@comment this file contains the mathematical background of decoding with Groebner bases

This section introduces some of the mathematical notions, definitions, and 
results for solving decoding problems and finding the minimum distance of
linear (and in particular cyclic) codes. The material presented here
should assist the user in working with @ref{decodegb_lib}.
More details can be obtained from @ref{[BP2008b]}.


@menu
* Codes and the decoding problem::
* Cooper philosophy::
* Generalized Newton identities::
* Fitzgerald-Lax method::
* Decoding method based on quadratic equations::
* References for decoding with Groebner bases::
@end menu

@c ---------------------------------------------------------------------------
@node Codes and the decoding problem, Cooper philosophy, , Decoding codes with Groebner bases 
@subsection Codes and the decoding problem
@cindex Codes and the decoding problem

@subsubheading Codes
@cindex Code
@c table @asis
@itemize @bullet
@item 
@tex 
  Let $F_q$ be a field with $q$ elements. A \emph{linear code} $C$ is a linear subspace of $F_q^n$ endowed with the
  {\bf Hamming metric}.  
@end tex
@ifinfo
  Let F_q be a field with q elements. A @strong{linear code} C is a linear subspace of F_q^n endowed with the
  @strong{Hamming metric}
@end ifinfo
@item 
@tex
  {\bf Hamming distance} between {\bf x,y}$\in F_q^n: d(x,y)=\#\{i|x_i\ne y_i\}$. 
  {\bf Hamming weight} of {\bf x}$\in F_q^n: wt(x)=\#\{i|x_i\ne 0\}$.
@end tex
@ifinfo
  @strong{Hamming distance} between x, y in F_q^n: d(x,y)=@{i|x_i<> y_i@},
  @strong{Hamming weight} of x in F_q^n: wt(x)=@{i|x_i<> 0@}.
@end ifinfo
@item 
@tex
  {\bf Minimum distance} of the code $C: d(C):=\min_{{\bf x,y}\in C, {\bf x}\ne {\bf y}}(d({\bf x,y}))$.
@end tex
@ifinfo
  @strong{Minimum distance} of the code C: d(C):=min(d(x,y)), where min is over x,y in C, x<>y.
@end ifinfo
@item 
@tex
  The code $C$ of dimension $k$ and minimum distance $d$ is denoted as $[n,k,d]$.
@end tex
@ifinfo
  The code C of dimension k and minimum distance d is denoted as [n,k,d].
@end ifinfo
@item 
@tex
  A matrix $G$ whose rows are the base vectors of $C$ is the {\bf generator matrix}.
@end tex
@ifinfo
  A matrix G whose rows are the basis vectors of C is the @strong{generator matrix}.
@end ifinfo
@item 
@tex
  A matrix $H$ with the property ${\bf c}\in C\iff H{\bf c}^T=0$ is the {\bf check matrix}.
@end tex
@ifinfo
  A matrix H with the property c in C if and only if Hc^T=0 is the @strong{check matrix}. 
@end ifinfo
@end itemize
@c @end table

@subsubheading Cyclic codes
@cindex cyclic code
@tex
  The code $C$ is {\bf cyclic}, if for every codeword ${\bf c}=(c_0,\dots,c_{n-1})$ in $C$ its
  cyclic shift $(c_{n-1},c_0,\dots,c_{n-2})$ is again a codeword in $C$.
  When working with cyclic codes, vectors are usually presented as polynomials.
  So ${\bf c}$ is represented by the polynomial $c(x)=\sum_{i=0}^{n-1}c_ix^i$ with $x^n=1$, more
  precisely $c(x)$ is an element of the factor ring $F_q[X]/\langle X^n-1 \rangle$.
  Cyclic codes over $F_q$ of length $n$ correspond one-to-one to ideals in this factor ring.
  We assume for cyclic codes that $(q,n)=1$. Let $F=F_{q^m}$ be
  the splitting field of $X^n-1$ over $F_q$. Then $F$ has a {\bf primitive $n$-th root of unity} which will be denoted by $a$.
  A cyclic code is uniquely given by a {\bf defining set} $S_C$ which is a subset of $Z\!\!\! Z_n$ such that
  $$
    c(x) \in C \ \hbox{ if } \ c(a ^i )=0 \ \hbox{ for all } \ i \in S_C.
  $$
  A cyclic code has several defining sets.
@end tex
@ifinfo
  The code C is @strong{cyclic}, if for every codeword c=(c_0,@dots{},c_(n-1)) in C its
  cyclic shift (c_(n-1),c_0,@dots{},c_(n-2)) is again a codeword in C.
  When working with cyclic codes, vectors are usually presented as polynomials.
  So c is represented by the polynomial c(x)=sum_(i=0)^(n-1)c_i x^i$ with x^n=1, more
  precisely c(x) is an element of the factor ring F_q[X]/<X^n-1>.
  Cyclic codes over F_q of length n correspond one-to-one to ideals in this factor ring.
  We assume for cyclic codes that (q,n)=1. Let F=F_(q^m) be
  the splitting field of X^n-1 over F_q. Then F has a @strong{primitive n-th root of unity} which will be denoted by a.
  A cyclic code is uniquely given by a @strong{defining set} S_C which is a subset of Z_n such that
  @display
    c(x) in C if c(a^i)=0 for all i in S_C.
  @end display
  A cyclic code has several defining sets.
@end ifinfo

