#############################################################################
##
#A  decoders.gi             GUAVA library                       Reinald Baart
#A                                                        &Jasper Cramwinckel
#A                                                           &Erik Roijackers
##                                                              &David Joyner
##  This file contains functions for decoding codes
##
#H  @(#)$Id: decoders.gi,v 1.4 2003/02/12 03:49:19 gap Exp $
##
## Decodeword (unrestricted) modified 11-3-2004
## Decodeword (linear) created 10-2004
## Hamming, permutation, and BCH decoders moved into this file 10-2004
## GeneralizedReedSolomonDecoderGao added 11-2004
## GeneralizedReedSolomonDecoder added 11-2004
## bug fix in GRS decoder 11-29-2004
## added GeneralizedReedSolomonListDecoder and GRS<functions>, 12-2004
## added PermutationDecodeNC, CyclicDecoder 5-2005
##
Revision.("guava/lib/decoders_gi") :=
    "@(#)$Id: decoders.gi,v 1.4 2003/02/12 03:49:19 gap Exp $";


#############################################################################
##
#F  Decode( <C>, <v> )  . . . . . . . . .  decodes the vector(s) v from <C>
##
##  v can be a "codeword" or a list of "codeword"s
## 

InstallMethod(Decode, "method for unrestricted code, codeword", true, 
	[IsCode, IsCodeword], 0, 
function(C, v) 
        local c;
	c := Codeword(v, C); 
	if HasSpecialDecoder(C) then
		return SpecialDecoder(C)(C, c);
	elif IsCyclicCode(C) or IsLinearCode(C) then 
		return Decode(C, c);
	else
		Error("no decoder present");
	fi;
end);

InstallOtherMethod(Decode, 
	"method for unrestricted code, list of codewords", 
	true, [IsCode, IsList], 0, 
function(C, L) 
        local i;
	return List(L, i->Decode(C, i)); 
end); 

InstallMethod(Decode, "method for linear code, codeword", true, 
	[IsLinearCode, IsCodeword], 0, 
function(C, c) 
    local S, syn, index, corr, Gt, i, x, F;
    c := Codeword(c, C); 
	if HasSpecialDecoder(C) then 
		return SpecialDecoder(C)(C, c); 
	fi; 
	F := LeftActingDomain(C);
    S := SyndromeTable(C);
    syn := Syndrome(C, c);
    index := 0;
    repeat
        index := index + 1;
    until S[index][2] = syn;
    corr := VectorCodeword(c - S[index][1]);	     # correct codeword
    x := SolutionMat(GeneratorMat(C), corr);
    return Codeword(x,LeftActingDomain(C));
end);



#############################################################################
##
#F  Decodeword( <C>, <v> )  . . . . . . . . .  decodes the vector(s) v from <C>
##
##  v can be a "codeword" or a list of "codeword"s
## 

InstallMethod(Decodeword, "method for unrestricted code, codeword", true, 
	[IsCode, IsCodeword], 0, 
function(C, v) 
        local r, c, Cwords, NearbyWords, dist;
	r := Codeword(v, C); 
        if r in C then return r; fi;
	if HasSpecialDecoder(C) then
		return SpecialDecoder(C)(C, r);
	elif IsCyclicCode(C) or IsLinearCode(C) then 
		return Decodeword(C, r);
	else
	     dist:=Int((MinimumDistance(C)-1)/2);
             Cwords:=Elements(C);
             NearbyWords:=[];
             for c in Cwords do
               if WeightCodeword(r-c) <= dist then
                   NearbyWords:=Concatenation(NearbyWords,[r]); 
               fi;
             od;
             return NearbyWords;	
	fi;
end);


InstallOtherMethod(Decodeword, 
	"method for unrestricted code, list of codewords", 
	true, [IsCode, IsList], 0, 
function(C, L) 
        local i;
	return List(L, i->Decodeword(C, i)); 
end); 


