Linear algebra for computing Gröbner bases of linear recursive multidimensional sequences
The so-called Berlekamp–Massey–Sakata algorithm computes a Gröbner basis of a 0-dimensional ideal of relations satisfied by an input table. It extends the Berlekamp–Massey algorithm to n-dimensional tables, for n>1. We investigate this problem and design several algorithms for computing such a Gr...
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| Vydáno v: | Journal of symbolic computation Ročník 83; číslo Supplement C; s. 36 - 67 |
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| Jazyk: | angličtina |
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Elsevier Ltd
01.11.2017
Elsevier |
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| ISSN: | 0747-7171, 1095-855X |
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| Abstract | The so-called Berlekamp–Massey–Sakata algorithm computes a Gröbner basis of a 0-dimensional ideal of relations satisfied by an input table. It extends the Berlekamp–Massey algorithm to n-dimensional tables, for n>1.
We investigate this problem and design several algorithms for computing such a Gröbner basis of an ideal of relations using linear algebra techniques. The first one performs a lot of table queries and is analogous to a change of variables on the ideal of relations.
As each query to the table can be expensive, we design a second algorithm requiring fewer queries, in general. This FGLM-like algorithm allows us to compute the relations of the table by extracting a full rank submatrix of a multi-Hankel matrix (a multivariate generalization of Hankel matrices).
Under some additional assumptions, we make a third, adaptive, algorithm and reduce further the number of table queries. Then, we relate the number of queries of this third algorithm to the geometry of the final staircase and we show that it is essentially linear in the size of the output when the staircase is convex. As a direct application to this, we decode n-cyclic codes, a generalization in dimension n of Reed Solomon codes.
We show that the multi-Hankel matrices are heavily structured when using the LEX ordering and that we can speed up the computations using fast algorithms for quasi-Hankel matrices. Finally, we design algorithms for computing the generating series of a linear recursive table. |
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| AbstractList | The so-called Berlekamp–Massey–Sakata algorithm computes a Gröbner basis of a 0-dimensional ideal of relations satisfied by an input table. It extends the Berlekamp–Massey algorithm to n-dimensional tables, for n>1.
We investigate this problem and design several algorithms for computing such a Gröbner basis of an ideal of relations using linear algebra techniques. The first one performs a lot of table queries and is analogous to a change of variables on the ideal of relations.
As each query to the table can be expensive, we design a second algorithm requiring fewer queries, in general. This FGLM-like algorithm allows us to compute the relations of the table by extracting a full rank submatrix of a multi-Hankel matrix (a multivariate generalization of Hankel matrices).
Under some additional assumptions, we make a third, adaptive, algorithm and reduce further the number of table queries. Then, we relate the number of queries of this third algorithm to the geometry of the final staircase and we show that it is essentially linear in the size of the output when the staircase is convex. As a direct application to this, we decode n-cyclic codes, a generalization in dimension n of Reed Solomon codes.
We show that the multi-Hankel matrices are heavily structured when using the LEX ordering and that we can speed up the computations using fast algorithms for quasi-Hankel matrices. Finally, we design algorithms for computing the generating series of a linear recursive table. The so-called Berlekamp~-- Massey~-- Sakata algorithmcomputes a Gröbner basis of a $0$-dimensional ideal of relations satisfied by an inputtable. It extends the Berlekamp~-- Massey algorithmto $n$-dimensional tables, for $n>1$.We investigate this problem and design several algorithms forcomputing such a Gröbner basis of an ideal of relations using linearalgebra techniques.The first one performs a lot of table queries andis analogous to a change of variables on the ideal of relations.As each query to the table can be expensive,we design a second algorithmrequiring fewer queries, in general.This \textsc{FGLM}-like algorithm allows us to compute the relations of thetable by extracting a full rank submatrix of a \emph{multi-Hankel}matrix (a multivariate generalization of Hankel matrices).Under someadditional assumptions, we make a third, adaptive, algorithm and reducefurther the number of table queries.Then, we relate the number of queries ofthis third algorithm to the\emph{geometry} of the final staircase and we show that it isessentially linear in the size of the output when the staircase is convex.As a direct application to this, we decode $n$-cyclic codes, ageneralization in dimension $n$ of Reed Solomon codes. We show that the multi-Hankelmatrices are heavily structured when using the \textsc{LEX} orderingand that we can speed up the computations using fast algorithms forquasi-Hankel matrices.Finally, we designalgorithms for computing the generating series of a linear recursivetable. |
| Author | Boyer, Brice Faugère, Jean-Charles Berthomieu, Jérémy |
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| Cites_doi | 10.1002/sapm1946251261 10.1145/2500122 10.1109/18.86974 10.1016/S0024-3795(99)00251-7 10.1109/18.476246 10.1016/S0012-365X(00)00147-3 10.1090/S0002-9947-1952-0049591-8 10.1016/0024-3795(89)90032-3 10.1017/S0305004100012354 10.1016/j.laa.2010.06.046 10.1016/j.jsc.2012.05.008 10.1006/jsco.1993.1051 10.1007/978-3-319-32859-1_11 10.1007/BF01876039 10.1016/j.laa.2013.06.016 10.1109/TIT.1968.1054109 10.1016/S0747-7171(88)80033-6 10.1109/TIT.1969.1054260 10.1109/18.59953 10.1016/0890-5401(90)90039-K |
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| Issue | Supplement C |
| Keywords | Multidimensional linear recursive sequence The FGLM algorithm The BMS algorithm Gröbner basis computation 0-dimensional ideal FGLM algorithm Berlekamp - Massey - Sakata algorithm multidimensional linear recurrent sequence 0$-dimensional ideal |
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| Snippet | The so-called Berlekamp–Massey–Sakata algorithm computes a Gröbner basis of a 0-dimensional ideal of relations satisfied by an input table. It extends the... The so-called Berlekamp~-- Massey~-- Sakata algorithmcomputes a Gröbner basis of a $0$-dimensional ideal of relations satisfied by an inputtable. It extends... |
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| SubjectTerms | 0-dimensional ideal Computer Science Gröbner basis computation Multidimensional linear recursive sequence Symbolic Computation The BMS algorithm The FGLM algorithm |
| Title | Linear algebra for computing Gröbner bases of linear recursive multidimensional sequences |
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