The (extended) rank weight enumerator and q-matroids

Size: px
Start display at page:

Download "The (extended) rank weight enumerator and q-matroids"

Transcription

1 The (extended) rank weight enumerator and q-matroids Relinde Jurrius Ruud Pellikaan Vrije Universiteit Brussel, Belgium Eindhoven University of Technology, The Netherlands Conference on Random network codes and Designs over GF(q) September 20, 2013

2 Field extension F q m/f q gives F q -isomorphism F n q m Fm n q, x m(x), so vectors over F q m are mapped to m n matrices over F q. Rank metric code is subspace of F q m subspace of F m n q.

3 q-analogues n finite set subset intersection union complement q n 1 q 1 F n q subspace intersection sum orthoplement size dimension ) [ n k] From q-analogue to normal : let q 1. ( n k q

4 C linear code supp(c) = coordinates of c that are non-zero wt H (c) = size of support Weight enumerator n W C (X, Y ) = A w X n w Y w w=0 with A w = number of words of weight w.

5 C rank metric code Rsupp(c) = row space of m(c) wt R (c) = dimension of support Rank weight enumerator n WC R (X, Y ) = A R w X n w Y w w=0 with A R w = number of words of weight w.

6 D C subcode supp(d) = union of supp(d) for all d D wt H (D) = size of support Generalized weight enumerators For all 0 r dim C: n WC r (X, Y ) = A r w X n w Y w w=0 with A r w = number of subcodes of dimension r and weight w. (Note: consistent with definition of generalized Hamming weights)

7 D C subcode Rsupp(D) = sum of Rsupp(d) for all d D wt R (D) = dimension of support Generalized rank weight enumerators For all 0 r dim C: W R,r C (X, Y ) = n w=0 A R,r w X n w Y w with A R,r w = number of subcodes of dimension r and weight w. (Note: consistent with definition of generalized rank weights)

8 F q e /F q field extension Extension code C F q e : code over F q e generated by words of C. Extended weight enumerator n W C (X, Y, T ) = A w (T )X n w Y n w=0 with A w (T ) polynomial such that A w (q e ) = number of words of weight w in C F q e.

9 F q me /F q m field extension Extension code C F q me : code over F q me generated by words of C. Extended rank weight enumerator n WC R (X, Y, T ) = A R w (T )X n w Y n w=0 with A R w (T ) polynomial such that A R w (q me ) = number of words of weight w in C F q me.

10 J subset of [n] C(J) = {c C : supp(c) J c } Lemma C(J) is a subspace of F n q l(j) = dim Fq C(J)

11 J subspace of F n q C(J) = {c C : Rsupp(c) J } Lemma C(J) is a subspace of F n q m l(j) = dim Fq m C(J)

12 Determining extended weight enumerator Determining generalized weight enumerators Determining l(j) for all J [n]

13 Determining extended rank weight enumerator Determining generalized rank weight enumerators Determining l(j) for all J F n q

14 Matroid E finite subset Independent sets I 2 E I If A I and B A then B I. If A, B I and A > B then there is an a A \ B such that B {a} I. Rank function r : 2 E N 0 r(a) A If A B then r(a) r(b). r(a B) + r(a B) r(a) + r(b) (semimodular)

15 Fact: a linear code gives a matroid with E = columns of generator matrix r(j) = dimension of subspace spanned by vectors of J Theorem r(j) = dim C l(j)

16 Rank generating function R M (X, Y ) = J E X r(e) r(j) Y J r(j) (Tutte polynomial: replace X by X 1 and Y by Y 1.) Theorem (Greene, 1976) The Tutte polynomial determines the weight enumerator. Theorem The extended weight enumerator determines the Tutte polynomial and vice versa.

17 q-matroid E = F n q q-independent sets I {subspaces of E} 0 I If A I and B A then B I. If A, B I and dim A > dim B then there is a 1-dimensional subspace a A, a B such that B + a I. q-rank function r : {subspaces of E} N 0 r(a) dim A If A B then r(a) r(b). r(a + B) + r(a B) r(a) + r(b) (semimodular)

18 Theorem Let r(j) = dim C l(j) for a rank metric code C. Then r(j) is the rank function of a q-matroid. Lemma l(a + B) + l(a B) l(a) + l(b)

19 q-rank generating function R q M (X, Y ) = X r(e) r(j) dim J r(j) Y J F n q Question: Are the extended rank weight enumerator and the q-rank generating function equivalent? Answer: Not sure, but probably yes.

20 Why study q-matroids? Matroids generalize: codes graphs some designs q-matroids generalize: rank metric codes q-graphs? q-designs?

21 Further work Equivalence between polynomials Various definetions of q-matroids Representable q-matroids Deletion and contraction

22 Thank you for your attention.

23 e element of finite set E {subsets containing e} {subsets of e c } = 2 E e 1-dimensional subspace of F n q {subspaces containing e} {subspaces of e } {all subspaces of F n q}

The extended coset leader weight enumerator

The extended coset leader weight enumerator The extended coset leader weight enumerator Relinde Jurrius Ruud Pellikaan Eindhoven University of Technology, The Netherlands Symposium on Information Theory in the Benelux, 2009 1/14 Outline Codes, weights

More information

A combinatorial view on derived codes

A combinatorial view on derived codes A combinatorial view on derived codes Relinde Jurrius (joint work with Philippe Cara) Vrije Universiteit Brussel, Belgium University of Neuchâtel, Switzerland Finite Geometries September 16, 2014 Relinde