InstallMethod(Decodeword, "method for linear code, codeword", true, 
	[IsLinearCode, IsCodeword], 0, 
function(C, v) 
    local c, S, syn, index, corr, Gt, i, x, F;
    c := Codeword(v, C); 
	if HasSpecialDecoder(C) then 
		return SpecialDecoder(C)(C, c); 
	fi; 
	F := LeftActingDomain(C);
    S := SyndromeTable(C);
    syn := Syndrome(C, c);
    index := 0;
    repeat
        index := index + 1;
    until S[index][2] = syn;
    corr := VectorCodeword(c - S[index][1]);	     # correct codeword
    return Codeword(corr,LeftActingDomain(C));
end);


##################################################################
##
##  PermutationDecode( <C>, <v> ) -  decodes the vector v  to <C>
##
##   Uses Leon's AutomorphismGroup in the binary case,
##   PermutationAutomorphismGroup in the non-binary case,
##   performs permutation decoding when possible.
##   Returns original vector and prints "fail" when not possible.
##

InstallMethod(PermutationDecode, "attribute method for linear codes", true, 
	[IsLinearCode,IsList], 0, 
function(C,v)
 #C is a linear [n,k,d] code,
 #v is a vector with <=(d-1)/2 errors
local M,G,perm,F,P,t,p0,p,pc,i,k,pv,c,H,d,G0,H0;
 G:=GeneratorMat(C);
 H:=CheckMat(C);
 d:=MinimumDistance(C);
 k:=Dimension(C);
 G0 := List(G, ShallowCopy);
 perm:=PutStandardForm(G0);
 H0 := List(H, ShallowCopy);
 PutStandardForm(H0,false);
 F:=LeftActingDomain(C);
 if F=GF(2) then 
   P:=AutomorphismGroup(C);
  else
   P:=PermutationAutomorphismGroup(C);
 fi;
 t:=Int((d-1)/2);
 p0:=0;
 if WeightCodeword(H0*v)=0 then return(v); fi;
 for p in P do
  pv:=Permuted(v,p);
  if WeightCodeword(Codeword(H0*pv))<=t then 
	p0:=p; 
	break; 
  fi;
 od;
 if p0=0 then Print("fail \n"); return(v); fi;
 pc:=TransposedMat(G0)*List([1..k],i->pv[i]);
 c:=Permuted(pc,p0^(-1));
 return Codeword(Permuted(c,perm));
end);


##################################################################
##
##
##  PermutationDecodeNC( <C>, <v>, <P> ) -  decodes the 
##                                vector v  to <C>
##
##   Performs permutation decoding when possible.
##   Returns original vector and prints "fail" when not possible.
##   NC version does Not Compute the aut gp so one must
##   input the permutation automorphism group  
##   <P> subset SymmGp(n), n=wordlength(<C>)
##
####  wdj,2-7-2005

InstallMethod(PermutationDecodeNC, "attribute method for linear codes", true, 
	[IsLinearCode,IsCodeword,IsGroup], 0, 
function(C,v,P)
 #C is a linear [n,k,d] code,
 #v is a vector with <=(d-1)/2 errors
local M,G,G0, perm,F,t,p0,p,pc,i,k,pv,c,H,H0, d;
 G:=GeneratorMat(C);
 H:=CheckMat(C);
 d:=MinimumDistance(C);
 k:=Dimension(C);
 G0 := List(G, ShallowCopy);
 perm:=PutStandardForm(G0);
 H0 := List(H, ShallowCopy);
 PutStandardForm(H0,false);
 F:=LeftActingDomain(C);
 t:=Int((d-1)/2);
 p0:=0;
 if WeightCodeword(H0*v)=0 then return(v); fi;
 for p in P do
  pv:=Permuted(v,p);
  if WeightCodeword(Codeword(H0*pv))<=t then 
	p0:=p; 
#	Print(p0," = p0 \n",pv,"\n",H*pv,"\n"); 
	break; 
  fi;
 od;
 if p0=0 then Print("fail \n"); return(v); fi;
 pc:=TransposedMat(G0)*List([1..k],i->pv[i]);
 c:=Permuted(pc,p0^(-1));
 return Codeword(Permuted(c,perm));
end);

