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Kosuke Sakata

Publications and source records attributed to Kosuke Sakata.

5 recordsLinked to original sources
arXiv2025

An Algorithm for Computing the Leading Monomials of a Minimal Groebner Basis of Generic Sequences

We present an efficient algorithm for computing the leading monomials of a minimal Groebner basis of a generic sequence of homogeneous polynomials. Our approach bypasses costly polynomial reductions by exploiting structural properties conjectured to hold for generic sequences-specifically, that their leading monomial ideals are weakly reverse lexicographic and that their Hilbert series follow a known closed-form expression. The algorithm incrementally constructs the set of leading monomials degree by degree by comparing Hilbert functions of monomial ideals with the expected Hilbert series of the input ideal. To enhance computational efficiency, we introduce several optimization techniques that progressively narrow the search space and reduce the number of divisibility checks required at each step. We also refine the loop termination condition using degree bounds, thereby avoiding unnecessary recomputation of Hilbert series. Experimental results confirm that the proposed method substantially reduces both computation time and memory usage compared to conventional Groebner basis computations for computing the leading monomials of a minimal Groebner basis of generic sequences.

cs.SC↗
arXiv2024

Superspecial genus-$4$ double covers of elliptic curves

In this paper we study genus-$4$ curves obtained as double covers of elliptic curves. Firstly we shall give explicit defining equations of such curves with explicit criterion for whether it is nonsingular, and show the irreducibility of the long polynomial determining whether the genus-4 curve is nonsingular or not, in any characteristic $\ne 2,3$. Secondly, as an application, we enumerate superspecial genus-$4$ double covers of elliptic curves in small characteristic.

math.AG↗
arXiv2018

An efficient reduction strategy for signature-based algorithms to compute Groebner basis

This paper introduces a strategy for signature-based algorithms to compute Groebner basis. The signature-based algorithms generate S-pairs instead of S-polynomials, and use s-reduction instead of the usual reduction used in the Buchberger algorithm. There are two strategies for s-reduction: one is the only-top reduction strategy which is the way that only leading monomials are s-reduced. The other is the full reduction strategy which is the way that all monomials are s-reduced. A new strategy, which we call selective-full strategy, for s-reduction of S-pairs is introduced in this paper. In the experiment, this strategy is efficient for computing the reduced Groebner basis. For computing a signature Groebner basis, it is the most efficient or not the worst of the three strategies.

cs.SC↗
arXiv2013

Spatially-Coupled Precoded Rateless Codes

Raptor codes are rateless codes that achieve the capacity on the binary erasure channels. However the maximum degree of optimal output degree distribution is unbounded. This leads to a computational complexity problem both at encoders and decoders. Aref and Urbanke investigated the potential advantage of universal achieving-capacity property of proposed spatially-coupled (SC) low-density generator matrix (LDGM) codes. However the decoding error probability of SC-LDGM codes is bounded away from 0. In this paper, we investigate SC-LDGM codes concatenated with SC low-density parity-check codes. The proposed codes can be regarded as SC Hsu-Anastasopoulos rateless codes. We derive a lower bound of the asymptotic overhead from stability analysis for successful decoding by density evolution. The numerical calculation reveals that the lower bound is tight. We observe that with a sufficiently large number of information bits, the asymptotic overhead and the decoding error rate approach 0 with bounded maximum degree.

cs.IT↗