The goal of the project is to write a program to perform list-decoding of Reed-Solomon codes using the Sudan Algorithm [S] and implement this program into the GAP coding-theory package known as GUAVA [GUA]. The Sudan Algorithm is Algorithm 12.1.1 in Justesen and Høholdt's recent book [JH]. GAP is a computer algebra package whose open source kernel is written in the C programming language [GAP]. However, the GUAVA package is written in GAP's own interpreted language. Neither GAP nor the GUAVA package contain a program for list-decoding of generalized Reed-Solomon codes. When implemented, this program will greatly increase GAP's speed in decoding Reed-Solomon codes. We will also discuss generalizations, both to the higher rate case (Algorithm 2) and the multivariate case (Algorithm 3).
Let denote a prime power. A finite field is a finite set of elements with operations of addition and multiplication which satisfy the properties of a field. Let denote a finite field with elements. A linear code is simply a finite dimensional vector space over a finite field, and its elements are called codewords. If , then we say has length . Moreover, if , then we call a code. A matrix whose rows form a basis of a linear code is called a generator matrix of the code.
Let be a linear code of length over with generator matrix , where is a power of a prime . If then the code is called binary. We assume that has been given the standard basis , , ..., . If the dimension of is , then the number of elements of is equal to . The quantity is called the rate of the code and measures the amount of information which the code can transmit. For instance, the code in the above example has rate .
Another important parameter associated to the code is the number of errors which it can, in principle, correct. For this notion, we need to introduce the Hamming metric. For any two , let denote the number of coordinates where these two vectors differ:
(1) |
We call the minimum distance of a code , denoted , the smallest distance between distinct codewords in . There exists and such that . Note that , where is the minimum weight of a codeword in . Also, for some other codeword , . Therefore, . Now, we see that the minimum distance of satisfies
(2) |
It is easy to check that is a linear code.
Since , by the assumption in Definition 2, the dimension of the Reed-Solomon code is equal to the same used in ([JH], p50). A code is MDS if its parameters satisfy the Singleton bound with equality ([JH], p49). Generalized Reed-Solomon codes are MDS codes, so is easily computed in terms of the other parameters. (Please see [McG] for more details on this.)
Reed-Solomon codes were discovered in 1959 and have applications in CDs, DVDs, and satellite communications among other things ([JH], p49). The GUAVA package does not contain any programs which provide fast list-decoding of Reed-Solomon codes. In fact, to our knowledge, this list decoder has not yet been implemented in any computer algebra system. A fast decoder for generalized Reed-Solomon codes has been recently implemented by J. McGowan [McG]. However, McGowan's program does not perform list-decoding. The optimal method of decoding these codes utilizes list decoding with the Sudan Algorithm. Instead of using brute force to examine all codewords to find the ones closest to the received vector, list decoding uses systems of linear equations and polynomial interpolation. This method of using brute force is known as nearest neighbor decoding for which the received vector is decoded as the codeword so that is the minimum distance ([HILL], p5). This method returns a list of codewords within some fixed distance of the received vector.
Let be a generalized Reed-Solomon code with parameters . Let .
Now we discuss a generalization of the algorithm implemented in [McG]. See his discussion for further details of the case .
We use the notation in Definition 2. Let be a received word where is a codeword in . Assume that the weight of the error vector is less than or equal to , . In other words, represents the maximum number of errors which the algorithm below can correct. The idea is to determine a non-zero polynomial
Such a polynomial (depending on ) is called an interpolating polynomial.
Here represents the number of codewords near which the algorithm below returns. (When , the codeword closest to is returned. When the two codewords closest to are returned, and so on.)
Our next aim will be to show that such a non-zero polynomial exists under certain conditions (to be made explicit below).
proof: By definition of and the fact that , the polynomial has degree at most . Since , except in at most cases, we have that , for at least of the in . This forces for all , since a non-zero polynomial of degree can have at most zeroes. Therefore, is a root of the polynomial . If we consider the polynomial as a polynomial in over , the division algorithm over the ring implies that divides .
This lemma means that any codeword as above is associated with a factor of .
Under what conditions on and does such an interpolating polynomial exist? Let us regard all the coefficients of as unknowns. The definition of tells us that the number of unknowns is determined by the conditions (2). Therefore, the number of unknowns is
The most interesting case is when the number of correctable errors, , is greater than , since otherwise the Nearest Neighbor Algorithm applies. (Please see [I] for a discussion of this.)
We now determine when . This condition forces
Another condition that arises from the condition (2) with , namely . This implies .
Summarizing the above, list decoding applies only when has ``low rate'' and moreover beats the Nearest Neighbor Algorithm when .
(3) |
The GAP implementation is below. This factorization routine is not the optimal one.
Examples of this algorithm are given in the next section.
In this section we give examples of a GUAVA implementation of Sudan's Algorithm 1.
