A Dynamical System-based Key Equation for Decoding One-Point Algebraic-Geometry Codes

TL;DR

This paper introduces a dynamical system framework for decoding one-point algebraic-geometry codes, showing that the syndrome array satisfies Cauchy's equations and that the Berlekamp-Massey-Sakata algorithm effectively solves these equations.

Contribution

It develops a novel dynamical system approach to decoding algebraic-geometry codes, linking syndrome arrays to Cauchy's equations and analyzing the algorithm's role.

Findings

Syndrome array is a linear recurring sequence.

Syndrome array solves Cauchy's homogeneous equations.

BMS algorithm solves these equations within the dynamical system context.

Abstract

A closer look at linear recurring sequences allowed us to define the multiplication of a univariate polynomial and a sequence, viewed as a power series with another variable, resulting in another sequence. Extending this operation, one gets the multiplication of matrices of multivariate polynomials and vectors of powers series. A dynamical system, according to U. Oberst is then the kernel of the linear mapping of modules defined by a polynomial matrix by this operation. Applying these tools in the decoding of the so-called one point algebraic-geometry codes, after showing that the syndrome array, which is the general transform of the error in a received word is a linear recurring sequence, we construct a dynamical system. We then prove that this array is the solution of Cauchy's homogeneous equations with respect to the dynamical system. The aim of the Berlekamp-Massey-Sakata Algorithm in the decoding process being the determination of the syndrome array, we have proved that in fact, this algorithm solves the Cauchy's homogeneous equations with respect to a dynamical system.

Equations

$$P_0{a}_n+P_1{a}_{n+1}\cdots +P_i{a}_{n+i}+\cdots +P_N{a}_{n+N}=0\text{for}n\in N,$$

$$a_{n+N}=-\frac{1}{P_N}(P_0{a}_n+P_1{a}_{n+1}\cdots +P_i{a}_{n+i}+\cdots +P_N{a}_{n+N-1}),$$

$$b_n=\sum _{i=0}^N{P}_i{a}_{n+i}\text{for}n\in N.$$

$$P(X)=\sum _{i=0}^N{P}_i{X}^i\in F[X],$$

$$a=a(Y)=\sum _{n=0}^{\infty }{a}_n{Y}^n,b=b(Y)=\sum _{n=0}^{\infty }{b}_n{Y}^n.$$

$$b(Y)=P(X)\circ a(Y).$$

$$P(X)\circ a(Y)=\sum_{n=0}^{\infty}\Bigl{(}\sum_{i=0}^{N}P_{i}a_{n+i}\Bigr{)}Y^{n}$$

$$X^{\alpha }=X_1^{{\alpha }_1}\cdots X_r^{{\alpha }_r}(resp.Y^{\alpha }=Y_1^{{\alpha }_1}\cdots Y_r^{{\alpha }_r}).$$

$$d(X_1,\dots ,X_r)=d(X)=\sum _{\alpha \in N^r}{d}_{\alpha }{X}^{\alpha }\text{ with}{d}_{\alpha }\in F\text{for all }\alpha \in N^r,$$

$$\mathrm{LE}(d(X))=\underset{{⩽}_T}{\max }\{\alpha \in N^r|d_{\alpha }\ne 0\}\in N^r.$$

$$W(Y_1,\dots ,Y_r)=W(Y)=\sum _{\alpha \in N^r}{W}_{\alpha }{Y}^{\alpha },$$

$$R(X)=(R_{ij}(X){)}_{1⩽i⩽k,1⩽j⩽l},$$

$$\begin{array}{rl}D\times A& ⟶A\\ (d(X),W(Y))& ⟼d(X)\circ W(Y)=\sum _{\alpha \in N^r}(\sum _{\beta \in N^r}{d}_{\beta }{W}_{\alpha +\beta })Y^{\alpha }.\end{array}$$

$$\begin{array}{rl}D\times A^l& ⟶A^l\\ (d(X),(W_j(Y){)}_{j=1,\dots ,l})& ⟼(d(X)\circ W_j(Y){)}_{j=1,\dots ,l}.\end{array}$$