#############################################################################
##
#F  CyclicDecoder( <C>, <w> )  . . . . . . . . . . . . decodes cyclic codes
##
InstallMethod(CyclicDecoder, "method for code, codeword", true, 
	[IsCode, IsCodeword], 0, 
function(C,w)
local d, g, wpol, s, ds, cpol, cc, c, i, m, e, x, n, ccc, r;
 if not(IsCyclicCode(C)) then
   Error("\n\n Code must be cyclic");
 fi;
 n:=WordLength(C);
 d:=MinimumDistance(C);
 g:=GeneratorPol(C);
 x:=IndeterminateOfUnivariateRationalFunction(g);
 wpol:=PolyCodeword(w);
 s:=wpol mod g;
 ds:=DegreeOfLaurentPolynomial(s);
 if ds<=Int((d-1)/2) then
   cpol:=wpol-s;
   cc:=CoefficientsOfUnivariatePolynomial(cpol);
   r:=Length(cc);
   ccc:=Concatenation(cc,List([1..(n-r)],i->0*cc[1]));
   c:=Codeword(ccc);
   return c;
 fi;
 for i in [1..(n-1)] do
   s:=x^i*wpol mod g;
   ds:=DegreeOfLaurentPolynomial(s);
   if ds<=Int((d-1)/2) then
     m:=i;
     e:=x^(n-m)*s mod (x^n-1);
     cpol:=wpol-e;
     cc:=CoefficientsOfUnivariatePolynomial(cpol);
     r:=Length(cc);
     ccc:=Concatenation(cc,List([1..(n-r)],i->0*cc[1]));
     c:=Codeword(ccc);
     return c;
   fi;
 od;
 return "fail";
end);

#############################################################################
##
#F  BCHDecoder( <C>, <r> )  . . . . . . . . . . . . . . . . decodes BCH codes
##
InstallMethod(BCHDecoder, "method for code, codeword", true, 
	[IsCode, IsCodeword], 0, 
function (C, r)
    local F, q, n, m, ExtF, x, a, t, ri_1, ri, rnew, si_1, si, snew,
          ti_1, ti, qi, sigma, i, cc, cl, mp, ErrorLocator, zero,
          Syndromes, null, pol, ExtSize, ErrorEvaluator, Fp;
    F := LeftActingDomain(C);
    q := Size(F);
    n := WordLength(C);
    m := OrderMod(q,n);
    t := QuoInt(DesignedDistance(C) - 1, 2);
    ExtF := GF(q^m);
    x := Indeterminate(ExtF);
    a := PrimitiveUnityRoot(q,n);
    zero := Zero(ExtF); 
    r := PolyCodeword(Codeword(r, n, F));
    if Value(GeneratorPol(C), a) <> zero then
        return Decode(C, r);  ##LR - inf loop !!! 
    fi;
    # Calculate syndrome: this simple line is faster than using minimal pols.
    Syndromes :=  List([1..2*QuoInt(DesignedDistance(C) - 1,2)],
                       i->Value(r, a^i));
    if Maximum(Syndromes) = Zero(F) then # no errors
        return Codeword(r / GeneratorPol(C), C);
    fi;
    # Use Euclidean algorithm:
    ri_1 := x^(2*t); 
	ri := LaurentPolynomialByCoefficients( 
			ElementsFamily(FamilyObj(ExtF)), 
			Syndromes, 0); 
	rnew := ShallowCopy(ri);
    si_1 := x^0; si := 0*x; snew := 0*x;
    ti_1 := 0*x; ti := x^0; sigma := x^0;
    while Degree(rnew) >= t do
        rnew := (ri_1 mod ri);
        qi := (ri_1 - rnew) / ri;
        snew := si_1 - qi*si;
        sigma := ti_1 - qi*ti;
        ri_1 := ri; ri := rnew;
        si_1 := si; si := snew;
        ti_1 := ti; ti := sigma;
    od;
    # Chien search for the zeros of error locator polynomial:
    ErrorLocator := []; 
    null := a^0;
    ExtSize := q^m-1;
    for i in [0..ExtSize-1] do
        if Value(sigma, null) = zero then
            AddSet(ErrorLocator, (ExtSize-i) mod n);
        fi;
        null := null * a;
    od;
    # And decode:
    if Length(ErrorLocator) = 0 then
        Error("not decodable");
    fi;
    x := Indeterminate(F);
    if q = 2 then # error locator is not necessary
        pol := Sum(List([1..Length(ErrorLocator)], i->x^ErrorLocator[i]));
        return Codeword((r - pol) / GeneratorPol(C), C);
    else
        pol := Derivative(sigma);
        Fp := One(F)*(x^n-1);
        ErrorEvaluator := List(ErrorLocator,i->
                              Value(rnew,a^-i)/Value(pol, a^-i));
        pol := Sum(List([1..Length(ErrorLocator)], i->
                       -ErrorEvaluator[i]*x^ErrorLocator[i]));
        return Codeword((r - pol) / GeneratorPol(C), C);
    fi;
end);