  
@subsubheading Decoding problem
@cindex decoding, decoding problem
@c table @asis
@itemize @bullet
@item 
@tex
  {\bf Complete decoding}: Given $y\in F_q^n$ and a code $C\subseteq F_q^n$, so that $y$ is at distance $d(y,C)$ from
  the code, find $c\in C: d(y,c)=d(y,C)$.
@end tex
@ifinfo
  @strong{Complete decoding}: Given y in F_q^n and a code C<=F_q^n, 
  so that y is at distance d(y,C) from the code, find cin C: d(y,c)=d(y,C).
@end ifinfo
@item 
@tex
  {\bf Bounded up to half the minimum distance}: With the additional assumption $d({\bf y},C)\le(d(C)-1)/2$, a codeword with the above property
  is unique.
@end tex
@ifinfo
  @strong{Bounded up to half the minimum distance}: Additional assumption d(y,C)<= (d(C)-1)/2.  
  Then a codeword with the above property is unique.
@end ifinfo
@end itemize

@c @end table

@subsubheading Decoding via systems solving
One distinguishes between two concepts:
@c table @asis
@itemize @bullet
@item 
@tex
  {\bf Generic decoding}: Solve some system $S(C)$ and obtain some "closed" formulas $CF$. Evaluating these formulas
  at data specific to a received word ${\bf r}$ should yield a solution to the decoding problem. For example for 
  $f\in CF: f(syndrome({\bf r}),x)=poly(x)$. The roots of $poly(x)=0$ yield error positions, see the section on the
  general error-locator polynomial.
@end tex
@ifinfo
  @strong{Generic decoding}: Solve some system S(C) and obtain some ``closed'' formulas CF.
  Evaluating these formulas at data specific to a received word r should yield a solution to the 
  decoding problem. For example for f in CF: f(syndrome(r),x)=poly(x). The roots of 
  poly(x)=0 yield error positions -- general error-locator polynomial f.
@end ifinfo
@item 
@tex
  {\bf Online decoding}: Solve some system $S(C,{\bf r})$. The solutions should solve the decoding problem.
@end tex
@ifinfo
  @strong{Online decoding}: Solve some system S(C,r). The solutions should solve the decoding problem.
@end ifinfo
@end itemize

@c @end table


@subsubheading Computational effort
@c table @asis
@itemize @bullet
@item Generic decoding. Here, preprocessing is very hard, whereas decoding is relatively simple (if the formulas are sparse).
@item Online decoding. In this case, decoding is the hard part.
@end itemize

@c @end table

@c ---------------------------------------------------------------------------
@node Cooper philosophy, Generalized Newton identities, Codes and the decoding problem, Decoding codes with Groebner bases 
@subsection Cooper philosophy
@cindex Cooper philosophy

@subsubheading Computing syndromes in cyclic code case
@tex
  Let $C$ be an $[n,k]$ cyclic code over $F_q$; $F$ is a splitting field with $a$ being a primitive n-th root of unity. Let 
  $S_C=\{i_1,\dots,i_{n-k}\}$ be the complete defining set of $C$. Let ${\bf r}={\bf c}+{\bf e}$ 
  be a received word with ${\bf c}\in C$ and ${\bf e}$ an error vector. 
  Denote the corresponding polynomials in $F_q[x]/\langle x^n-1 \rangle$ by $r(x)$, $c(x)$ and $e(x)$, resp. Compute syndromes
  $$
    s_{i_m}=r(a^{i_m})=e(a^{i_m})= \sum_{l=1}^{t}e_{j_l}(a^{i_m})^{j_l}, \ \ 1\le m \le n-k,
  $$
  where $t$ is the number of errors, $j_1,\dots,j_t$ are the {\bf error positions} and $e_{j_1},\dots,e_{j_t}$ are the {\bf
  error values}.
@end tex
@ifinfo
Let C be an [n,k] cyclic code over F_q; F is a splitting field with a being a primitive n-th root of unity. Let 
S_C=@{i_1,@dots{},i_(n-k)@} be the complete defining set of C. 
Let r=c+e be the received word with c in C and e an error vector. 
Denote the corresponding polynomials in F_q[x]/<x^n-1> by r(x),c(x) and e(x), resp. 
Compute syndromes 
@display
  s_(i_m)=r(a^(i_m))=e(a^(i_m))=sum_(l=1)^t e_(j_l)(a^(i_m))^(j_l), 1<= m <= n-k,
@end display
where t is the number of errors, @{ j_1 , @dots{} , j_t @} are the @strong{error positions} and
@{e_(j_1),@dots{},e_(j_t)@} are the @strong{error values}.
@end ifinfo
@tex
  Define $z_l=a^{j_l}$ and $y_l=e_{j_l}$. Then $z_1,\dots,z_t$ are the error locations and $y_1,\dots,y_t$ are the error values and
  the syndromes above become {\bf generalized power sum functions} $s_{i_m}=\sum_{l=1}^{t}y_lz_l^{i_m}, \ \ 1\le m \le n-k. $
@end tex
@ifinfo
Define z_l=a^(j_l) and y_l=e_(j_l). Then z_1,@dots{},z_t are the error locations and 
y_1,@dots{},y_t are the error values and
the syndromes above become @strong{generalized power sum functions} 
s_(i_m)=sum_(l=1)^t y_l z_l^(i_m), 1\le m \le n-k.
@end ifinfo