$$\begin{array}{rl}R(X):A^l& ⟶A^k\\ W(Y)& ⟼R(X)\circ W(Y)\end{array}$$

$$\displaystyle\begin{split}R(X)\circ W(Y)&=\left(\begin{array}[]{c}\sum_{j=1}^{l}R_{1j}(X)\circ W_{j}(Y)\\ \vdots\\ \sum_{j}^{l}R_{kj}(X)\circ W_{j}(Y)\\ \end{array}\right)\\ &=\left(\begin{array}[]{c}\sum_{\rho\in\operatorname{\mathbb{N}}^{r}}(\sum_{j=1}^{l}\sum_{\alpha\in\operatorname{\mathbb{N}}^{r}}R_{1j\alpha}W_{j(\alpha+\rho)})Y^{\rho}\\ \vdots\\ \sum_{\rho\in\operatorname{\mathbb{N}}^{r}}(\sum_{j=1}^{l}\sum_{\alpha\in\operatorname{\mathbb{N}}^{r}}R_{ij\alpha}W_{j(\alpha+\rho)})Y^{\rho}\\ \vdots\\ \sum_{\rho\in\operatorname{\mathbb{N}}^{r}}(s\sum_{j=1}^{l}\sum_{\alpha\in\operatorname{\mathbb{N}}^{r}}R_{kj\alpha}W_{j(\alpha+\rho)})Y^{\rho}\\ \end{array}\right)\end{split}$$

$$S=\ker R(X)=\{W(Y)\in A^l|R(X)\circ W(Y)=0\}$$

$$\ker P(X)=\{a(Y)=\sum _{n=0}^{\infty }{a}_n{Y}^n\in F[|Y]]|P(X)\circ W(Y)=0\}.$$

$$P^{\perp }=\{W(Y)\in A^l|d(X)\circ W(Y)=0\text{for}d(X)\in P\}\subset A^l$$

$$Q^{\perp }=\{d(X)\in D^l|d(X)\circ W(Y)=0\text{for}W(Y)\in A^l\}\subset D^l.$$

$$S=\Biggl{\{}\left(\begin{array}[]{c}U\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr V\end{array}\right)\in\operatorname{\boldsymbol{A}}^{m}\times\operatorname{\boldsymbol{A}}^{p}\;|\;P(X)\circ V=Q(X)\circ U\Biggr{\}},$$

$$l=m+p,P\in D^{k,p},Q\in D^{k,m},$$

$$\mathrm{rank}(P)=\mathrm{rank}(R)=p.$$

$$[p]=\{1,\dots ,p\},$$

$$\{\begin{array}{l}P(X)\circ V=0,\\ V_{|\Gamma }=V_0,V_0\in F^{\Gamma },\end{array}$$

$$\mathcal{X}=\{P=(a_0,\dots ,a_r)\in {\mathbb{P}}^r(F)|F(P)=0\text{for}F\in \mathcal{F}\},$$

$$I(\mathcal{X})=\{F\in F[X_0,X_1,\dots ,X_r]|F(P)=0\text{for}P\in \mathcal{X}\}.$$

$$F[\mathcal{X}]=F[X_0,X_1,\dots ,X_r]/I(\mathcal{X}).$$

$$\begin{array}{r}F(\mathcal{X})=\{f(X_0,X_1,\dots ,X_r)/g(X_0,X_1,\dots ,X_r)|f,g\in F[X_0,X_1,\dots ,X_r]\\ \text{and}g(X_0,X_1,\dots ,X_r)\notin I(\mathcal{X})\}.\end{array}$$

$${\mathcal{X}}_{aff}=\{P=(a_1,\dots ,a_r)\in F^r|F(P)=0\text{for}P\in \mathcal{G}\},$$

$$I({\mathcal{X}}_{aff})=\{F\in F[X_1,\dots ,X_r]|F(P)=0\text{for}P\in {\mathcal{X}}_{aff}\}.$$

Full text

1. 1 Introduction
2. 2 Oberst’s algebraic dynamical systems and the Cauchy’s homogeneous problem
3. 3 On point algebraic-geometry codes
4. 4 Cauchy’s equations for the syndrome array

1 Introduction

S. Sakata, in [11, 12], generalized the famous Berlekamp-Massey algorithm ([9]) to the two and multidimensional case. The result was the (again) famous Berlekamp-Massey-Sakata (BMS) algorithm, whose aim is to find a Gröbner basis of the set of characteristic polynomials of a multidimensional sequence. He also used the algorithm to decode algebraic-geometry (AG) codes ([13, 14]). The main difficulties is that Sakata’s papers involves many difficult notations and calculations.

Heegard and Saints ([7]) gave a shorter and clearer version of this algorithm, and explained that, in the framework of the decoding process, the algorithm computes a sufficiently number of terms of the syndrome array and construct sets of polynomials which “converges” to a Gröbner basis, which allows the calculation of the syndrome array.