#############################################################################
##
#F  HammingDecoder( <C>, <r> )  . . . . . . . . . . . . decodes Hamming codes
##
##  Generator matrix must have all unit columns
## 
InstallMethod(HammingDecoder, "method for code, codeword", true, 
	[IsCode, IsCodeword], 0, 
function(C, r) 
    local H, S,p, F, fac, e,z,x,ind, i,Sf;
    S := VectorCodeword(Syndrome(C,r));
    r := ShallowCopy(VectorCodeword(r));
    F := LeftActingDomain(C);
    p := PositionProperty(S, s->s<>Zero(F)); 
    if p <> fail then
        z := Z(Characteristic(S[p]))^0;
        if z = S[p] then
            fac := One(F); 
        else
            fac := S[p]/z;
        fi;
        Sf := S/fac;
        H := CheckMat(C);
        ind := [1..WordLength(C)];
        for i in [1..Redundancy(C)] do
            ind := Filtered(ind, j-> H[i][j] = Sf[i]);
        od;
        e := ind[1];
        r[e] := r[e]-fac;     # correct error
    fi;
    x := SolutionMat(GeneratorMat(C), r);
    return Codeword(x);
end);	

#############################################################################
##
#F  GeneralizedReedSolomonDecoderGao( <C>, <v> )   . . decodes 
##                                           generalized Reed-Solomon codes
##                                           using S. Gao's method
## 
InstallMethod(GeneralizedReedSolomonDecoderGao,"method for code, codeword", true, 
	[IsCode, IsCodeword], 0, 
function(C,vec)
 local vars,a,b,n,i,g0,g1,geea,u,q,v,r,g,f,F,R,x,k,GcdExtEuclideanUntilBound; 
 if C!.name<>" generalized Reed-Solomon code" then
   Error("\N This method only applies to GRS codes.\n");
 fi;

GcdExtEuclideanUntilBound:=function(F,f,g,d,R)
## S Gao's version
## R is a poly ring
local vars,u,v,r,q,i,num,x;
 vars:=IndeterminatesOfPolynomialRing(R);
 x:=vars[1]; # define local x
 u:=List([1..3],i->Zero(F)); ## Need u[3] as a temporary variable
 v:=List([1..3],i->Zero(F));
 r:=List([1..3],i->Zero(F));
 q:=Zero(F);
 u[1]:=One(F); u[2]:=Zero(F); v[1]:=Zero(F); v[2]:=One(F);
 r[2]:=f; r[1]:=g;           ### applied with r2=f=g1, r1=g=g0
 while DegreeIndeterminate(r[2],x)> d-1 do    ### assume Degree(0) < 0.
  q   := EuclideanQuotient(R,r[1],r[2]);
  r[3]:=EuclideanRemainder(R,r[1],r[2]);
  u[3]:=u[1] - q*u[2];
  v[3]:=v[1] - q*v[2];
  r[1]:=r[2]; r[2] :=r[3];
  u[1]:=u[2]; u[2] :=u[3];
  v[1]:=v[2]; v[2] :=v[3];
 od;
 return([r[2],u[2],v[2]]);
end;