@subsubheading CRHT-ideal
@cindex CRHT-ideal
Replace the concrete values above by variables and add some natural restrictions. Introduce 
@c table @asis
@itemize @bullet  
@item 
@tex 
  $f_u := \sum_{l=1}^e Y_lZ_l^{i_u}-X_u=0, 1\le u\le n-k$;
@end tex
@ifinfo
  f_u := sum_(l=1)^e Y_l Z_l^(i_u)-X_u=0, 1<= u<= n-k;
@end ifinfo
@item 
@tex 
  $\epsilon_j:=X_j^{q^m}-X_j=0, 1\le j\le n-k,$ since $s_j\in F$;
@end tex
@ifinfo
  epsilon_j:=X_j^(q^m)-X_j=0, 1<= j<= n-k, since s_j in F;
@end ifinfo
@item 
@tex 
  $\eta_i:= Z_i^{n+1}-Z_i=0, 1\le i\le e$, since $a^{j_i}$ are either $n$-th roots of unity or zero; 
@end tex
@ifinfo
  eta_i:= Z_i^(n+1)-Z_i=0, 1<= i<= e, since a^(j_i) are either n-th roots of unity or zero; 
@end ifinfo
@item 
@tex 
  $\lambda_i:=Y_i^{q-1}-1=0, 1\le i\le e$, since $y_l\in F_q \setminus\{0\}$.
@end tex
@ifinfo
  lambda_i:=Y_i^(q-1)-1=0, 1<= i<= e, since y_l F_q\ @{0@};
@end ifinfo
@end itemize
@c @end table

@tex
  We obtain the following set of polynomials in the variables $X=(X_1,\dots,X_{n-k})$, $Z=(Z_1,\dots,Z_e)$  and  $Y=(Y_1,\dots,Y_e)$:
  $$ F_C=\{f_j,\epsilon_j,\eta_i,\lambda_i:1\le j\le n-k,1\le i\le e\}\subset F_q[X,Z,Y].$$  
  The zero-dimensional ideal $I_C$ generated by $F_C$ is the {\bf CRHT-syndrome ideal}
  associated to the code $C$, and the variety $V(F_C)$ defined by $F_C$ is the {\bf CRHT-syndrome variety},
  after Chen, Reed, Helleseth and Truong.
@end tex
@ifinfo
We obtain the following set of polynomials in the variables X=(X_1,@dots{},X_(n-k)), Z=(Z_1,@dots{},Z_e)
and Y=(Y_1,@dots{},Y_e): 
@display
  F_C=@{f_j,epsilon_j,eta_i,lambda_i: 1<= j<= n-k, 1<= i<= e@}< F_q[X,Z,Y].
@end display
The zero-dimensional ideal I_C generated by F_C is the @strong{CRHT-syndrome ideal}
associated to the code C, and the variety V(F_C) defined by F_C is the @strong{CRHT-syndrome variety},
after Chen, Reed, Helleseth and Truong.
@end ifinfo

@subsubheading General error-locator polynomial
@cindex general error-locator polynomial
@tex
  Adding some more polynomials to $F_C$, thus obtaining some $F'_C$, it is possible to prove the following {\bf Theorem}: 
@end tex
@ifinfo
  Adding some additional polynomials to F_C, thus obtaining some F'_C, it is possible to prove the following @strong{Theorem}: 
@end ifinfo

@tex
  Every cyclic code $C$ possesses a {\bf general error-locator polynomial} $L_C$ from $F_q[X_1,\ldots ,X_{n-k},Z]$ that satisfies the following 
  two properties:
@end tex
@ifinfo
  Every cyclic code C possesses a @strong{general error-locator polynomial} L_C from F_q[X_1,@dots{},X_(n-k),Z] that satisfies the following 
  two properties:
@end ifinfo
@c table @asis
@itemize @bullet 
@item
@tex
  $L_C=Z^e+a_{t-1}Z^{e-1}+\dots +a_0$  with $a_j\in F_q[X_1,\dots ,X_{n-k}],\ 0\le j\le e-1$, where $e$ is the error-correcting
  capacity;
@end tex
@ifinfo
  L_C=Z^e+a_(t-1)Z^(e-1)+@dots{}+a_0,  a_j in F_q[X_1,@dots{},X_(n-k)], 0<= j<= e-1, where e is the error-correcting
  capacity;
@end ifinfo
@item  
@tex
    given a syndrome ${\bf s}=(s_{i_1},\dots,s_{i_{n-k}})\in F^{n-k}$ corresponding to an error of weight $t\le e$
    and error locations $\{k_1,\dots,k_{t}\}$, if we evaluate the $X_u=s_{i_u}$ for all $1\le u \le n-k$,
    then the roots of $L_C({\bf s},Z)$ are exactly $a^{k_1},\dots,a^{k_t}$ and 0 of multiplicity $e-t$, in other words
    $L_C({\bf s},Z)=Z^{e-t}\prod_{i=1}^{t}(Z-a ^{k_i}).$
@end tex
@ifinfo
    given a syndrome s=(s_(i_1),@dots{},s_(i_(n-k))) in F^(n-k)@} corresponding to an error of weight t<= e
    and error locations @{k_1,@dots{},k_t@}, if we evaluate the X_u=s_(i_u) for all 1<= u <= n-k,
    then the roots of L_C(s,Z) are exactly a^(k_1),@dots{},a^(k_t) and 0 of multiplicity e-t, in other words
    L_C(s,Z)=Z^(e-t) prod_(i=1)^t (Z-a^(k_i)).
@end ifinfo
@end itemize
@c @end table  

@tex
  The general error-locator polynomial  actually is an element of the reduced Gr\"obner basis of $\langle F'_C \rangle$. Having this polynomial,
  decoding of the cyclic code $C$ reduces to univariate factorization.
@end tex
@ifinfo
The general error-locator polynomial actually is an element of the reduced Groebner basis of 
<F'_C>. Having this polynomial, decoding of the cyclic code C reduces to univariate factorization.
@end ifinfo

For an example see @code{sysCRHT} in @ref{decodegb_lib}.
More on Cooper's philosophy and the general error-locator polynomial can be found in @ref{[OS2005]}.