Since then, the BMS algorithm has been refined and widely used by many authors, see [2, 4, 5, 8, 15], and also D. Augot, “Les codes algébriques principaux et leur décodage”, Journées nationales du calcul formel. Luminy, mai 2010 and J. Bertomieux and J. C. Faugère, “In-depth comparison of the Berlekamp-Massey-Sakata and the Scalar-FGLM algorithms: the non adaptive variants”,

arXiv:1709.07168 [cs.SC] (2017).

Therefore, due to its importance, we present here a new explanation of the BMS algorithm, in the framework of the decoding process of one point algebraic-geometry codes, as in [7]. To construct these codes, one starts from a smooth irreducible projective curve which have a unique point only at the hyperplane at infinity, and a finite set of points of the curve, distinct from the point at infinity.  The code is the defined as evaluations of certain rational functions ((14) and (15)) on the curve on the set of points ((17)). The conditions these functions have to satisfy is that they have a unique pole, which is the point at the infinity, and moreover, the order of this pole is less than an appropriate number, which satisfies an inequality involving the genus of the curve and the number of evaluation points ((16)).

An important tool we use is the general transform (Definition (20)). The crucial starting point of our result is that the general transform of the error in a received word is a linear recurring sequence (Corollary 2). Here is where the notion of dynamical system can be introduced : the orthogonal of the syndrome array is a polynomial module, and therefore has a Gröbner basis. We consider the dynamical system defined by this basis.

We prove in our main theorem (Theorem 1) that the syndrome array of a received word is the solution of the *Cauchy’s homogeneous problem *(Definition 2) with respect to the above dynamical system, under the input/output representation ((9) and (26)), with an appropriate initial data, defined on a “Delta-set”((24)).

Our theorem provides a new equation for the decoding problem. We hope that our equation is a good starting point for understanding the BMS algorithm and decoding one point AG codes because it provides a clean and elegant algebraic presentation of the algorithm and the decoding problem.

This paper is organized as follows: in section 2, we introduce Oberst’s dynamical systems theory and the Cauchy’s homogeneous problem. In section 3, we present results about projective curves and one-point algebraic-geometry codes. In the last section 4, we state and prove our main theorem.

As we already mentioned in the abstract, the simple notion of *linear recurring sequence * is useful to understand the operation denoted by “$\circ$” in Section 2. A sequence $a=(a_{n})_{n\in\operatorname{\mathbb{N}}}$ of elements of a commutative field $\operatorname{\mathbb{F}}$ is said to be a linear recurring sequence (LRS) if the following equality holds:

[TABLE]

where $N\geqslant 1$ is an integer, $P_{i}\in\operatorname{\mathbb{F}}$ for $i=0,\ldots,N$ with $P_{N}\neq 0$. Using equation (1), we have that

[TABLE]

so that we can calculate $a_{n+N}$ using the $N$ previous terms of the sequence, which are $a_{n},\ldots,a_{n+N-1}$.

We observe that the left hand side of (1) is the $n$-th term of a new sequence of elements of $\operatorname{\mathbb{F}}$. Denoting this sequence by $b=(b_{n})_{n\in\operatorname{\mathbb{N}}}$, we have

[TABLE]

Now, construct the univariate polynomial

[TABLE]

and write the sequences $a$ and $b$ as power series in another variable, say $Y$:

[TABLE]

We say that $b(Y)$ is the product of $P(X)$ and $a(Y)$ and write

[TABLE]

Using (2), we have

[TABLE]

(compare with (5)). The polynomial $P(X)$ is called a characteristic polynomial of the sequence $a$.

2 Oberst’s algebraic dynamical systems and the Cauchy’s homogeneous problem

Let $\operatorname{\mathbb{F}}$ be a commutative field. For an integer $r\geqslant 1$, let $X_{1},\ldots,X_{r}$ and $Y_{1},\ldots,Y_{r}$ distinct variables. The letter $X$ (resp. $Y$) will denote the set of variables $(X_{1},\ldots,X_{r})$ (resp. $(Y_{1},\ldots,Y_{r}$)). For $\alpha=(\alpha_{1},\ldots,\alpha_{r})\in\operatorname{\mathbb{N}}^{r}$, we define $X^{\alpha}$ (resp. $Y^{\alpha}$) as the product

[TABLE]