 a:=C!.points;
 R:=C!.ring;
 k:=C!.degree;
 F:=CoefficientsRing(R);
 b:=VectorCodeword(vec);
 vars:=IndeterminatesOfPolynomialRing(R);
 x:=vars[1]; # define local x
 n:=Length(a);
 if Size(Set(a)) < n then    
   Error("`Points in 1st vector must be distinct.`\n\n");  
 fi;
 g0:=One(F)*Product([1..n],i->x-a[i]);
 g1:=InterpolatedPolynomial(F,a,b);
 geea:=GcdExtEuclideanUntilBound(F,g1,g0,(n+k)/2,R);
 u:=geea[2]; v:=geea[3]; g:=geea[1];
 if v=Zero(F) then return("fail"); fi;   
 if v=One(F) then 
     q:=g; 
     r:=Zero(F); 
   else 
     q:=EuclideanQuotient(R,g,v);  
     r:=EuclideanRemainder(R,g,v);
 fi;
 if ((r=Zero(F) or LeadingCoefficient(r)=Zero(F)) and (Degree(q) < k)) then
     f:=q;
   else
     f:="fail";  ## this does not occur if num errors < (mindist - 1)/2
 fi;
 if f="fail" then return(f); else
    return Codeword(List(a,i->Value(f,i)),C);
 fi;
end);

##########################################################
#
# Input: Pts=[x1,..,xn], l = highest power,
#       L=[h_1,...,h_ell] is list of powers
#       r=[r1,...,rn] is received vector
# Output: Computes matrix described in Algor. 12.1.1 in [JH]
#
GRSLocatorMat:=function(r,Pts,L)
  local a,j,b,ell,DiagonalPower,add_col_mat,add_row_mat,block_matrix;

 add_col_mat:=function(M,N) ## "AddColumnsToMatrix"
  #N is a matrix with same rowdim as M 
  #the fcn adjoins N to the end of M
  local i,j,S,col,NT;
  col:=MutableTransposedMat(M);  #preserves M
  NT:=MutableTransposedMat(N);   #preserves N
  for j in [1..DimensionsMat(N)[2]] do
      Add(col,NT[j]);
  od;
  return MutableTransposedMat(col);
 end;

 add_row_mat:=function(M,N) ## "AddRowsToMatrix"
  #N is a matrix with same coldim as M 
  #the fcn adjoins N to the bottom of M
  local i,j,S,row;
  row:=ShallowCopy(M);#to preserve M;
  for j in [1..DimensionsMat(N)[1]] do
    Add(row,N[j]);
  od;
  return row;
 end;

 block_matrix:=function(L) ## slightly simpler syntax IMHO than "BlockMatrix"
  # L is an array of matrices of the form
  # [[M1,...,Ma],[N1,...,Na],...,[P1,...,Pa]]
  # returns the associated block matrix 
  local A,B,i,j,m,n;
  n:=Length(L[1]);
  m:=Length(L);
  A:=[];
  if n=1 then
     if m=1 then return L[1][1]; fi;
     A:=L[1][1];
     for i in [2..m] do
         A:=add_row_mat(A,L[i][1]);
     od;
     return A;
  fi;
  for j in [1..m] do
    A[j]:=L[j][1];
  od;
  for j in [1..m] do
   for i in [2..n] do
    A[j]:=add_col_mat(A[j],L[j][i]);
   od;
  od;
  B:=A[1];
  for j in [2..m] do
   B:= add_row_mat(B,A[j]);
  od;
  return B;
 end;

 DiagonalPower:=function(r,j)
  local R,n,i;
  n:=Length(r);
  R:=DiagonalMat(List([1..n],i->r[i]^j));
  return R;
 end;

  ell:=Length(L);
  a:=List([1..ell],j->DiagonalPower(r,(j-1))*VandermondeMat(Pts,L[j]));
  b:=List([1..ell],j->[1,j,a[j]]);
  return (block_matrix([a]));  
end; 