@subsubheading Finding the minimum distance
The method described above can be adapted to find the minimum distance of a code.
More concretely, the following holds:

@tex
  Let $C$ be the binary $[n,k,d]$ cyclic code with the defining set $S_C=\{i_1,\dots,i_v\}$. Let $1\le w\le n$ and let $J_C(w)$ denote the system:
  $$
  Z_1^{i_1}+\dots+Z_w^{i_1}=0,
  $$
  $$
  \vdots
  $$
  $$
  Z_1^{i_v}+\dots+Z_w^{i_v}=0,
  $$
  $$
  Z_1^n-1=0,
  $$
  $$
  \vdots
  $$
  $$
  Z_w^n-1=0,
  $$
  $$
  p(n,Z_i,Z_j)=0, 1\le i<j\le w.     
  $$
@end tex
@ifinfo
  Let C be the binary [n,k,d] cyclic code with the defining set 
  S_C=@{i_1,@dots{},i_v@}. Let 1<= w<= n and let J_C(w) denote the system:
  @display
  Z_1^(i_1)+@dots{}+Z_w^(i_1)=0,  
  Z_1^(i_v)+@dots{}+Z_w^(i_v)=0,  
  Z_1^n-1=0,  
  Z_w^n-1=0,  
  p(n,Z_i,Z_j)=0,1<= i<j<= w.  
  @end display
@end ifinfo
@tex
  Then the number of solutions of $J_C(w)$ is equal to $w!$ times the number of codewords of weight $w$. And for $1\le w\le d$, either $J_C(w)$
  has no solutions, which is equivalent to $w<d$, or $J_C(w)$ has some solutions, which is equivalent to $w=d$.  
@end tex
@ifinfo
Then the number of solutions of J_C(w) is equal to w! 
times the number of codewords of weight w. And for 1<= w<= d: 
either J_C(w) has no solutions, which is equivalent to w<d, or 
J_C(w) has some solutions, which is equivalent to w=d.  
@end ifinfo

For an example see @code{sysCRHTMindist} in @ref{decodegb_lib}.
More on finding the minimum distance with Groebner bases can be found in @ref{[S2007]}.
@xref{[OS2005]}, for the definition of the polynomial @math{p} above.


@c ---------------------------------------------------------------------------
@node Generalized Newton identities, Fitzgerald-Lax method, Cooper philosophy,  Decoding codes with Groebner bases 
@subsection Generalized Newton identities
@cindex Generalized Newton identities

@tex
  The {\bf error-locator polynomial} is defined by
  $$
    \sigma(Z)=\prod_{l=1}^{t}(Z-z_l).
  $$
  If this product is expanded,
  $$
    \sigma(Z)=Z^t+\sigma_1Z^{t-1}+\dots+\sigma_{t-1}Z+\sigma_t,
  $$
  then the coefficients $\sigma_i$ are the {\bf elementary symmetric functions} in
  the error locations $z_1,\dots,z_t$
  $$
    \sigma_i=(-1)^i\sum_{1\le j_1<j_2<\dots<j_i\le t}z_{j_1}z_{j_2}\dots z_{j_i}, \ 1\le i\le t.
  $$
@end tex
@ifinfo
The @strong{error-locator polynomial} is defined by
@display
  sigma(Z)=prod_(l=1)^t (Z-z_l).
@end display
If this product is expanded 
@display
  sigma(Z)=Z^t+sigma_1 Z^(t-1)+@dots{}+sigma_(t-1)Z+sigma_t,
@end display
then the coefficients sigma_i are the @strong{elementary symmetric functions} in
the error locations z_1,@dots{},z_t: 
@display
  sigma_i=(-1)^i sum_(1<=j_1<j_2<@dots{}<j_i<=t) z_(j_1) z_(j_2)@dots{}z_(j_i),1<= i<= t.
@end display
@end ifinfo

@subsubheading Generalized Newton identities

@tex
  The syndromes $s_i=r(a^i)=e(a^i)$ and the coefficients $\sigma_i$ satisfy
  the following {\bf generalized Newton identities}:
  $$
    s_i+\sum_{j=1}^{t}\sigma_js_{i-j}=0,\ \  \hbox{ for all } \ i\in Z\!\!\! Z_n.    
  $$
@end tex
@ifinfo
The syndromes s_i=r(a^i)=e(a^i) and the coefficients sigma_i satisfy
the following @strong{generalized Newton identities}: 
@display
  s_i+sum_(j=1)^t sigma_j s_(i-j)=0, forall i in Z_n.    
@end display
@end ifinfo