Let $\operatorname{\boldsymbol{D}}=\operatorname{\mathbb{F}}[X_{1},\ldots,X_{r}]=\operatorname{\mathbb{F}}[X]$ be the $\operatorname{\mathbb{F}}$-vector space of the polynomials with the $r$ variables $X_{1},\ldots,X_{r}$ and entries in $\operatorname{\mathbb{F}}$. An element of $\operatorname{\boldsymbol{D}}$ can be uniquely written as

[TABLE]

where $d_{\alpha}=0$ except for a finite number of $\alpha$’s. We fix a monomial ordering $\leqslant_{T}$ on $\operatorname{\mathbb{N}}^{r}$, ([3, 10]) which is then a well ordering. For a non-zero element $d(X)\in\operatorname{\boldsymbol{D}}$, we define the leading exponent of $d(X)$ by

[TABLE]

Let $\operatorname{\boldsymbol{A}}=\operatorname{\mathbb{F}}[[Y_{1},\ldots,Y_{r}]]=\operatorname{\mathbb{F}}[[Y]]$ be $\operatorname{\mathbb{F}}$-vector space of the formal power series with the variables $Y_{1},\ldots,Y_{r}$ and entries in $\operatorname{\mathbb{F}}$. An element of $\operatorname{\boldsymbol{A}}$ can be uniquely written as

[TABLE]

where $W_{\alpha}\in\operatorname{\mathbb{F}}$ for all $\alpha\in\operatorname{\mathbb{N}}^{r}$.

For integers $k,l\geqslant 1$, the set of matrices with $k$ rows and $l$ columns with entries in $\operatorname{\boldsymbol{D}}$ is denoted by $\operatorname{\boldsymbol{D}}^{k,l}$. An element $R(X)\in\operatorname{\boldsymbol{D}}^{k,l}$ is of the form

[TABLE]

where $R_{ij}(X)\in\operatorname{\boldsymbol{D}}$ for $i=1,\dots,k$ and $j=1,\ldots,l$. With the multiplication by polynomials as external operation of $\operatorname{\boldsymbol{D}}$ on $\operatorname{\boldsymbol{D}}^{k,l}$, this latter becomes $\operatorname{\boldsymbol{D}}$-module. The notation $\operatorname{\boldsymbol{D}}^{l}$ (resp. $\operatorname{\boldsymbol{A}}^{l}$) will be for the set of polynomials with one row and $l$ columns (resp. power series in $\operatorname{\boldsymbol{A}}$ with $l$ rows and one column).

The external operation, (also called multiplication) of $\operatorname{\boldsymbol{D}}$ on $\operatorname{\boldsymbol{A}}$ is defined by

[TABLE]

This operation provides $\operatorname{\boldsymbol{A}}$ with a $\operatorname{\boldsymbol{D}}$-module structure. The set $\operatorname{\boldsymbol{A}}^{l}$ becomes a $\operatorname{\boldsymbol{D}}$-module too, with the external operation

[TABLE]

More generally, given $R(X)\in\operatorname{\boldsymbol{D}}^{k,l}$, the following mapping, also denoted by $R(X)$, is a $\operatorname{\boldsymbol{D}}$-linear mapping of modules

[TABLE]

where

[TABLE]

is $\operatorname{\boldsymbol{D}}$-linear ([1, 10]. Note that this expression of $R(X)\circ W(Y)$ is similar to that of the usual matrix-vector multiplication). Its kernel is then a $\operatorname{\boldsymbol{D}}$-submodule of $\operatorname{\boldsymbol{A}}^{l}$. This legitimates the following definition:

Definition 1** (Oberst, [10]).**

An algebraic dynamical system (or simply a system) is a $\operatorname{\boldsymbol{D}}$-submodule of $\operatorname{\boldsymbol{A}}^{l}$ of the form

[TABLE]

where $R(X)\in\operatorname{\boldsymbol{D}}^{k,l}$ and also denotes the $\operatorname{\boldsymbol{D}}$-linear mapping of $\operatorname{\boldsymbol{D}}$-modules defined by (8).

The integer $r$ is the dimension of the system. Willems treated the one-dimensional case only. An element $W$ of a system $S$ is called a trajectory.

Example 1** (Linear recurring sequence).**

Take $r=1$. Then $\operatorname{\mathbb{F}}[X]$ is the set of univariate polynomials in $X$ and $\operatorname{\mathbb{F}}[[Y]]$ the set of power series in the unique variable $Y$. A polynomial $P(X)\in\operatorname{\mathbb{F}}[X]$ defines the dynamical system

[TABLE]

If $P(X)=0$, then $\ker P(X)=\operatorname{\mathbb{F}}[[Y]]$, otherwise, using (5), for $r=1$, we are in the situation in (3), so that the elements of $\operatorname{Ker}P(X)$ are the linear recurring sequences having $P(X)$ as a characteristic polynomial.

