scholarly journals On the (29, 5)-Arcs in PG(2, 7) and Some Generalized Arcs in PG(2, q)

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 320
Author(s):  
Iliya Bouyukliev ◽  
Eun Ju Cheon ◽  
Tatsuya Maruta ◽  
Tsukasa Okazaki
Keyword(s):  
Linear Codes ◽  
Computer Search ◽  
Blocking Sets ◽  
Classification Result ◽  
Geometrical Structures ◽  
The Relationship

Using an exhaustive computer search, we prove that the number of inequivalent ( 29 , 5 ) -arcs in PG ( 2 , 7 ) is exactly 22. This generalizes a result of Barlotti (see Barlotti, A. Some Topics in Finite Geometrical Structures, 1965), who constructed the first such arc from a conic. Our classification result is based on the fact that arcs and linear codes are related, which enables us to apply an algorithm for classifying the associated linear codes instead. Related to this result, several infinite families of arcs and multiple blocking sets are constructed. Lastly, the relationship between these arcs and the Barlotti arc is explored using a construction that we call transitioning.

scholarly journals Full Characterization of Minimal Linear Codes as Cutting Blocking Sets

2021 ◽  
pp. 1-1
Author(s):  
Chunming Tang ◽  
Yan Qiu ◽  
Qunying Liao ◽  
Zhengchun Zhou
Keyword(s):  
Linear Codes ◽  
Blocking Sets ◽  
Full Characterization

scholarly journals Combinatorial Reductions for the Stanley Depth of $I$ and $S/I$

10.37236/6783 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Mitchel T. Keller ◽  
Stephen J. Young
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Strict Inequality ◽  
Polynomial Rings ◽  
Computer Search ◽  
Stanley Depth ◽  
The Relationship

We develop combinatorial tools to study the relationship between the Stanley depth of a monomial ideal $I$ and the Stanley depth of its compliment, $S/I$. Using these results we are able to prove that if $S$ is a polynomial ring with at most 5 indeterminates and $I$ is a square-free monomial ideal, then the Stanley depth of $S/I$ is strictly larger than the Stanley depth of $I$. Using a computer search, we are able to extend this strict inequality up to polynomial rings with at most 7 indeterminates. This partially answers questions asked by Propescu and Qureshi as well as Herzog.

Linear codes from vectorial Boolean functions in the context of algebraic attacks

2020 ◽  
pp. 2150032
Author(s):  
M. Boumezbeur ◽  
S. Mesnager ◽  
K. Guenda
Keyword(s):  
Boolean Functions ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Weight Enumerator ◽  
Direct Link ◽  
Algebraic Attacks ◽  
Vectorial Boolean Functions ◽  
Lcd Codes ◽  
Vectorial Function ◽  
The Relationship

In this paper, we study the relationship between vectorial (Boolean) functions and cyclic codes in the context of algebraic attacks. We first derive a direct link between the annihilators of a vectorial function (in univariate form) and certain [Formula: see text]-ary cyclic codes (which we show that they are LCD codes). We also present some properties of those cyclic codes as well as their weight enumerator. In addition, we generalize the so-called algebraic complement and study its properties.

ADSORPTION REACTION OF POLAR ORGANIC MOLECULES ON ${\rm Si}^+_n$ CLUSTER IONS

2005 ◽  
Vol 19 (15n17) ◽  
pp. 2502-2507
Author(s):  
WAKANA NAKAGAWARA ◽  
HIRONORI TSUNOYAMA ◽  
ARI FURUYA ◽  
FUMINORI MISAIZU ◽  
KOICHI OHNO
Keyword(s):  
Density Functional Theory ◽  
Density Functional ◽  
Organic Molecules ◽  
Theoretical Calculations ◽  
Cluster Ions ◽  
Silicon Cluster ◽  
Geometrical Structures ◽  
Functional Theory ◽  
Relationship Of ◽  
The Relationship

We have examined chemical reactions of small silicon cluster ions [Formula: see text] for n = 7 - 16 with polar organic molecules M ( M = CH 3 CN , CD 3 OD , C 2 H 5 CN , and C 2 H 5 OH ). The intensities of the adsorption products [Formula: see text] for m = 1 and 2 were investigated as a function of n. We found for all polar molecules that the relative intensity of Si n M + to the unreacted [Formula: see text] is smaller for n = 11, 13, and 14, that is, the adsorption reactivity is smaller for these n than others. It was also commonly observed that the [Formula: see text] ion are more intense than neighboring n. We discussed the relationship of the reactivity with the geometrical structures and the stabilities of the bare [Formula: see text] ions and adsorbed [Formula: see text] ions, from theoretical calculations based on density functional theory.