@subsubheading Decoding up to error-correcting capacity
  
@tex 
  We have $ s_{i+n}=s_i, \hbox{ for all }  i\in Z\!\!\! Z_n$, since $s_{i+n}=r(a^{i+n})=r(a^i)$. Furthermore
  $$
    s_i^q=(e(a^i))^q=e(a^{iq})=s_{qi}, \hbox{ for all } i \in Z_n,
  $$
  and $\sigma_i^{q^m}=\sigma_i, \hbox{ for all } 1\le i \le t$.
  Replace the syndromes by variables and obtain the following set of polynomials $Newton_t$ in the variables
  $S_1,\dots,S_n$ and $\sigma_1,\dots,\sigma_t$:
  $$  
  \sigma_i^{q^m}-\sigma_i, \ \  \forall \ 1\le i \le t,
  $$
  $$
  S_{i+n}-S_i, \ \  \forall \  i\in Z\!\!\! Z_n,
  $$
  $$
  S_i^q-S_{qi}, \ \  \forall \  i\in Z\!\!\! Z_n,
  $$
  $$
  S_i+\sum_{j=1}^{t}\sigma_jS_{i-j}, \ \  \forall \  i\in Z\!\!\! Z_n,
  $$
  $$
  S_i-s_i(r) \ \ \forall \ i\in S_C.  
  $$
@end tex
@ifinfo
We have s_(i+n)=s_i, forall  i in Z_n, since s_(i+n)=r(a^(i+n))=r(a^i). Furthermore
@display
  s_i^q=(e(a^i))^q=e(a^(iq))=s_(qi), forall i in Z_n 
@end display 
and sigma_i^(q^m)=sigma_i, forall 1<= i <= t.
Replace the syndromes by variables and obtain following set of polynomials @code{Newton_t} in the variables
S_1,@dots{},S_n and sigma_1,@dots{},sigma_t:
@display
  sigma_i^(q^m)-sigma_i, forall  1<= i <= t,
  S_(i+n)-S_i, forall i in Z_n,  
  S_i^q-S_(qi), forall i in Z_n, 
  S_i+sum_(j=1)^t sigma_j S_(i-j), forall i in Z_n,  
  S_i-s_i(r) forall i in S_C.    
@end display
@end ifinfo

For an example see @code{sysNewton} in @ref{decodegb_lib}. More on this method and the method based on
Waring function can be found in @ref{[ABF2002]}. See also @ref{[ABF2008]}.


@c ---------------------------------------------------------------------------
@node Fitzgerald-Lax method, Decoding method based on quadratic equations, Generalized Newton identities, Decoding codes with Groebner bases 
@subsection Fitzgerald-Lax method
@cindex Fitzgerald-Lax method

@subsubheading Affine codes
@cindex affine code

@tex
  Let $I=\langle g_1,\dots,g_m \rangle \subseteq F_q[X_1,\dots,X_s]$ be an
  ideal. Define 
  $$
    I_q:=I+\langle X_1^q-X_1,\dots,X_s^q-X_s\rangle .
  $$ 
  So $I_q$ is a zero-dimensional ideal. Define also $V(I_q)=:\{P_1,\dots,P_n\}$.
  Every $q$-ary linear code $C$ with parameters $[n,k]$ can be seen as an
  {\bf affine variety code} $C(I,L)$, that is, the image of a vector space $L$ of the {\bf evaluation map}
  $$  
  \phi :R\to F_q ^n
  $$
  $$
  \bar{f}\mapsto(f(P_1),\dots,f(P_n)),  
  $$
  where $R:=F_q[U_1,\dots,U_s]/I_q$, $L$ is a vector subspace of $R$ and $\bar{f}$ the coset of $f$ in $F_q[U_1,\dots,U_s]$ modulo $I_q$.
@end tex
@ifinfo
Let I=<g_1,@dots{},g_m> <= F_q[X_1,@dots{},X_s] be an ideal. 
Define 
@display
  I_q:=I+<X_1^q-X_1,@dots{},X_s^q-X_s>.
@end display
So I_q is a zero-dimensional ideal. Define also V(I_q)=:@{P_1,@dots{},P_n@}.
Every q-ary linear code C with parameters [n,k] can be seen as an
@strong{affine variety code} C(I,L), that is the image of a vector space L of the 
@strong{evaluation map}
@display
 phi:R --> F_q ^n,
 ^f^ |-> (f(P_1),@dots{},f(P_n)),    
@end display
where R:=F_q[U_1,@dots{},U_s]/I_q, L is a vector subspace of R and ^f^ 
the coset of f in F_q[U_1,@dots{},U_s] modulo I_q.
@end ifinfo

@subsubheading Decoding affine variety codes

@tex
  Given a $q$-ary $[n,k]$ code $C$ with a generator matrix $G=(g_{ij})$:
@end tex
@ifinfo
  Given a q-ary [n,k] code C with the generator matrix G=(g_(ij)):
@end ifinfo
@enumerate
@c @bullet
@c @table @asis
@item 
@tex
  choose $s$, such that $q^s\ge n$, and construct $s$ distinct points $P_1,\dots,P_s$ in $F_q^s$. 
@end tex
@ifinfo
  choose s, such that q^s>= n, and construct s distinct points P_1,@dots{},P_s in F_q^s. 
@end ifinfo
@item 
@tex
  Construct a Gr\"obner basis $\{g_1,\dots,g_m\}$ for an ideal $I$ of polynomials from $F_q[X_1,\dots,X_s]$ that vanish at the points 
  $P_1,\dots,P_s$. Define $\xi_i\in F_q[X_1,\dots,X_s]$ such that $\xi_i(P_i)=1,\xi_i(P_j)=0,i\ne j$. 
@end tex
@ifinfo
  Construct a Groebner basis @{g_1,@dots{},g_m@} for an ideal I of polynomials from F_q[X_1,@dots{},X_s]
  that vanish at the points P_1,@dots{},P_s @ref{vanishId}. Denote by
  xi_i in F_q[X_1,@dots{},X_s] such that xi_i(P_i)=1,xi_i(P_j)=0,i<>j. 
@end ifinfo
@item 
@tex
  Then $f_i=\sum_{i=1}^ng_{ij}\xi_j$ span the space $L$, so that $g_{ij}=f_i(P_j)$.
@end tex
@ifinfo
  Then f_i=sum_(i=1)^n g_(ij) xi_j span the space L, so that g_(ij)=f_i(P_j).
@end ifinfo
@end enumerate
@c @end table  