For a subset $P$ of $\operatorname{\boldsymbol{D}}^{l}$ and a subset $Q$ of $\operatorname{\boldsymbol{A}}^{l}$, we define their orthogonals by

[TABLE]

$P^{\perp}$ is a $\operatorname{\boldsymbol{D}}$-submodule of $\operatorname{\boldsymbol{A}}^{l}$ and $Q^{\perp}$ is a $\operatorname{\boldsymbol{D}}$-submodule of $\operatorname{\boldsymbol{D}}^{l}$ ([10]).

Example 2**.**

For a non-zero polynomial $P(X)\in\operatorname{\boldsymbol{D}}$, the set $P(X)^{\perp}=\{P(X)\}^{\perp}$ is those of the LRS having $P(X)$ as a characteristic polynomial. For a power series $W(Y)\in\operatorname{\boldsymbol{A}}$, the set $W(Y)^{\perp}=\{W(Y)\}^{\perp}$ is those of the characteristic polynomials of $W(Y)$ and the zero polynomial.

In [10], it is proven that every system $S$ admits an Input/Output representation

[TABLE]

where $m,p\geqslant 1$ are integers with

[TABLE]

the columns of $P$ being $\mathbf{K}$-linearly independent with $\mathbf{K}=\operatorname{\mathbb{F}}(X_{1},\ldots,X_{r})$ and

[TABLE]

The system written in the form (9) is called an I/O system.

Now, we need some notations for an integer $p\geqslant 0$, we write

[TABLE]

and $\Gamma$ denotes a subset of $[p]\times\operatorname{\mathbb{N}}^{r}$ (If $p=1$, then we identify $[p]\times\operatorname{\mathbb{N}}^{r}$ with $\operatorname{\mathbb{N}}^{r}$). We may identify $\;\;\operatorname{\mathbb{F}}^{[p]\times\operatorname{\mathbb{N}}^{r}}$ with $\operatorname{\boldsymbol{A}}^{p}$ and consider $\operatorname{\mathbb{F}}_{p}^{\Gamma}$ as a subset of $\operatorname{\boldsymbol{A}}^{p}$, where $\operatorname{\mathbb{F}}^{\Gamma}$ is the set of mappings from $\Gamma$ to $\operatorname{\mathbb{F}}$.

Definition 2** (Oberst, [10]).**

The homogeneous Cauchy problem $(P(X),0,\Gamma)$ for the I/O system (9) is the system of equations

[TABLE]

where the unknown is $V\in\operatorname{\boldsymbol{A}}^{p}$, the initial data being $V_{0}\in\operatorname{\mathbb{F}}^{\Gamma}$.

3 On point algebraic-geometry codes

For algebraic geometry, we refer to [3, 6] and the construction of one point AG codes, we refer to [7]. We recall here the basic notations and ideas for the construction of such codes. From now on, $\operatorname{\mathbb{F}}_{q}$ denotes the Galois field with $q$ elements, where $q$ is a power of a positive prime integer. Let $\operatorname{\mathbb{F}}$ be the algebraic closure of $\operatorname{\mathbb{F}}_{q}$ and $r\geqslant 1$ and integer.

We write $X=(X_{1},\ldots,X_{r})$ as in section 2. We will use the polynomial rings $\operatorname{\mathbb{F}}_{q}[x_{1},\ldots,X_{r}],\operatorname{\mathbb{F}}[X_{1},\ldots,X_{r}]$ and $\operatorname{\mathbb{F}}[X_{0},\ldots,X_{r}]$, where $X_{0}$ is another variable. We denote by $\mathbb{P}^{r}(\operatorname{\mathbb{F}})$ the $r$-dimensional projective space over $\operatorname{\mathbb{F}}$. An element of $\mathbb{P}^{r}(\operatorname{\mathbb{F}})$ is of the form $P=(a_{0}:a_{1}:\ldots:a_{r})$, where $a_{i}\in\operatorname{\mathbb{F}}$. The hyperplane at infinity is the set $\mathbb{H}^{r}_{\infty}$ of the points of the form $(0:a_{1}:\ldots:a_{r})\in\operatorname{\mathbb{F}}^{r+1}$. One may then write (up to an isomorphism) $\mathbb{P}^{r}(\operatorname{\mathbb{F}})=\operatorname{\mathbb{F}}^{r}\cup\mathbb{H}^{r}_{\infty}$, and identify a point $P=(a_{1}:\ldots:a_{r})\in\operatorname{\mathbb{F}}_{q}$ with the point $P=(1:a_{1}:\ldots:a_{r})\in\mathbb{P}^{r}(\operatorname{\mathbb{F}})$.