##############################################################
#
#
# Input: Pts=[x1,..,xn], 
#       L=[h_1,...,h_ell] is list of powers
#       r=[r1,...,rn] is received vector
#
# Compute kernel of matrix in alg 12.1.1 in [JH].
# Choose a basis vector in kernel.  
# Output: the lists of coefficients of the polynomial Q(x,y) in alg 12.1.1.  
#
GRSErrorLocatorCoeffs:=function(r,Pts,L)
  local a,j,b,vec,e,QC,i,lengths,ker,ell;
  ell:=Length(L); 
  e:=GRSLocatorMat(r,Pts,L);
  ker:=TriangulizedNullspaceMat(TransposedMat(e));
  if ker=[] then Print("Decoding fails.\n"); return []; fi;
  vec:=ker[Length(ker)];
  QC:=[];
  lengths:=List([1..ell],i->Sum(List([1..i],j->1+L[j])));
  QC[1]:=List([1..lengths[1]],j->vec[j]);
  for i in [2..ell] do
  QC[i]:=List([(lengths[i-1]+1)..lengths[i]],j->vec[j]);
  od;
  return QC;
end;


#######################################################
#
# Input: List L of coefficients ell, R = F[x]
#        Pts=[x1,..,xn], 
# Output: list of polynomials Qi as in Algor. 12.1.1 in [JH]
# 
GRSErrorLocatorPolynomials:=function(r,Pts,L,R)
  local q,p,i,ell;
  ell:=Length(L)+1; ##  ?? Length(L) instead ??
  q:=GRSErrorLocatorCoeffs(r,Pts,L);
  if q=[] then Print("Decoding fails.\n"); return []; fi;
  p:=[];
  for i in [1..Length(q)] do 
     p:=Concatenation(p,[CoefficientToPolynomial(q[i],R)]);
  od;
  return p;
end;

##########################################################
#
# Input: List L of coefficients ell, R = F[x]
#       Pts=[x1,..,xn], 
# Output: interpolating polynomial Q as in Algor. 12.1.1 in [JH]
# 
GRSInterpolatingPolynomial:=function(r,Pts,L,R)
  local poly,i,Ry,F,y,Q,ell;
  ell:=Length(L)+1; ##  ?? Length(L) instead ??  ##### not used ???
Q:=GRSErrorLocatorPolynomials(r,Pts,L,R);
  if Q=[] then Print("Decoding fails.\n"); return 0; fi;
  F:=CoefficientsRing(R);
  y:=IndeterminatesOfPolynomialRing(R)[2];
# Ry:=PolynomialRing(F,[y]);
# poly:=CoefficientToPolynomial(Q,Ry);
  poly:=Sum(List([1..Length(Q)],i->Q[i]*y^(i-1)));
  return poly;
end;

#############################################################################
##
#F  GeneralizedReedSolomonDecoder( <C>, <v> )   . . decodes 
##                                           generalized Reed-Solomon codes
##                                           using the interpolation algorithm
## 
InstallMethod(GeneralizedReedSolomonDecoder,"method for code, codeword", true, 
	[IsCode, IsCodeword], 0, 
function(C,vec)
local v,R,k,P,z,F,f,s,t,L,n,Qpolys,vars,x,c,y;

 v:=VectorCodeword(vec);
 R:=C!.ring;
 P:=C!.points;
 k:=C!.degree;
 F:=CoefficientsRing(R);
 vars:=IndeterminatesOfPolynomialRing(R);
 x:=vars[1];
#y:=vars[2];
 n:=Length(v);
 t:=Int((n-k)/2);
 L:=[n-1-t,n-t-k];
 Qpolys:=GRSErrorLocatorPolynomials(v,P,L,R);
 f:=-Qpolys[2]/Qpolys[1];
 c:=List(P,s->Value(f,[x],[s]));
 return Codeword(c,n,F);
end);