@tex
  In this way we obtain that the code $C$ is the image of the evaluation above, thus $C=C(I,L)$. In the
  same way by considering a parity check matrix instead of a generator matrix we have that the dual code is also an affine variety code.
  
  The method of decoding is a generalization of CRHT. One needs to add polynomials
  $(g_l(X_{k1},\dots,X_{ks}))_{l=1,\dots,m; k=1,\dots,t}$ for every error position. We also assume that field equations on $X_{ij}$'s are included
  among the polynomials above. Let $C$ be a $q$-ary $[n,k]$ linear code such that its
  dual is written as an affine variety code of the form $C^{\bot}=C(I,L)$.
  Let ${\bf r}={\bf c}+{\bf e}$ as usual and $t\le e$. Then the syndromes are computed by $s_i=\sum_{j=1}^nr_jf_i(P_j)=\sum_{j=1}^ne_jf_i(P_j) 
  \hbox{ for } i=1,\dots,n-k$.  
@end tex
@ifinfo
In this way we obtain that the code C is the image of the evaluation above, so C=C(I,L). In the
same way by considering a parity check matrix instead of a generator matrix we have that the dual code is also an affine variety code.

The method of decoding is a generalization of CRHT. One needs to add polynomials
(g_l(X_(k1),@dots{},X_(ks)))_@{l=1,@dots{},m; k=1,@dots{},t@} for every error position. 
We also assume that field equations on X_(ij)'s are included
among the polynomials above. Let C be a q-ary [n,k] linear code such that its
dual is written as an affine variety code of the form C^perpend=C(I,L). 
Let r=c+e as usual and t<= e. Then the syndromes are computed by 
s_i=sum_(j=1)^n r_j f_i(P_j)=sum_(j=1)^n e_j f_i(P_j),  i=1,@dots{},n-k.
@end ifinfo

@tex
  Consider the ring $F_q[X_{11},\dots,X_{1s},\ldots,X_{t1},\dots,X_{ts},E_1,\dots,E_t]$, where $(X_{i1},\dots,X_{is})$ correspond to
  the $i$-th error position and $E_i$ to the $i$-th error value. Consider the ideal $Id_C$ generated by
  $$  
  \sum_{j=1}^tE_jf_i(X_{j1},\dots,X_{js})-s_i, 1\le i\le n-k,
  $$
  $$
  g_l(X_{j1},\dots,X_{js}), 1\le l\le m,
  $$
  $$
  E_k^{q-1}-1.  
  $$
@end tex
@ifinfo
Consider the ring F_q[X_(11),@dots{},X_(1s),@dots{},X_(t1),@dots{},X_(ts),E_1,@dots{},E_t], where 
(X_(i1),@dots{},X_(is)) correspond to
the i-th error position and E_i to the i-th error value. 
Consider the ideal Id_C generated by
@display
 sum_(j=1)^t E_j f_i(X_(j1),@dots{},X_(js))-s_i, 1<= i<= n-k,  
 g_l(X_(j1),@dots{},X_(js)), 1<= l<= m,
 E_k^(q-1)-1.    
@end display
@end ifinfo


@table @code
@item @strong{Theorem:}
@tex
  Let $G$ be the reduced Gr\"obner basis for $Id_C$ with respect to an elimination order $X_{11}<\dots<X_{1s}<E_1$.
  Then we may solve for the error locations and values by applying elimination theory to the polynomials in $G$.
@end tex
@ifinfo
  Let G be the reduced Groebner basis for Id_C with respect to an elimination order induced by
  X_(11),@dots{},X_(1s),E_1, such that X_(11)<@dots{}<X_(1s)<E_1. 
  Then we may solve for the error locations and values by applying elimination theory to the polynomials in G.
@end ifinfo
@end table

For an example see @code{sysFL} in @ref{decodegb_lib}. More on this method can be found in @ref{[FL1998]}.


@c ---------------------------------------------------------------------------
@node Decoding method based on quadratic equations, References for decoding with Groebner bases, Fitzgerald-Lax method, Decoding codes with Groebner bases 
@subsection Decoding method based on quadratic equations
@cindex Decoding method based on quadratic equations

@subsubheading Preliminary definitions
@cindex unknown syndrome

@tex
  Let ${\bf b}_1,\dots,{\bf b}_n$ be a basis of $F_q^n$ and let $B$ be the $n \times n$ matrix with
  ${\bf b}_1,\dots,{\bf b}_n$ as rows. The {\bf unknown syndrome} ${\bf u}(B,{\bf e})$ of a word ${\bf e}$ w.r.t $B$
  is the column vector ${\bf u}(B,{\bf e})=B{\bf e}^T$ with entries $u_i(B,{\bf e})={\bf b}_i\cdot {\bf e}$ for $i=1,\dots,n$.