We will consider a smooth irreducible projective curve $\mathcal{X}$ *defined over

$\operatorname{\mathbb{F}}_{q}[X_{1},\ldots,X_{r}]$*. It is an affine variety of dimension $1$, defined by

[TABLE]

where $\mathcal{F}$ is a set of homogeneous polynomials of $\operatorname{\mathbb{F}}[X_{0},X_{1},\ldots,X_{r}]$. The ideal of $\mathcal{X}$ is

[TABLE]

The coordinate ring of $\mathcal{X}$ is the ring

[TABLE]

The $\operatorname{\mathbb{F}}[\mathcal{X}]$ is an integral domain and its field of fractions is called the *field of rational functions *on $\mathcal{X}$ and denoted by $\operatorname{\mathbb{F}}(\mathcal{X})$. We may write

[TABLE]

The curve $\mathcal{X}$ is constructed from a smooth irreducible affine curve $\mathcal{X}_{aff}$ defined over $\operatorname{\mathbb{F}}_{q}[X_{1},\ldots,X_{r}]$, which is of the form

[TABLE]

where $\mathcal{G}$ is a set of polynomials in $\operatorname{\mathbb{F}}[X_{1},\ldots,X_{r}]$. The ideal of $\mathcal{X}_{aff}$ is

[TABLE]

The terminology “$\mathcal{X}$ (or $\mathcal{X}_{aff}$) defined over $\operatorname{\mathbb{F}}_{q}[X_{1},\ldots,X_{r}]$” means that the ideal $I(\mathcal{X}_{aff})$ is generated by polynomials in $\operatorname{\mathbb{F}}_{q}[X_{1},\ldots,X_{r}]$. As in (14) and (15), we define the coordinate ring (resp. the field of rational functions) of $\mathcal{X}_{aff}$ :

[TABLE]

The field of rational functions $\operatorname{\mathbb{F}}(\mathcal{X})$ is birationally equivalent to $\operatorname{\mathbb{F}}(\mathcal{X}_{aff})$, so we may use this latter only. Moreover, the projective curve we consider will have a unique point $Q$ lying at the hyperplane at infinity and is in special position with respect to $Q$. Let $a$ be an integer verifying

[TABLE]

where $g$ is the genus of $\mathcal{X}$. Let $\mathcal{L}(aQ)$ be the set of the functions $\phi$ on $\operatorname{\mathbb{F}}(\mathcal{X}_{aff})$ which have a unique pole at $Q$, of order less than $a$.

Let $\mathcal{P}=\{P_{1},\ldots,P_{n}\}$ a set of points of $\mathcal{X}$. The code $\mathcal{C}_{L}(\mathcal{P},aQ)$ is the evaluation of the functions of the vector space $\mathcal{L}(aQ)$

[TABLE]

and its dual is

[TABLE]

There exists $o_{1},\ldots,o_{r}\in\operatorname{\mathbb{N}}\setminus\{0\}$ such that for a monomial $M=X_{1}^{i_{1}}\cdots X_{r}^{i_{r}}$, the *pole order *of $M$ at $Q$ is

[TABLE]

thus $v_{Q}(X_{i})=-o_{i}$ for $i=1,\ldots,r$. We may define the monomial order

[TABLE]

A generating family of $\mathcal{C}_{L}(\mathcal{P},aQ)$ is then

[TABLE]

with $X^{\alpha}(P)=x_{1}^{\alpha_{1}}\cdots x_{r}^{\alpha_{r}}$, where $P=(1:x_{1}:\ldots:x_{r})$. As a consequence, one has a much simpler form of the code $\mathcal{C}_{L}(\mathcal{P},aQ)^{\perp}$:

[TABLE]

Now, we use the sets $\operatorname{\boldsymbol{A}}$ and $\operatorname{\boldsymbol{D}}$, defined as in Section 2, using the field $\operatorname{\mathbb{F}}$.

Definition 3** ([7]).**

The generalized transform is

[TABLE]

This transform defines an $\operatorname{\mathbb{F}}_{q}$-injective linear mapping.

