ Z(3)^0, Z(3), Z(3)^0, 0*Z(3), Z(3)^0, Z(3)^0, Z(3)^0, Z(3)^0 ][0X
    [4Xgap> G:=GeneratorMat(C);[0X
    [4X[ [ Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ],[0X
    [4X  [ 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ],[0X
    [4X  [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ],[0X
    [4X  [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ],[0X
    [4X  [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3) ],[0X
    [4X  [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0 ] ][0X
    [4Xgap> AClosestVectorCombinationsMatFFEVecFFE(G,F,v,1,1);[0X
    [4X[ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0 ][0X
  [4X------------------------------------------------------------------[0X
  
  [1X2.1-2 AClosestVectorComb..MatFFEVecFFECoords[0X
  
  [2X> AClosestVectorComb..MatFFEVecFFECoords( [0X[3Xmat, F, vec, r, st[0X[2X ) _____[0Xfunction
  
  [10XAClosestVectorCombinationsMatFFEVecFFECoords[0X  returns  a  two  element  list
  containing      (a)      the      same      closest     vector     as     in
  [10XAClosestVectorCombinationsMatFFEVecFFE[0X,  and  (b)  a vector [3Xv[0X with exactly [3Xr[0X
  non-zero entries, such that v*mat is the closest vector.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> F:=GF(3);;[0X
    [4Xgap> x:= Indeterminate( F );; pol:= x^2+1;[0X
    [4Xx_1^2+Z(3)^0[0X
    [4Xgap> C := GeneratorPolCode(pol,8,F);[0X
    [4Xa cyclic [8,6,1..2]1..2 code defined by generator polynomial over GF(3)[0X
    [4Xgap> v:=Codeword("12101111"); v:=VectorCodeword(v);;[0X
    [4X[ 1 2 1 0 1 1 1 1 ][0X
    [4Xgap> G:=GeneratorMat(C);;[0X
    [4Xgap> AClosestVectorCombinationsMatFFEVecFFECoords(G,F,v,1,1);[0X
    [4X[ [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0 ],[0X
    [4X  [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ] ][0X
  [4X------------------------------------------------------------------[0X
  
  [1X2.1-3 DistancesDistributionMatFFEVecFFE[0X
  
  [2X> DistancesDistributionMatFFEVecFFE( [0X[3Xmat, f, vec[0X[2X ) _________________[0Xfunction
  
  [10XDistancesDistributionMatFFEVecFFE[0X  returns the distances distribution of the
  vector  [3Xvec[0X  to the vectors in the vector space generated by the rows of the
  matrix  [3Xmat[0X  over the finite field [3Xf[0X. All vectors must have the same length,
  and all elements must lie in a common field. The distances distribution is a
  list  d  of  length Length(vec)+1, such that the value d[i] is the number of
  vectors in vecs that have distance i+1 to [3Xvec[0X.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> v:=[ Z(3)^0, Z(3), Z(3)^0, 0*Z(3), Z(3)^0, Z(3)^0, Z(3)^0, Z(3)^0 ];;[0X
    [4Xgap> vecs:=[ [ Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ],[0X
    [4X>   [ 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ],[0X
    [4X>   [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ],[0X
    [4X>   [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ],[0X
    [4X>   [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3) ],[0X
    [4X>   [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0 ] ];;[0X
    [4Xgap> DistancesDistributionMatFFEVecFFE(vecs,GF(3),v);[0X
    [4X[ 0, 4, 6, 60, 109, 216, 192, 112, 30 ][0X
  [4X------------------------------------------------------------------[0X
  
  [1X2.1-4 DistancesDistributionVecFFEsVecFFE[0X
  
  [2X> DistancesDistributionVecFFEsVecFFE( [0X[3Xvecs, vec[0X[2X ) __________________[0Xfunction
  
  [10XDistancesDistributionVecFFEsVecFFE[0X returns the distances distribution of the
  vector  [3Xvec[0X  to the vectors in the list [3Xvecs[0X. All vectors must have the same
  length,  and  all  elements  must  lie  in  a  common  field.  The distances
  distribution  is  a list d of length Length(vec)+1, such that the value d[i]
  is the number of vectors in [3Xvecs[0X that have distance i+1 to [3Xvec[0X.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> v:=[ Z(3)^0, Z(3), Z(3)^0, 0*Z(3), Z(3)^0, Z(3)^0, Z(3)^0, Z(3)^0 ];;[0X
    [4Xgap> vecs:=[ [ Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ],[0X
    [4X>   [ 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ],[0X
    [4X>   [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ],[0X
    [4X>   [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ],[0X
    [4X>   [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3) ],[0X
    [4X>   [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0 ] ];;[0X
    [4Xgap> DistancesDistributionVecFFEsVecFFE(vecs,v);[0X
    [4X[ 0, 0, 0, 0, 0, 4, 0, 1, 1 ][0X
  [4X------------------------------------------------------------------[0X
  
  [1X2.1-5 WeightVecFFE[0X
  
  [2X> WeightVecFFE( [0X[3Xvec[0X[2X ) ______________________________________________[0Xfunction
  
  [10XWeightVecFFE[0X  returns  the  weight  of the finite field vector [3Xvec[0X, i.e. the
  number of nonzero entries.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> v:=[ Z(3)^0, Z(3), Z(3)^0, 0*Z(3), Z(3)^0, Z(3)^0, Z(3)^0, Z(3)^0 ];;[0X
    [4Xgap> WeightVecFFE(v);[0X
    [4X7[0X
  [4X------------------------------------------------------------------[0X
  
  [1X2.1-6 DistanceVecFFE[0X
  
  [2X> DistanceVecFFE( [0X[3Xvec1, vec2[0X[2X ) _____________________________________[0Xfunction
  
  The [13XHamming metric[0X on GF(q)^n is the function
  
  
       dist((v_1,...,v_n),(w_1,...,w_n)) =|\{i\in [1..n]\ |\ v_i\not=
       w_i\}|.
  