  For two vectors ${\bf x,y} \in F_q^n$ define
  ${\bf x*y}=(x_1y_1,\dots,x_ny_n)$. Then ${\bf b}_i*{\bf b_j}$ is a linear combination of 
  ${\bf b}_1,\dots,{\bf b}_n$, so there are constants $\mu^{ij}_l\in F_q $ such that
  ${\bf b}_i*{\bf b}_j = \sum_{l=1}^n \mu^{ij}_l{\bf b}_l.$
  The elements $\mu^{ij}_l\in F_q$ are the {\bf structure constants} of the basis ${\bf b}_1,\dots,{\bf b}_n$.

  Let $B_s$ be the $s \times n$ matrix with ${\bf b}_1,\dots,{\bf b}_s$ as rows ($B=B_n$).
  Then ${\bf b}_1,\dots,{\bf b}_n$ is an {\bf ordered MDS basis} and $B$ an {\bf MDS matrix} if all
  the $s\times s$ submatrices of $B_s$ have rank $s$ for all $s=1 ,\dots ,n$. 
@end tex
@ifinfo
Let b_1, @dots{} ,b_n be a basis of F_q^n and let B be the n x n matrix with
b_1, @dots{} ,b_n as rows. The @strong{unknown syndrome} u(B,e) of a word e w.r.t B
is the column vector u(B,e)=Be^T with entries u_i(B,e)=b_i·e for i=1,@dots{},n.

For two vectors x,y in F_q^n define
x*y=(x_1 y_1,@dots{},x_n y_n). Then b_i*b_j is a linear combination of 
b_1,@dots{},b_n, so there are constants mu^(ij)_l in F_q such that
b_i*b_j = sum_(l=1)^n mu^(ij)_l b_l.
The elements mu^(ij)_l in F_q are the @strong{structure constants} of the basis b_1,@dots{},b_n.

Let B_s be the s x n matrix with b_1,@dots{},b_s as rows (B=B_n).
Then b_1,@dots{},b_n is an @strong{ordered MDS basis} and B an @strong{MDS matrix} if all
the s x s submatrices of B_s have rank s for all s=1,@dots{},n. 
@end ifinfo
  
@subsubheading Expressing known syndromes
@tex
  Let $C$ be an $F_q$-linear code with parameters $[n,k,d]$. W.l.o.g $n\le q$. $H$ is a check matrix of $C$.
  Let ${\bf h}_1,\dots,{\bf h}_{n-k}$ be the rows of $H$.
  One can express ${\bf h}_i = \sum _{j=1}^n a_{ij}{\bf b}_j$ with some $a_{ij}\in F_q$.
  In other words $H=AB$ where $A$ is the $(n-k) \times n$ matrix with entries $a_{ij}$.

  Let ${\bf r}={\bf c}+{\bf e}$ be a received word with ${\bf c}\in C$ and ${\bf e}$ an error vector.
  The syndromes of ${\bf r}$ and ${\bf e}$ w.r.t $H$ are equal and known:
  $$
    s_i({\bf r}):={\bf h}_i \cdot {\bf r}={\bf h}_i\cdot {\bf e}=s_i({\bf e}).
  $$
  They can be expressed in the unknown syndromes of ${\bf e}$ w.r.t $B$:
  $$
    s_i({\bf r})=s_i({\bf e}) =\sum _{j=1}^n a_{ij}u_j({\bf e})
  $$
  since ${\bf h}_i = \sum _{j=1}^n a_{ij}{\bf b}_j$ and ${\bf b}_j \cdot {\bf e}=u_j({\bf e})$.
@end tex
@ifinfo
Let C be an F_q-linear code with parameters [n,k,d]. W.l.o.g n<= q. 
H is a check matrix of C.
Let h_1,@dots{},h_(n-k) be the rows of H.
One can express h_i = sum_(j=1)^n a_(ij)b_j for some a_(ij) in F_q.
In other words H=AB where A is the (n-k) x n matrix with entries a_(ij).
Let r=c+e be the received word with c in C and e an error vector.
The syndromes of r and e w.r.t H are equal and known:
@display
  s_i(r):=h_i·r=h_i·e=s_i(e).
@end display
They can be expressed in the unknown syndromes of e w.r.t B: 
@display
  s_i(r)=s_i(e) =sum_(j=1)^n a_(ij)u_j(e)
@end display
since h_i=sum_(j=1)^n a_(ij)b_j and b_j·e=u_j(e).
@end ifinfo

