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#############################################################################
##
#A util2.gi GUAVA library Reinald Baart
#A &Jasper Cramwinckel
#A &Erik Roijackers
## &David Joyner
##
## This file contains miscellaneous functions
##
## minor bug in SortedGaloisFieldElements fixed 9-24-2004
## several functions added 12-16-2005 MostCommonInList,
## MultiplicityInList, VandermondeMat
## added 5-13-2005:
## RotateList, CirculantMatrix, ZechLog,
## MatrixRepresentationOfElement,IrreduciblePolynomialsNr,
## PrimitivePolynomialsNr, TraceCode, CodeLength
## added 12-4-2005, 1-3-206: GuavaVersion. guava_version
## 22 May 2006 (CJ): modified GuavaVersion so that the version
## information is obtained directly from "PackageInfo"
## 26 Dec 2007 (CJ): added RightRotateList and NegacirculantMatrix
## functions
##
########################################################################
##
#F AllOneVector( <n> [, <field> ] )
##
## Return a vector with all ones.
##
InstallMethod(AllOneVector, "length, Field", true, [IsInt, IsField], 0,
function(n, F)
if n <= 0 then
Error( "AllOneVector: <n> must be a positive integer" );
fi;
return List( [ 1 .. n ], x -> One(F) );
end);
InstallOtherMethod(AllOneVector, "length, fieldsize", true, [IsInt, IsInt], 0,
function(n, q)
return AllOneVector(n, GF(q));
end);
InstallOtherMethod(AllOneVector, "length", true, [IsInt], 0,
function(n)
return AllOneVector(n, Rationals);
end);
########################################################################
##
#F AllOneCodeword( <n>, <field> )
##
## Return a codeword with <n> ones.
##
InstallMethod(AllOneCodeword, "wordlength, field", true, [IsInt, IsField], 0,
function(n, F)
if n <= 0 then
Error( "AllOneCodeword: <n> must be a positive integer" );
fi;
return Codeword( AllOneVector( n, F ), F );
end);
InstallOtherMethod(AllOneCodeword, "wordlength, fieldsize", true,
[IsInt, IsInt], 0,
function(n, q)
return AllOneCodeword(n, GF(q));
end);
InstallOtherMethod(AllOneCodeword, "wordlength", true, [IsInt], 0,
function(n)
return AllOneCodeword(n, GF(2));
end);
#############################################################################
##
#F IntCeiling( <r> )
##
## Return the smallest integer greater than or equal to r.
## 3/2 => 2, -3/2 => -1.
##
InstallMethod(IntCeiling, "method for integer", true, [IsInt], 0,
function(r)
# don't round integers
return r;
end);
InstallMethod(IntCeiling, "method for rational", true, [IsRat], 0,
function(r)
if r > 0 then
# round positive numbers to smallest integer
# greater than r (3/2 => 2)
return Int(r)+1;
else
# round negative numbers to smallest integer
# greater than r (-3/2 => -1)
return Int(r);
fi;
end);
########################################################################
##
#F IntFloor( <r> )
##
## Return the greatest integer smaller than or equal to r.
## 3/2 => 1, -3/2 => -2.
##
InstallMethod(IntFloor, "method for integer", true, [IsInt], 0,
function(r)
# don't round integers
return r;
end);
InstallMethod(IntFloor, "method for rational", true, [IsRat], 0,
function(r)
if r > 0 then
# round positive numbers to largest integer
# smaller than r (3/2 => 1)
return Int(r);
else
# round negative numbers to largest integer
# smaller than r (-3/2 => -2)
return Int(r-1);
fi;
end);
########################################################################
##
#F KroneckerDelta( <i>, <j> )
##
## Return 1 if i = j,
## 0 otherwise
##
InstallMethod(KroneckerDelta, true, [IsInt, IsInt], 0,
function ( i, j )
if i = j then
return 1;
else
return 0;
fi;
end);
########################################################################
##
#F BinaryRepresentation( <elements>, <length> )
