Computing column bases of polynomial matrices

Zhou, Wei; Labahn, George

doi:10.1145/2465506.2465947

Cited by 12 publications

(31 citation statements)

References 16 publications

(35 reference statements)

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“…The situation here is similar to the situation in computing a left kernel basis in the column basis computation from [15]. That is, the order b + 1, or equivalently, the s-row degrees of M T may be unbalanced and can have degree as large as s. We therefore follow the same process as in the computation of column bases [15].…”

Section: Efficient Computationmentioning

confidence: 98%

“…Proof. This follows from [15,Lemma 3.3] which tells us that if U is a column basis of A then A = UB.…”

Section: Unimodular Completionmentioning

confidence: 99%

“…The cost of column basis computation for F ∈ K[z] m×n is given in [15] with the cost given as O ∼ nm ω−1ŝ field operations in K, whereŝ is the average column degree of F.…”

Section: Kernel Bases and Column Basesmentioning

confidence: 99%

“…In this paper we also will need the following lemma from [15,13] which shows that a left kernel basis of a right kernel basis is contained in an order basis of a right kernel basis.…”

Section: Order Basesmentioning

confidence: 99%

“…For simplicity and without loss of generality, we focus our discussion on the situation where the unimodular completion always exists. For an input matrix that cannot be completed to a unimodular matrices, we can always factor it as F = TG using the column basis algorithm from [15], something which can be done efficiently. Then the right factor G can be unimodularly completed.…”

Section: Introductionmentioning

confidence: 99%

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Unimodular completion of polynomial matrices

Zhou

Labahn

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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Given a rectangular matrix F ∈ K[x] m×n with m < n of univariate polynomials over a field K we give an efficient algorithm for computing a unimodular completion of F. Our algorithm is deterministic and computes such a completion, when it exists, with cost O ∼ (n ω s) field operations from K. Here s is the average of the m largest column degrees of F and ω is the exponent on the cost of matrix multiplication. Here O ∼ is big-O but with log factors removed. If a unimodular completion does not exist for F, our algorithm computes a unimodular completion for a right cofactor of a column basis of F, or equivalently, computes a completion that preserves the generalized determinant.

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Section: Efficient Computationmentioning

confidence: 98%

“…Proof. This follows from [15,Lemma 3.3] which tells us that if U is a column basis of A then A = UB.…”

Section: Unimodular Completionmentioning

confidence: 99%

“…The cost of column basis computation for F ∈ K[z] m×n is given in [15] with the cost given as O ∼ nm ω−1ŝ field operations in K, whereŝ is the average column degree of F.…”

Section: Kernel Bases and Column Basesmentioning

confidence: 99%

“…In this paper we also will need the following lemma from [15,13] which shows that a left kernel basis of a right kernel basis is contained in an order basis of a right kernel basis.…”

Section: Order Basesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Unimodular completion of polynomial matrices

Zhou

Labahn

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix

Labahn

Neiger

Zhou

2017

Journal of Complexity

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International audienceGiven a nonsingular $n \times n$ matrix of univariate polynomials over a field$\mathbb{K}$, we give fast and deterministic algorithms to compute itsdeterminant and its Hermite normal form. Our algorithms use$\widetilde{\mathcal{O}}(n^\omega \lceil s \rceil)$ operations in$\mathbb{K}$, where $s$ is bounded from above by both the average of thedegrees of the rows and that of the columns of the matrix and $\omega$ is theexponent of matrix multiplication. The soft-$O$ notation indicates thatlogarithmic factors in the big-$O$ are omitted while the ceiling functionindicates that the cost is $\widetilde{\mathcal{O}}(n^\omega)$ when $s =o(1)$. Our algorithms are based on a fast and deterministic triangularizationmethod for computing the diagonal entries of the Hermite form of a nonsingularmatrix

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Fast Decoding of AG Codes

Beelen

Rosenkilde

Solomatov

2022

IEEE Trans. Inform. Theory

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We present an efficient list decoding algorithm in the style of Guruswami-Sudan for algebraic geometry codes. Our decoder can decode any such code using Õps ω µ ω´1 pn `gqq operations in the underlying finite field, where n is the code length, g is the genus of the function field used to construct the code, s is the multiplicity parameter, is the designed list size and µ is the smallest positive element in the Weierstrass semigroup at some chosen place; the "soft-O" notation Õp¨q is similar to the "big-O" notation Op¨q, but ignores logarithmic factors. For the interpolation step, which constitutes the computational bottleneck of our approach, we use known algorithms for univariate polynomial matrices, while the root-finding step is solved using existing algorithms for root-finding over univariate power series.

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Computing column bases of polynomial matrices

Cited by 12 publications

References 16 publications

Unimodular completion of polynomial matrices

Unimodular completion of polynomial matrices

Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix

Fast Decoding of AG Codes

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