Now, we consider the situation in which a codeword $c$ of our code has been sent through a communication channel. The received word, say $w\in\operatorname{\mathbb{F}}_{q}^{n}$ is not necessarily equal to $c$, because of a possible error $e$ produced by the channel. We may write

[TABLE]

Of course, the receiver does not know either $c$ or $e$. The problem is to find $e$ in order to know $c=w-e$. Instead of finding $e$ directly, one constructs the syndrome array.

Definition 4** ([7]).**

The syndrome array is

[TABLE]

Definition 5** ([7]).**

The errors locator ideal is

[TABLE]

We are going to show that if $E\neq 0$, then $E^{\perp}\neq\{0\}$, which means that $E$ is a linear recurring sequence (1). Using (5), this yields

[TABLE]

where $F(X)=\sum_{\beta\in\operatorname{\mathbb{N}}^{r}}F_{\beta}X^{\beta}$. For this purpose, we will need the following lemma:

Lemma 1** ([7]).**

For an AG code, one has

[TABLE]

*where $\operatorname{Supp}(e)=\{P_{i}\in\mathcal{P}\;|\;(i\in\{1,\ldots,n\})\;\ e_{i}\neq 0\}$.

We then have what we need :

Corollary 1**.**

*If $E\neq 0$, then $E^{\perp}\neq\{0\}$.

Proof.

If $\operatorname{Supp}(e)=\{Q_{1},\ldots,Q_{m}\}\subset\mathcal{P}$ where

[TABLE]

then the polynomial

[TABLE]

is non-zero and verifies

[TABLE]

Thus $F(X)\in\mathbf{I}(\operatorname{Supp}(e))$ and by lemma 1, it follows that $F(X)\in E^{\perp}$.∎∎

We have obtained what we need :

Corollary 2**.**

. The syndrome array $E$ is a linear recurring sequence.

4 Cauchy’s equations for the syndrome array

By Corollary 1, if $E\neq 0$, the ideal $E^{\perp}$ is non zero. Let $\leqslant_{+}$ be the partial order defined on $\operatorname{\mathbb{N}}^{r}$ by

[TABLE]

for $\alpha$ and $\beta\in\operatorname{\mathbb{N}}^{r}$. Then $E^{\perp}$ has a Gröbner basis $\mathcal{G}=\{G_{1}(X),...,G_{k}(X)\}$ (with respect to the monomial order $\leqslant_{T}$ in Section 2) where $G_{i}(X)\in\operatorname{\boldsymbol{D}}$ for $i=1,\ldots k$ ([3, 10]). Consider the “Delta-sets” ([5, 7, 11])

[TABLE]

and the set

[TABLE]

Since $\mathcal{G}$ is a Gröbner basis of $E^{\perp}$, we have

[TABLE]

so that

[TABLE]

([3, 7, 10]). Let $G(X)$ be the matrix

[TABLE]

and consider the system

[TABLE]

The (unique) column of the matrix $G(X)$ is obviously $\mathbf{K}$-linearly independent, where $\mathbf{K}$ is the field of fractions of $\operatorname{\boldsymbol{D}}$. Thus, according to 9, $S$ is a I/O system, with $p=m=1,Q=0\in\operatorname{\boldsymbol{D}}^{k,1}$ and $U=0\in\operatorname{\boldsymbol{A}}$. Therefore, we may, as in 2, consider the Cauchy’s homogeneous equations with respect to $S$.

Here is our main theorem:

Theorem 1**.**

*The syndrome $E$ is the unique solution of the Cauchy’s homogeneous equations

$(G(X),0,\Delta(\mathcal{G}))$:*

[TABLE]

*where $V_{0}\in\operatorname{\mathbb{F}}_{q}^{\Delta(\mathcal{G})}$ is an arbitrary element.

Proof.

We are going to prove that (27) is verified by all element $W$ of $S$, hence true for the particular case $W=E$. The first equation of (27) follows from the construction of $S$. Now, write $\Gamma=\Delta(\mathcal{G})$. Each trajectory $W$ of $S$ is then uniquely determined by its restriction to $\Gamma$, which is $V_{0}$. Indeed, suppose that $W_{\alpha}$ is known and is equal to $V_{0\alpha}$ for $\alpha\in\Gamma$. We are going to calculate $W_{\alpha}$ by nœtherian or transfinite induction (see [10]) on $\alpha\in\operatorname{\mathbb{N}}^{r}\setminus\Gamma$. Let $\alpha_{0}=\min_{\leqslant_{T}}(\operatorname{\mathbb{N}}^{r}\setminus\Gamma)$. Using (25), there exists $G_{k}(X)\in\mathcal{G}$ such that $\alpha_{0}$ is an entry with respect to  $G_{k}(X)$, i.e. there exists $t\in\operatorname{\mathbb{N}}^{r}$ such that $\alpha_{0}=t+d$ with $d=\operatorname{LE}G_{k}(X)$. Since $G_{k}(X)\circ W=0$, we then have