Triple Cyclic Codes Over 𝔽q + u𝔽q

2021 ◽  
pp. 1-21
Author(s):  
Ting Yao ◽  
Shixin Zhu ◽  
Binbin Pang
Keyword(s):  
Special Class ◽  
Linear Codes ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Cyclic Shift ◽  
Generating Sets ◽  
Optimal Linear ◽  
Number Formula ◽  
The Relationship ◽  
Optimal Linear Codes

Let [Formula: see text], where [Formula: see text] is a power of a prime number [Formula: see text] and [Formula: see text]. A triple cyclic code of length [Formula: see text] over [Formula: see text] is a set that can be partitioned into three parts that any cyclic shift of the coordinates of the three parts leaves the code invariant. These codes can be viewed as [Formula: see text]-submodules of [Formula: see text]. In this paper, we study the generator polynomials and the minimum generating sets of this kind of codes. Some optimal or almost optimal linear codes are obtained from this family of codes. We present the relationship between the generators of triple cyclic codes and their duals. As a special class of triple cyclic codes, separable codes over [Formula: see text] are discussed briefly in the end.

scholarly journals Codes from Cubic Curves and their Extensions

10.37236/766 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
T. L. Alderson ◽  
A. A. Bruen
Keyword(s):  
Linear Code ◽  
Linear Codes ◽  
Mds Codes ◽  
Blocking Sets ◽  
Linear Extensions ◽  
Cubic Curves ◽  
Cubic Curve ◽  
Rédei Blocking Sets

We study the linear codes and their extensions associated with sets of points in the plane corresponding to cubic curves. Instead of merely studying linear extensions, all possible extensions of the code are studied. In this way several new results are obtained and some existing results are strengthened. This type of analysis was carried out by Alderson, Bruen, and Silverman [J. Combin. Theory Ser. A, 114(6), 2007] for the case of MDS codes and by the present authors [Des. Codes Cryptogr., 47(1-3), 2008] for a broader range of codes. The methods cast some light on the question as to when a linear code can be extended to a nonlinear code. For example, for $p$ prime, it is shown that a linear $[n,3,n-3]_p$ code corresponding to a non-singular cubic curve comprising $n> p+4$ points admits only extensions that are equivalent to linear codes. The methods involve the theory of Rédei blocking sets and the use of the Bruen-Silverman model of linear codes.

scholarly journals On Zeros of a Polynomial in a Finite Grid

2018 ◽  
Vol 27 (3) ◽  
pp. 310-333 ◽  
Author(s):  
ANURAG BISHNOI ◽  
PETE L. CLARK ◽  
ADITYA POTUKUCHI ◽  
JOHN R. SCHMITT
Keyword(s):  
Upper Bound ◽  
Integral Domain ◽  
Hamming Distance ◽  
Finite Geometry ◽  
Blocking Sets ◽  
Multivariate Polynomial ◽  
Number Of Zeros ◽  
Minimum Hamming Distance ◽  
Affine Variety Codes ◽  
The Relationship

A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain ‘Condition (D)’ on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further generalized Alon–Füredi theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon–Füredi. We then discuss the relationship between Alon–Füredi and results of DeMillo–Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon–Füredi theorem and its generalization in terms of Reed–Muller-type affine variety codes is shown, which gives us the minimum Hamming distance of these codes. Then we apply the Alon–Füredi theorem to quickly recover – and sometimes strengthen – old and new results in finite geometry, including the Jamison–Brouwer–Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.

scholarly journals Algebraic entropy of endomorphisms of M-sets

2021 ◽  
Vol 9 (1) ◽  
pp. 53-71
Author(s):  
Nicolò Zava
Keyword(s):  
The Self ◽  
Classification Result ◽  
Algebraic Entropy ◽  
A Value ◽  
The Relationship

Abstract The usual notion of algebraic entropy associates to every group (monoid) endomorphism a value estimating the chaos created by the self-map. In this paper, we study the extension of this notion to arbitrary sets endowed with monoid actions, providing properties and relating it with other entropy notions. In particular, we focus our attention on the relationship with the coarse entropy of bornologous self-maps of quasi-coarse spaces. While studying the connection, an extension of a classification result due to Protasov is provided.

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