##
## Return a binary representation of an element
## of GF( 2^k ), where k <= length.
##
## The representation is actually the binary
## representation of k+1, where k is the exponent
## of the element, taken in the field 2^length.
## For example, Z(16)^10 = Z(4)^2. If length = 4,
## the binary representation 1011 = 11(base 10)
## is returned. If length = 2, the binary
## representation 11 = 3(base 10) is returned.
##
## If elements is a list, then return the binary
## representation of every element of the list.
##
## This function is used to make to Gabidulin codes.
## It is not intended to be a global function, but including
## it in all five Gabidulin codes is a bit over the top
##
## Therefore, no error checking is done.
##
BinaryRepresentation := function ( elementlist, length )
local field, i, log, vector, element;
if IsList( elementlist ) then
return( List( elementlist,
x -> BinaryRepresentation( x, length ) ) );
else
element := elementlist;
field := Field( element );
vector := NullVector( length, GF(2) );
if element = Zero(field) then
# exception, log is not defined for zeroes
return vector;
else
log := LogFFE( element, Z(2^length) ) + 1;
for i in [ 1 .. length ] do
if log >= 2^( length - i ) then
vector[ i ] := One(GF(2));
log := log - 2^( length - i );
fi;
od;
return vector;
fi;
fi;
end;
########################################################################
##
#F SortedGaloisFieldElements( <size> )
##
## Sort the field elements of size <size> according to
## their log.
##
## This function is used to make to Gabidulin codes.
## It is not intended to be a global function, but including
## it in all five Gabidulin codes is not a good idea.
##
SortedGaloisFieldElements := function ( size )
local field, els, sortlist, alpha;
if IsInt( size ) then
field := GF( size );
else
field := size;
size := Size( field );
fi;
alpha:=PrimitiveRoot( field );
# this line was moved from immed after the local statement 9-2004
els := ShallowCopy(AsSSortedList( field ));
sortlist := NullVector( size );
# log 1 = 0, so we add one to each log to avoid
# conflicts with the 0 for zero.
sortlist := List( els, function( x )
if x = Zero(field) then
return 0;
else
return LogFFE( x, alpha ) + 1;
fi;
end );
sortlist{ [ 2 .. size ] } := List( els { [ 2 .. size ] },
x -> LogFFE( x, alpha ) + 1 );
SortParallel( sortlist, els );
return els;
end;
########################################################################
##
#F VandermondeMat( <Pts> , <a> )
##
## Input: Pts=[x1,..,xn], a >0 an integer
## Output: Vandermonde matrix (xi^j),
## for xi in Pts and 0 <= j <= a
## (an nx(a+1) matrix)
##
InstallMethod(VandermondeMat, true, [IsList, IsInt], 0,
function(Pts,a)
local V,n,i,j;
n:=Length(Pts);
V:=List([1..(a+1)],j->List([1..n],i->Pts[i]^(j-1)));
return TransposedMat(V);
end);
###########################################################
##
#F MultiplicityInList(L,a)
##
## Input: a list L
## an element a of L
## Output: the multiplicity a occurs in L
##
MultiplicityInList:=function(L,a)
local mult,b;
mult:=0;
for b in L do
if b=a then mult:=mult+1; fi;
od;
return mult;
end;
###########################################################
##
#F MostCommonInList(L,a)
##
## Input: a list L
## Output: an a in L which occurs at least as much as any other in L
##
MostCommonInList:=function(L)
local mults,max,maxi,x;
mults:=List(L,x->MultiplicityInList(L,x));
max:=Maximum(mults);
maxi:=Position(mults,max);
return L[maxi];
end;
###########################################################
##
#F RotateList(L)
##
## Input: a list L
## Output: cyclic rotation of the list (to the right)
##
RotateList:=function(L)
local rL,i,n;
n:=Length(L);
rL:=[];
for i in [1..n] do
if i<n then rL[i]:=L[i+1]; fi;
if i=n then rL[i]:=L[1]; fi;
od;
return rL;
end;
###########################################################
##
#F RightRotateList(L)