#############################################################################
##
#F  GeneralizedReedSolomonListDecoder( <C>, <v> , <ell> )   . . ell-list decodes 
##                                           generalized Reed-Solomon codes
##                                           using M. Sudan's algorithm
## 
##
## Input: v is a received vector (a GUAVA codeword)
##        C is a GRS code
##        ell>0 is the length of the decoded list (should be at least
##                  2 to beat GeneralizedReedSolomonDecoder
##                  or Decoder with the special method of interpolation decoding)
##
InstallMethod(GeneralizedReedSolomonListDecoder,"method for code, codeword, integer", 
    true, [IsCode, IsCodeword, IsInt], 0, 
function(C,v,ell)
local f,h,g,x,R,R2,L,F,t,i,c,Pts,k,n,tau,Q,divisorsf,div,
      CodewordList,p,vars,y,degy, divisorsdeg1;
 R:=C!.ring;
 F:=CoefficientsRing(R);
 vars:=IndeterminatesOfPolynomialRing(R);
 x:=vars[1]; 
 Pts:=C!.points;
 n:=Length(Pts);
 k:=C!.degree; 
 tau:=Int((n-k)/2);
 L:=List([0..ell],i->n-tau-1-i*(k-1));
 y:=X(F,vars);;
 R2:=PolynomialRing(F,[x,y]);
 vars:=IndeterminatesOfPolynomialRing(R2);
 Q:=GRSInterpolatingPolynomial(v,Pts,L,R2); 
 divisorsf:=DivisorsMultivariatePolynomial(Q,R2);
 divisorsdeg1:=[];
 CodewordList:=[];
 for div in divisorsf do
  degy:=DegreeIndeterminate(div,y);
  if degy=1 then ######### div=h*y+g
    g:=Value(div,vars,[x,Zero(F)]);
    h:=Derivative(div,y);
   if DegreeIndeterminate(h,x)=0 then
      f:= -h^(-1)*g*y^0;
      divisorsdeg1:=Concatenation(divisorsdeg1,[f]);
    if g=Zero(F)*x then
       c:=List(Pts,p->Zero(F));
     else
       c:=List(Pts,p->Value(f,[x,y],[p,Zero(F)]));
    fi;
    CodewordList:=Concatenation(CodewordList,[Codeword(c,C)]);
   fi;
  fi;
 od;
 return CodewordList;
end);

#############################################################################
##
#F  NearestNeighborGRSDecodewords( <C>, <v> , <dist> )   . . . finds all 
##                                           generalized Reed-Solomon codewords
##                                           within distance <dist> from v
##                                           *and* the associated polynomial,
##                                           using "brute force"
##
## Input: v is a received vector (a GUAVA codeword)
##        C is a GRS code
##        dist>0 is the distance from v to search
## Output: a list of pairs [c,f(x)], where wt(c-v)<dist
##         and c = (f(x1),...,f(xn))
##
## ****** very slow*****
##
InstallMethod(NearestNeighborGRSDecodewords,"method for code, codeword, integer", 
    true, [IsCode, IsCodeword, IsInt], 0, 
function(C,r,dist)
# "brute force" decoder
local k,F,Pts,v,p,x,f,NearbyWords,c,a;
 k:=C!.degree;
 Pts:=C!.points;
 F:=LeftActingDomain(C);
 NearbyWords:=[];
 for v in F^k do
   a := Codeword(v); 
   f:=PolyCodeword(a);
   x:=IndeterminateOfLaurentPolynomial(f);
   c:=Codeword(List(Pts,p->Value(f,[x],[p])));
   if WeightCodeword(r-c) <= dist then
   NearbyWords:=Concatenation(NearbyWords,[[c,f]]); 
 fi;
od;
return NearbyWords;
end);

#############################################################################
##
#F  NearestNeighborDecodewords( <C>, <v> , <dist> )   . . . finds all 
##                                           codewords in a linear code C 
##                                           within distance <dist> from v
##                                           using "brute force"
##
## Input: v is a received vector (a GUAVA codeword)
##        C is a linear code
##        dist>0 is the distance from v to search
## Output: a list of c in C, where wt(c-v)<dist
##
## ****** very slow*****
##
InstallMethod(NearestNeighborDecodewords,"method for code, codeword, integer", 
    true, [IsCode, IsCodeword, IsInt], 0, 
function(C,r,dist)
# "brute force" decoder
local k,F,Pts,v,p,x,f,NearbyWords,c,a;
 F:=LeftActingDomain(C);
 NearbyWords:=[];
 for v in C do
   c := Codeword(v); 
   if WeightCodeword(r-c) <= dist then
   NearbyWords:=Concatenation(NearbyWords,[c]); 
 fi;
od;
return NearbyWords;
end);