  
  This  is  also  called  the  (Hamming)  distance between v=(v_1,...,v_n) and
  w=(w_1,...,w_n). [10XDistanceVecFFE[0X returns the distance between the two vectors
  [3Xvec1[0X  and  [3Xvec2[0X, which must have the same length and whose elements must lie
  in  a common field. The distance is the number of places where [3Xvec1[0X and [3Xvec2[0X
  differ.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> v1:=[ Z(3)^0, Z(3), Z(3)^0, 0*Z(3), Z(3)^0, Z(3)^0, Z(3)^0, Z(3)^0 ];;[0X
    [4Xgap> v2:=[ Z(3), Z(3)^0, Z(3)^0, 0*Z(3), Z(3)^0, Z(3)^0, Z(3)^0, Z(3)^0 ];;[0X
    [4Xgap> DistanceVecFFE(v1,v2);[0X
    [4X2[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X2.2 Other functions[0X
  
  We basically repeat, with minor variation, the material in the GAP manual or
  from                 Frank                 Luebeck's                 website
  [7Xhttp://www.math.rwth-aachen.de:8001/~Frank.Luebeck/data/ConwayPol[0X  on Conway
  polynomials.  The  [12Xprime  fields[0X: If p>= 2 is a prime then GF(p) denotes the
  field Z}/pZ}, with addition and multiplication performed mod p.
  
  The  [12Xprime  power  fields[0X:  Suppose  q=p^r  is  a  prime power, r>1, and put
  F=GF(p).  Let  F[x]  denote  the ring of all polynomials over F and let f(x)
  denote  a monic irreducible polynomial in F[x] of degree r. The quotient E =
  F[x]/(f(x))=  F[x]/f(x)F[x]  is  a  field with q elements. If f(x) and E are
  related  in  this way, we say that f(x) is the [12Xdefining polynomial[0X of E. Any
  defining polynomial factors completely into distinct linear factors over the
  field it defines.
  
  For any finite field F, the multiplicative group of non-zero elements F^x is
  a  cyclic  group.  An  alpha  in  F is called a [12Xprimitive element[0X if it is a
  generator of F^x. A defining polynomial f(x) of F is said to be [12Xprimitive[0X if
  it has a root in F which is a primitive element.
  
  [1X2.2-1 ConwayPolynomial[0X
  
  [2X> ConwayPolynomial( [0X[3Xp, n[0X[2X ) _________________________________________[0Xfunction
  
  A   standard   notation   for  the  elements  of  GF(p)  is  given  via  the
  representatives  0, ..., p-1 of the cosets modulo p. We order these elements
  by  0  <  1  < 2 < ... < p-1. We introduce an ordering of the polynomials of
  degree r over GF(p). Let g(x) = g_rx^r + ... + g_0 and h(x) = h_rx^r + ... +
  h_0  (by  convention, g_i=h_i=0 for i > r). Then we define g < h if and only
  if  there is an index k with g_i = h_i for i > k and (-1)^r-k g_k < (-1)^r-k
  h_k.
  
  The  [12XConway  polynomial[0X  f_p,r(x)  for GF(p^r) is the smallest polynomial of
  degree r with respect to this ordering such that:
  
  --    f_p,r(x) is monic,
  
  --    f_p,r(x)  is  primitive,  that  is,  any  zero  is  a generator of the
        (cyclic) multiplicative group of GF(p^r),
  
  --    for each proper divisor m of r we have that f_p,m(x^(p^r-1) / (p^m-1))
        = 0 mod f_p,r(x); that is, the (p^r-1) / (p^m-1)-th power of a zero of
        f_p,r(x) is a zero of f_p,m(x).
  
  [10XConwayPolynomial(p,n)[0X returns the polynomial f_p,r(x) defined above.
  
  [10XIsCheapConwayPolynomial(p,n)[0X  returns  true if [10XConwayPolynomial( p, n )[0X will
  give  a  result  in  reasonable  time.  This  is  either  the case when this
  polynomial is pre-computed, or if n,p are not too big.
  
  [1X2.2-2 RandomPrimitivePolynomial[0X
  
  [2X> RandomPrimitivePolynomial( [0X[3XF, n[0X[2X ) ________________________________[0Xfunction
  
  For  a  finite  field  [3XF[0X  and  a  positive integer [3Xn[0X this function returns a
  primitive  polynomial  of degree [3Xn[0X over [3XF[0X, that is a zero of this polynomial
  has maximal multiplicative order |F|^n-1.
  
  [10XIsPrimitivePolynomial(f)[0X  can  be used to check if a univariate polynomial [3Xf[0X
  is primitive or not.
  