##
## Input: a list L
## Output: cyclic rotation of the list (to the right)
## RotateList function seems to rotate to the left ??
## Am I losing my direction now? left, right?
##
RightRotateList:=function(L)
local rL,i,n;
n :=Length(L);
rL:=[];
rL[1] := L[n];
for i in [1..(n-1)] do
rL[n-i+1] := L[n-i];
od;
return rL;
end;
###########################################################
##
#F CirculantMatrix(k,L)
##
## Input: integer k, a list L of length n
## Output: kxn matrix whose rows are cyclic rotations of the list L
##
CirculantMatrix:=function(k,L)
local M,i,rL;
rL:=L;
M:=[L];
for i in [1..(k-1)] do
rL:=RightRotateList(rL);
M:=Concatenation(M,[rL]);
od;
return M;
end;
###########################################################
##
#F NegacirculantMatrix(k,L)
##
## Input: integer k, a list L of length n
## Output: kxk matrix whose rows are cyclic (right) rotations
## of the list L
## NOTE: RotateList function is not used as it generates
## cyclic rotation to the left. RightRotateList
## function is used instead.
##
NegacirculantMatrix:=function(k,L)
local M, rL, i;
if k <> Size(L) then
Error("Negacirculant matrix must be a square matrix\n");
fi;
rL:= L;
M := [L];
for i in [1..(k-1)] do;
# generate nega cyclic shift (to the right)
rL := RightRotateList(rL);
rL[1] := -rL[1];
M := Concatenation(M, [rL]);
od;
return M;
end;
###############################################
### some "convenience functions":
CodeLength:= function (C)
return WordLength(C);
end;
TraceCode:= function (C,F)
# Input: C is a linear code defined over an extension E of F
# (F is base field)
# Output: The linear code generated by Tr_{E/F}(c), c in C
# ****extremely slow**** so hesitant to include in codesfun.gi
local FF, TC, TrC, i, n, c;
n:= WordLength(C);
FF:= LeftActingDomain(C);
TC:= List(C,c->Codeword(List([1..n],i->Trace(FF,F,c[i]))));
TrC:=ElementsCode(TC, F);
IsLinearCode(TrC);
return TrC;
end;
###################### finite field functions
## see also ConwayPolynomial
## IsPrimitivePolynomial, for p < 256
## IsCheapConwayPolynomial
## RandomPrimitivePolynomial
##
PrimitivePolynomialsNr:=function(n,q)
#n is an integer>1
#F is a finite field
#FACT (Golomb): The number of irreducible polynomials mod p of degree n
#with (maximum) period p^n-1 is lambda(n,p)=phi(p^n-1)/n).
local period;
#q:=Size(F);
if n<2 then
Error("\n\n First arg must be > 1.\n");
fi;
return Phi(q^n-1)/n;
end;
IrreduciblePolynomialsNr:=function(n,q)
#FACT (Golomb): The number of irreducible polynomials in GF(q)[x] of degree n
#is Psi0(n,p).
local d;
return (1/n)*Sum(DivisorsInt(n),d->q^d*MoebiusMu(n/d));
end;
MatrixRepresentationOfElement:=function(a,F)
#returns the matrix representation of
#the element a
local q,p,d,A,i,j,f,coeffs,c0,Id;
p:=Characteristic(F);
if p>0 then
q:=Size(F);
# d:=LogInt(q,p);
f:=MinimalPolynomial(GF(p),a);
A:=CompanionMat(f);
# Id:=IdentityMat(d,F);
return A;
fi;
if p=0 then
f:=MinimalPolynomial(Rationals,a);
A:=CompanionMat(f);
# Id:=IdentityMat(d,F);
return A;
fi;
end;
ZechLog:=function(x,b,F)
#Zech log of x to base b, ie the i such that x+1=b^i,
# so y+z=y*(1+z/y)=b^k, where k=Log(y,b)+ZechLog(z/y,b)
# b must be a primitive element of F
return LogFFE(x+One(F),b);
end;
GuavaVersion:=function()
# prints current version
#local version;
# version:="2.5";
# return version;
return PackageInfo("guava")[1].Version;
end;
guava_version:=function()
# prints current version
return GuavaVersion();
end;
InstallMethod(SupportToCodeword, "set, wordlength", true, [IsSet, IsInt], 0,
function(s, n)
local v,i;
v := NullVector(n,GF(2));
for i in s do
if (i > n) or (i<1) then
Error("Support elements should lie between 1 and n");
fi;
v[i]:=1;
od;
return Codeword(v,n,GF(2));
end);
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