Let be a primitive element of where and consider the Reed-Solomon code obtained by evaluating polynomials of degree at most 2 in the powers of . The code has minimum distance 13 and thus is 6-error correcting. With it is possible to decode up to seven errors with list size at most 2. Suppose is the received vector . Solving the system in step 2 then plugging the results into the polynomial in step 3 gives . Since the linear factors of are and , the functions in step 4 are and . Therefore, by step 5, the corresponding two codewords with 7 errors corrected are:
We used our GAP code to test the above example. The following parameters were used when testing the example.
gap> F:=GF(16); GF(2^4) gap> a:=PrimitiveRoot(F);; b:=a^7; b^4+b^3+1; ## alpha in JH Ex 12.1.1, pg 129 Z(2^4)^7 0*Z(2) gap> Pts:=List([0..14],i->b^i); [ Z(2)^0, Z(2^4)^7, Z(2^4)^14, Z(2^4)^6, Z(2^4)^13, Z(2^2), Z(2^4)^12, Z(2^4)^4, Z(2^4)^11, Z(2^4)^3, Z(2^2)^2, Z(2^4)^2, Z(2^4)^9, Z(2^4), Z(2^4)^8 ] gap> R1:=PolynomialRing(F,1);; gap> vars:=IndeterminatesOfPolynomialRing(R1);; gap> x:=vars[1]; x_1 gap> y:=Indeterminate(F,vars);; gap> R2:=PolynomialRing(F,[x,y]);; gap> C:=GeneralizedReedSolomonCode(Pts,3,R1); MinimumDistance(C); a linear [15,3,1..13]10..12 generalized Reed-Solomon code over GF(16) 13 gap> z:=Zero(F); 0*Z(2) gap> r:=[z,z,z,z,z,z,z,z,b^6,b^2,b^5,b^14,b,b^7,b^11];; ## as in JH Ex 12.1.1 gap> r:=Codeword(r); [ 0 0 0 0 0 0 0 0 a^12 a^14 a^5 a^8 a^7 a^4 a^2 ] gap> cs1:=NearestNeighborGRSDecodewords(C,r,7); time; [ [ [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ], 0*Z(2) ], [ [ 0 a^9 a^3 a^13 a^6 a^10 a^11 a a^12 a^14 a^5 a^8 a^7 a^4 a^2 ], x_1+Z(2)^0 ] ] 1556 gap> cs2:=GeneralizedReedSolomonListDecoder(C,r,2); time; [ [ 0 a^9 a^3 a^13 a^6 a^10 a^11 a a^12 a^14 a^5 a^8 a^7 a^4 a^2 ], [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] ] 151
This verifies the above example.
For the next example we use a slightly different received vector. One of the codewords listed will actually correct 8 errors.
gap> F:=GF(16); GF(2^4) gap> a:=PrimitiveRoot(F);; b:=a^7; b^4+b^3+1; ## alpha in JH Ex 12.1.1, pg 129 Z(2^4)^7 0*Z(2) gap> Pts:=List([0..14],i->b^i); [ Z(2)^0, Z(2^4)^7, Z(2^4)^14, Z(2^4)^6, Z(2^4)^13, Z(2^2), Z(2^4)^12, Z(2^4)^4, Z(2^4)^11, Z(2^4)^3, Z(2^2)^2, Z(2^4)^2, Z(2^4)^9, Z(2^4), Z(2^4)^8 ] gap> R1:=PolynomialRing(F,1);; gap> vars:=IndeterminatesOfPolynomialRing(R1);; gap> x:=vars[1]; x_1 gap> y:=Indeterminate(F,vars);; gap> R2:=PolynomialRing(F,[x,y]);; gap> C:=GeneralizedReedSolomonCode(Pts,3,R1); MinimumDistance(C); a linear [15,3,1..13]10..12 generalized Reed-Solomon code over GF(16) 13 gap> z:=Zero(F); 0*Z(2) gap> r:=[z,z,z,z,z,z,z,b^13,b^6,b^2,b^5,b^14,b,b^7,b^11];; gap> r:=Codeword(r); [ 0 0 0 0 0 0 0 a a^12 a^14 a^5 a^8 a^7 a^4 a^2 ] gap> cs1:=NearestNeighborGRSDecodewords(C,r,7); time; [ [ [ 0 a^9 a^3 a^13 a^6 a^10 a^11 a a^12 a^14 a^5 a^8 a^7 a^4 a^2 ], x_1+Z(2)^0 ] ] 1570 gap> cs2:=GeneralizedReedSolomonListDecoder(C,r,2); time; [ [ 0 a^9 a^3 a^13 a^6 a^10 a^11 a a^12 a^14 a^5 a^8 a^7 a^4 a^2 ], [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] ] 147 gap> c1:=cs2[1]; c1 in C; [ 0 a^9 a^3 a^13 a^6 a^10 a^11 a a^12 a^14 a^5 a^8 a^7 a^4 a^2 ] true gap> c2:=cs2[2]; c2 in C; [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] true gap> WeightCodeword(c1-r); 6 gap> WeightCodeword(c2-r); 8
The time to run the program that finds the nearest neighbor codewords by brute force (1570 ``gapstones'') is much longer than the time to run the program which uses Sudan's algorithm (147 ``gapstones'') for this example.
