Minimum distance functions of complete intersections

We study the minimum distance function of a complete intersection graded ideal in a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial ideal is a complete intersection, we use the footprint function to give a sharp lower bound for the minimum distance function. Then we show some applications to coding theory.

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The Stanley regularity of complete intersections and ideals of mixed products

Journal of Algebra and Its Applications ◽

10.1142/s0219498817501225 ◽

2016 ◽

Vol 16 (07) ◽

pp. 1750122

Author(s):

Lizhong Chu ◽

V. H. Jorge Pérez

Keyword(s):

Complete Intersection ◽

Polynomial Ring ◽

Monomial Ideal ◽

Monomial Ideals ◽

Complete Intersections

Let [Formula: see text] be a polynomial ring over a field [Formula: see text] and [Formula: see text] a monomial ideal. We give some inequalities on Stanley regularity of monomial ideals. As consequences, we prove that [Formula: see text] and [Formula: see text] hold in the following cases: (1) [Formula: see text] is a complete intersection; (2) [Formula: see text] is an ideal of mixed products.

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The Erdős–Burgess constant of the multiplicative semigroup of a factor ring of 𝔽q[x]

International Journal of Number Theory ◽

10.1142/s1793042119500015 ◽

2019 ◽

Vol 15 (01) ◽

pp. 131-136 ◽

Cited By ~ 2

Author(s):

Haoli Wang ◽

Jun Hao ◽

Lizhen Zhang

Keyword(s):

Lower Bound ◽

Finite Field ◽

Prime Ideal ◽

Polynomial Ring ◽

Prime Power ◽

Quotient Ring ◽

Prime Ideals ◽

Multiplicative Semigroup ◽

Associative Operation ◽

Sharp Lower Bound

Let [Formula: see text] be a commutative semigroup endowed with a binary associative operation [Formula: see text]. An element [Formula: see text] of [Formula: see text] is said to be idempotent if [Formula: see text]. The Erdős–Burgess constant of [Formula: see text] is defined as the smallest [Formula: see text] such that any sequence [Formula: see text] of terms from [Formula: see text] and of length [Formula: see text] contains a nonempty subsequence, the sum of whose terms is idempotent. Let [Formula: see text] be a prime power, and let [Formula: see text] be the polynomial ring over the finite field [Formula: see text]. Let [Formula: see text] be a quotient ring of [Formula: see text] modulo any ideal [Formula: see text]. We gave a sharp lower bound of the Erdős–Burgess constant of the multiplicative semigroup of the ring [Formula: see text], in particular, we determined the Erdős–Burgess constant in the case when [Formula: see text] is the power of a prime ideal or a product of pairwise distinct prime ideals in [Formula: see text].

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Invariant subrings which are complete intersections, I: (Invariant subrings of finite Abelian groups)

Nagoya Mathematical Journal ◽

10.1017/s0027763000018687 ◽

1980 ◽

Vol 77 ◽

pp. 89-98 ◽

Cited By ~ 13

Author(s):

Keiichi Watanabe

Keyword(s):

Abelian Group ◽

Complete Intersection ◽

Polynomial Ring ◽

Abelian Groups ◽

Finite Abelian Group ◽

Finite Subgroup ◽

Complete Intersections ◽

Complex Numbers ◽

Finite Abelian Groups

Let G be a finite subgroup of GL(n, C) (C is the field of complex numbers). Then G acts naturally on the polynomial ring S = C[X1, …, Xn]. We consider the followingProblem. When is the invariant subring SG a complete intersection?In this paper, we treat the case where G is a finite Abelian group. We can solve the problem completely. The result is stated in Theorem 2.1.

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Some classes of concatenated quantum codes: constructions and lower bounds

Quantum Information and Computation ◽

10.26421/qic15.5-6-2 ◽

2015 ◽

pp. 385-405

Author(s):

Hachiro Fujita

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Polynomial Time ◽

Coding Theory ◽

Minimum Distance ◽

Concatenated Codes ◽

Quantum Codes ◽

Decoding Algorithm ◽

Quantum Analogue

In classical coding theory code concatenation is successfully used to construct good errorcorrecting codes and most of the asymptotically good codes known so far are based on concatenation. In this paper we present some classes of asymptotically good concatenated quantum codes, which are a quantum analogue of classical concatenated codes, and derive lower bounds on the minimum distance and the rate of the codes. Our bounds improve on the best lower bound of Ashikhmin–Litsyn–Tsfasman and Matsumoto for rates smaller than about one half. We also give a polynomial-time decoding algorithm for the codes that can decode up to one fourth of the lower bound on the minimum distance of the codes.

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Monomial multiplicities in explicit form

Journal of Algebra and Its Applications ◽

10.1142/s0219498820501819 ◽

2019 ◽

Vol 19 (09) ◽

pp. 2050181

Author(s):

Guillermo Alesandroni

Keyword(s):

Explicit Formula ◽

Explicit Form ◽

Complete Intersection ◽

Polynomial Ring ◽

Monomial Ideal ◽

The Other ◽

Complete Intersections ◽

New Class ◽

Other Hand ◽

The Family

Denote by [Formula: see text] a polynomial ring over a field, and let [Formula: see text] be a monomial ideal of [Formula: see text]. If [Formula: see text], we prove that the multiplicity of [Formula: see text] is given by [Formula: see text] On the other hand, if [Formula: see text] is a complete intersection, and [Formula: see text] is an almost complete intersection, we show that [Formula: see text] We also introduce a new class of ideals that extends the family of monomial complete intersections and that of codimension 1 ideals, and give an explicit formula for their multiplicity.

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Bounds for the minimum distance function

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2021-0042 ◽

2021 ◽

Vol 29 (3) ◽

pp. 229-242

Author(s):

Luis Núñez-Betancourt ◽

Yuriko Pitones ◽

Rafael H. Villarreal

Keyword(s):

Asymptotic Behavior ◽

Polynomial Ring ◽

Distance Function ◽

Minimum Distance ◽

Monomial Ideal ◽

Homogeneous Ideal

Abstract Let I be a homogeneous ideal in a polynomial ring S. In this paper, we extend the study of the asymptotic behavior of the minimum distance function δI of I and give bounds for its stabilization point, rI , when I is an F -pure or a square-free monomial ideal. These bounds are related with the dimension and the Castelnuovo–Mumford regularity of I.

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The least distance between extremums and minimal period of solutions to autonomous vector differential equations

Доклады Академии наук ◽

10.31857/s0869-56524852142-144 ◽

2019 ◽

Vol 485 (2) ◽

pp. 142-144

Author(s):

A. A. Zevin

Keyword(s):

Lower Bound ◽

Differential Equations ◽

Periodic Solutions ◽

Lipschitz Constant ◽

Minimal Period ◽

Arbitrary Vector ◽

Vector Norm ◽

Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

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