@subsubheading Contructing the system

@tex
  Let $B$ be an MDS matrix with structure constants $\mu^{ij}_l$.
  Define $U_{ij}$ in the variables $U_1, \dots ,U_n$ by
  $$
    U_{ij}=\sum _{l=1}^n \mu^{ij}_lU_l.
  $$
  The ideal $J({\bf r})$ in $F_q[U_1,\dots,U_n]$ is generated by
  $$
     \sum_{l=1}^n a_{jl}U_l-s_j({\bf r})  \hbox{ for }   j=1,\dots,n-k.
  $$
  The ideal $I(t,U,V)$ in $F_q[U_1,\dots,U_n,V_1,\dots,V_t]$
  is generated by 
  $$  
    \sum_{j=1}^t U_{ij}V_j-U_{i,t+1} \hbox{ for } i=1,\dots,n  
  $$
  Let $J(t,{\bf r})$ be the ideal in $F_q[U_1,\dots,U_n,V_1,\dots,V_t]$
  generated by $J({\bf r})$ and $I(t,U,V)$.
@end tex
@ifinfo
Let B be an MDS matrix with structure constants mu^(ij)_l.
Define U_(ij) in the variables U_1,@dots{},U_n by
@display
  U_(ij)=sum_(l=1)^n mu^(ij)_l U_l.
@end display  
The ideal J(r) in F_q [U_1,@dots{},U_n] is generated by
@display
  sum_(l=1)^n a_(jl) U_l-s_j(r),j=1,@dots{},n-k
@end display
The ideal I(t,U,V) in F_q[U_1,@dots{},U_n,V_1,@dots{},V_t]
is generated by 
@display
  sum_(j=1)^t U_(ij) V_j-U_(i,t+1), i=1,@dots{},n.
@end display
Let J(t,r) be the ideal in F_q[U_1,@dots{},U_n,V_1,@dots{},V_t]
generated by J(r) and I(t,U,V).
@end ifinfo

@subsubheading Main theorem

@tex
  Let $B$ be an MDS matrix with structure constants $\mu^{ij}_l$.  Let $H$ be a check matrix of the code $C$ such that $H=AB$ as above.
  Let ${\bf r}={\bf c}+{\bf e}$ be a received word with ${\bf c} \in C$ the codeword sent
  and ${\bf e}$ the error vector. Suppose that  $wt({\bf e})\ne 0$ and $wt({\bf e})\le\lfloor (d(C)-1)/2\rfloor$.
  Let $t$ be the smallest positive integer such that $J(t,{\bf r})$ has a solution $({\bf u,v})$ over the algebraic closure of
  $F_q$. Then   
@end tex
@ifinfo
Let B be an MDS matrix with structure constants mu^(ij)_l. 
Let H be a check matrix of the code C such that H=AB as above.
Let r=c+e be a received word with c in C the codeword sent
and e the error vector. Suppose that wt(e)<>0 and wt(e)<=(d(C)-1)/2.
Let t be the smallest positive integer such that J(t,r) has a solution (u,v) over
algebraic closure of F_q. Then
@end ifinfo
@c table @asis
@itemize @bullet
@item 
@tex 
  $wt({\bf e})=t$ and the solution is unique and of multiplicity one satisfying ${\bf u}={\bf u}({\bf e})$.
@end tex
@ifinfo
  wt(e)=t and the solution is unique and of multiplicity one satisfying u=u(e).
@end ifinfo
@item 
@tex
  the reduced Gr\"{o}bner basis $G$ for the ideal $J(t,{\bf r})$ w.r.t any
  monomial ordering is
  $$
  U_i-u_i({\bf e}), i=1,\dots,n,
  $$
  $$
  V_j-v_j, j=1,\dots,t,  
  $$
  where $({\bf u}({\bf e}),{\bf v})$ is the unique solution.
@end tex
@ifinfo
  the reduced Groebner basis G for the ideal J(t,r) w.r.t any monomial ordering is
  @display
  U_i-u_i(e), i=1,@dots{},n,
  V_j-v_j, j=1,@dots{},t,  
  @end display
  where (u(e),v) is a unique solution.
@end ifinfo
@end itemize

For an example see @code{sysQE} in @ref{decodegb_lib}. More on this method can be found in @ref{[BP2008a]}.


@node References for decoding with Groebner bases, ,Decoding method based on quadratic equations, Decoding codes with Groebner bases
@subsection References for decoding with Groebner bases

@itemize @bullet
@item [ABF2002]
Augot D.; Bardet M.; Faug@'ere J.-C.: @anchor{[ABF2002]}
Efficient Decoding of (binary) Cyclic Codes beyond the correction capacity of the code using Gr@"obner bases.
INRIA Report (2002) 4652

@item [ABF2008]
Augot D.; Bardet M.; Faug@'ere, J.-C.: @anchor{[ABF2008]}
On the decoding of cyclic codes with Newton identities.
to appear in Special Issue ``Gr@"obner Bases Techniques in Cryptography and Coding Theory'' of Journ. Symbolic Comp. (2008)

@item [BP2008a]
Bulygin S.; Pellikaan R.: @anchor{[BP2008a]}
Bounded distance decoding of linear error-correcting codes with Gr@"obner bases.
to appear in Special Issue ``Gr@"obner Bases Techniques in Cryptography and Coding Theory'' of Journ. Symbolic Comp. (2008)

@item [BP2008b]
Bulygin S.; Pellikaan R.: @anchor{[BP2008b]}
Decoding and finding the minimum distance with Gr@"obner bases: history and new insights.
to appear in ``Selected topics of information and coding theory'', World Scientific (2008)

@item [FL1998]
Fitzgerald J.; Lax R.F.: @anchor{[FL1998]}
Decoding affine variety codes using Gr@"obner bases.
Designs, Codes and Cryptography (1998) 13, 147-158

@item [OS2005]
Orsini E.; Sala M.: @anchor{[OS2005]}
Correcting errors and erasures via the syndrome variety.
J. Pure and Appl. Algebra (2005) 200, 191-226

@item [S2007]
Sala M.: @anchor{[S2007]}
Gr@"obner basis techniques to compute weight distributions of shortened cyclic codes.
J. Algebra Appl. (2007) 6, 3, 403-414

@end itemize