Now let us try to use the standard command Decodeword
to decode the
received vector
. We type into GUAVA the following command:
gap> Decodeword(C,r); time; Error, Denominator evaluates as zero called from Value( rf, inds, vals, One( CoefficientsFamily( FamilyObj( rf ) ) ) ) called from Value( f, [ x ], [ s ] ) called from func( elm ) called from List( P, function ( s ) return Value( f, [ x ], [ s ] ); end ) called from SpecialDecoder( C )( C, c ) called from ... Entering break read-eval-print loop ... you can 'quit;' to quit to outer loop, or you can 'return;' to continue brk>
The error that GUAVA gives us here indicates that it is not possible to use ordinary decoding with a received word with errors, because . However, list decoding does work in this case.
This section contains related algorithms which will be briefly discussed but not implemented in this project.
(4) |
(5) |
This is an improvement on Algorithm 1 since the Guruswami-Sudan Algorithm works for codes with any rate, but the Sudan Algorithm only works for RS codes with low rates ([JH], p130).
In the remainder of this section we speculate on Sudan's generalization to polynomials in more than one variable. No proofs will be given. For details please see §5 of [S].
In general terms, the idea of the higher-dimensional generalization is the following:
Input: a finite field , a subset , the dimension , a function representing the received vector , the number of coordinates where the received vector is correct , and the ``weighted degree'' of the code .
Output: All multivariate polynomials , , such that where denotes a weighted degree.
A more precise version is stated below.
Consider the following type of evaluation code
Here is the more precise version of Sudan's Algorithm, as it applies to this case.
Let L denote the list of all polynomials with weighted degree such that divides .
Output: The list of codewords If represents the received word then each satisfies .
Finally, note that the curves [Crv] GAP package command
SolveLinearEquations
can be used to solve for the coefficients of
in the system of equations
.
# List decoder for RS codes using Sudan-Guruswami algorithm # (this implementation only works for low rate codes) # ######################################################## #Input: List coeffs of coefficients, R = F[x] #Output: polynomial L[0]+L[1]x+...+L[d]x^d # CoefficientToPolynomial:=function(coeffs,R) local p,i,j, lengths, F,xx; xx:=IndeterminatesOfPolynomialRing(R)[1]; F:=Field(coeffs); p:=Zero(F); # lengths:=List([1..Length(coeffs)],i->Sum(List([1..i],j->1+coeffs[j]))); for i in [1..Length(coeffs)] do p:=p+coeffs[i]*xx^(i-1); od; return p; end; #Input: Pts=[x1,..,xn], a = element of L #Output: Vandermonde matrix (xi^j) # VandermondeMat:=function(Pts,a) ## returns an nx(a+1) matrix 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; #Input: Xlist=[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] # LocatorMat:=function(r,Pts,L) ## returns an nx(ell+sum(L)) matrix local a,j,b,ell,add_col_mat,add_row_mat,block_matrix,diagonal_power; 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) ## "MakeBlockMatrix" #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; diagonal_power:=function(r,j) ## returns an nxn matrix local A,n,i; n:=Length(r); A:=DiagonalMat(List([1..n],i->r[i]^j)); return A; end; ell:=Length(L); a:=List([1..ell],j->diagonal_power(r,(j-1))*VandermondeMat(Pts,L[j])); b:=List([1..ell],j->[1,j,a[j]]); return block_matrix([a]); end; # Compute kernel of matrix in alg 12.1.1 in [JH]. # Choose a basis vector in kernel. # Construct the polynomial Q(x,y) in alg 12.1.1. # ErrorLocatorCoeffs:=function(r,Pts,L) local a,j,b,vec,e,QC,i,lengths,ker,ell; ell:=Length(L); e:=LocatorMat(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] # Xlist=[x1,..,xn], #Output: list of polynomials Qi as in Algor. 12.1.1 in [JH] # ErrorLocatorPolynomials:=function(r,Pts,L,R) local q,p,i,ell; ell:=Length(L)+1; ## ?? Length(L) instead ?? q:=ErrorLocatorCoeffs(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] # InterpolatingPolynomialGRS:=function(r,Pts,L,R) local poly,i,Ry,F,y,Q,ell; ell:=Length(L)+1; Q:=ErrorLocatorPolynomials(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; GeneralizedReedSolomonListDecoder:=function(C,v,ell) # # 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) # 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:=InterpolatingPolynomialGRS(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); # h:=(div-g)/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)])); CodewordList:=Concatenation(CodewordList,[Codeword(c,C)]); fi; fi; od; return CodewordList; end; ###################################################### NearestNeighborGRSDecodewords:=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; NearestNeighborDecodewords:=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 F^k do c := Codeword(v); if WeightCodeword(r-c) <= dist then NearbyWords:=Concatenation(NearbyWords,[c]); fi; od; return NearbyWords; end;
The command line arguments were:
latex2html -t 'C. Lennon Math Honors Thesis 2004-2005' -split 0 lennon_mathhonors2004-2005.tex
The translation was initiated by David Joyner on 2005-04-06