[TABLE]

and

[TABLE]

But, since

[TABLE]

and by the choice of $\alpha_{0}$, we necessarily have $\alpha+t\in\Gamma$. Thus, $W_{\alpha+t}=V_{\alpha+t}$ is already known and $W_{\alpha_{0}}$ can be calculated by (29) for $\alpha_{0}=\min_{\leqslant_{T}}(\operatorname{\mathbb{N}}^{r}\setminus\Gamma)$. Now, let $\alpha\in\operatorname{\mathbb{N}}^{r}\setminus\Gamma$ and suppose, by the recurrence hypothesis that $W_{v}$ is already calculated for $v$ with $\alpha_{0}\leqslant_{T}v<\alpha$. Using again (25) there exists $t\in\operatorname{\mathbb{N}}^{r}$ and $G_{l}(X)\in\mathcal{G}$ such that $\alpha=t+d$, with $d=\operatorname{LE}(G_{l})$. As in (29), we have

[TABLE]

and $W_{\alpha+t}$ is already known by the recurrence hypothesis, since we have $\alpha+t<_{T}\alpha+d=\alpha$. Thus $W_{\alpha}$ can be uniquely calculated by(30). Therefore, by nœtherian recurrence, we can calculate $W_{\alpha}$ for $\alpha\in\operatorname{\mathbb{N}}^{r}\setminus\Gamma$. ∎

Now, consider the one dimensional case $r=1$. Let :

$\bullet$ $E$ be the generalized transform of the error $e$,

$\bullet$ $F(X)$ be the characteristic polynomial of $E^{\perp}$ and $d=\deg F(X)>1$,

$\bullet$ $S=\operatorname{Ker}F(X)$.

Then $\Gamma$ and $\Delta(E^{\perp})$ are of the following forms

[TABLE]

and we have a simpler version of lemma 27:

Every element $W\in S$ is the unique solution of the Cauchy’s equations

[TABLE]

We can directly calculate $W$ with $F(X)$ and $V$. Indeed, write $F(X)=\sum_{i=1}^{d}F_{i}X^{i}$ with $d=\deg F(X)$ et $F_{d}=1$. We have $W_{h}=V_{h}$ for $h\leqslant d-1$. For $k\in\operatorname{\mathbb{N}}$, we have :

[TABLE]

and this defines $W_{n}$ using $W_{h}$, with $h<n$.∎

We may consider (27) as the fundamental equation which lies behind the BMS algorithm in the decoding process. However, at the beginning, the matrix $G(X)$ in (27), is of course unknown, because it is constructed from the unknown syndrome $E$. But, by (21), we have $\operatorname{\boldsymbol{GT}}(w)=\operatorname{\boldsymbol{GT}}(c+e)=\operatorname{\boldsymbol{GT}}(c)+\operatorname{\boldsymbol{GT}}(e)$. Using (19) and (20), we have $[\operatorname{\boldsymbol{GT}}(c)]_{\alpha}=0$ whenever $\operatorname{wdeg}(\alpha)\leqslant a$ (where $[W]_{\alpha}$ also denotes the coefficient of the power series $W\in\operatorname{\mathbb{F}}_{q}[Y]$ with respect to $Y^{\alpha}$). Let $Z$ be the set

[TABLE]

We then have $[\operatorname{\boldsymbol{GT}}(w)]_{\alpha}=[\operatorname{\boldsymbol{GT}}(e)]_{\alpha}=E_{\alpha}$ for $\alpha\in Z$, so that $E_{\alpha}$ is known on the set $Z$ only since it is equal to $[\operatorname{\boldsymbol{GT}}(w)]_{\alpha}$ and $w$ is known.

The general idea of the BMS algorithm is to use these known terms of $E$ to construct some polynomials, which are valid recurrence relations for theses terms. Then, using these polynomials, the algorithm calculates more terms of $E$ and so on. Finally, the algorithm finds a Gröbner basis of the ideal $E^{\perp}$, which, in turn, by (27), allows to calculate $E$, and $e$, using the inverse of the $\operatorname{\boldsymbol{GT}}$ transform.