More information

Application of hyperplane arrangements to weight enumeration

Application of hyperplane arrangements to weight enumeration Application of hyperplane arrangements to weight enumeration Relinde Jurrius (joint work with Ruud Pellikaan) Vrije Universiteit Brussel Joint Mathematics Meetings January 15, 2014 Relinde Jurrius (VUB)

More information

The coset leader and list weight enumerator

The coset leader and list weight enumerator Contemporary Mathematics The coset leader and list weight enumerator Relinde Jurrius and Ruud Pellikaan In Topics in Finite Fields 11th International Conference on Finite Fields and their Applications

More information

Arrangements, matroids and codes

Arrangements, matroids and codes Arrangements, matroids and codes first lecture Ruud Pellikaan joint work with Relinde Jurrius ACAGM summer school Leuven Belgium, 18 July 2011 References 2/43 1. Codes, arrangements and matroids by Relinde

More information

An Introduction to Transversal Matroids

An Introduction to Transversal Matroids An Introduction to Transversal Matroids Joseph E Bonin The George Washington University These slides and an accompanying expository paper (in essence, notes for this talk, and more) are available at http://homegwuedu/

More information

Assignment 1 Math 5341 Linear Algebra Review. Give complete answers to each of the following questions. Show all of your work.

Assignment 1 Math 5341 Linear Algebra Review. Give complete answers to each of the following questions. Show all of your work. Assignment 1 Math 5341 Linear Algebra Review Give complete answers to each of the following questions Show all of your work Note: You might struggle with some of these questions, either because it has

More information

Linear Algebra problems

Linear Algebra problems Linear Algebra problems 1. Show that the set F = ({1, 0}, +,.) is a field where + and. are defined as 1+1=0, 0+0=0, 0+1=1+0=1, 0.0=0.1=1.0=0, 1.1=1.. Let X be a non-empty set and F be any field. Let X

More information

Exam 2 Solutions. (a) Is W closed under addition? Why or why not? W is not closed under addition. For example,

Exam 2 Solutions. (a) Is W closed under addition? Why or why not? W is not closed under addition. For example, Exam 2 Solutions. Let V be the set of pairs of real numbers (x, y). Define the following operations on V : (x, y) (x, y ) = (x + x, xx + yy ) r (x, y) = (rx, y) Check if V together with and satisfy properties

More information

6.4 BASIS AND DIMENSION (Review) DEF 1 Vectors v 1, v 2,, v k in a vector space V are said to form a basis for V if. (a) v 1,, v k span V and

6.4 BASIS AND DIMENSION (Review) DEF 1 Vectors v 1, v 2,, v k in a vector space V are said to form a basis for V if. (a) v 1,, v k span V and 6.4 BASIS AND DIMENSION (Review) DEF 1 Vectors v 1, v 2,, v k in a vector space V are said to form a basis for V if (a) v 1,, v k span V and (b) v 1,, v k are linearly independent. HMHsueh 1 Natural Basis

More information

TUTTE POLYNOMIALS OF q-cones

TUTTE POLYNOMIALS OF q-cones TUTTE POLYNOMIALS OF q-cones JOSEPH E. BONIN AND HONGXUN QIN ABSTRACT. We derive a formula for the Tutte polynomial t(g ; x, y) of a q-cone G of a GF (q)-representable geometry G in terms of t(g; x, y).

More information

Math 369 Exam #2 Practice Problem Solutions

Math 369 Exam #2 Practice Problem Solutions Math 369 Exam #2 Practice Problem Solutions 2 5. Is { 2, 3, 8 } a basis for R 3? Answer: No, it is not. To show that it is not a basis, it suffices to show that this is not a linearly independent set.

More information

Elementary 2-Group Character Codes. Abstract. In this correspondence we describe a class of codes over GF (q),

Elementary 2-Group Character Codes. Abstract. In this correspondence we describe a class of codes over GF (q), Elementary 2-Group Character Codes Cunsheng Ding 1, David Kohel 2, and San Ling Abstract In this correspondence we describe a class of codes over GF (q), where q is a power of an odd prime. These codes

More information

Math 4153 Exam 1 Review

Math 4153 Exam 1 Review The syllabus for Exam 1 is Chapters 1 3 in Axler. 1. You should be sure to know precise definition of the terms we have used, and you should know precise statements (including all relevant hypotheses)

More information

Vertex subsets with minimal width and dual width

Vertex subsets with minimal width and dual width Vertex subsets with minimal width and dual width in Q-polynomial distance-regular graphs University of Wisconsin & Tohoku University February 2, 2011 Every face (or facet) of a hypercube is a hypercube...

More information

1 Matroid intersection

1 Matroid intersection CS 369P: Polyhedral techniques in combinatorial optimization Instructor: Jan Vondrák Lecture date: October 21st, 2010 Scribe: Bernd Bandemer 1 Matroid intersection Given two matroids M 1 = (E, I 1 ) and

More information

MATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible.

MATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible. MATH 2331 Linear Algebra Section 2.1 Matrix Operations Definition: A : m n, B : n p ( 1 2 p ) ( 1 2 p ) AB = A b b b = Ab Ab Ab Example: Compute AB, if possible. 1 Row-column rule: i-j-th entry of AB:

More information

Second Midterm Exam April 14, 2011 Answers., and

Second Midterm Exam April 14, 2011 Answers., and Mathematics 34, Spring Problem ( points) (a) Consider the matrices all matrices. Second Midterm Exam April 4, Answers [ Do these matrices span M? ] [, ] [, and Lectures & (Wilson) ], as vectors in the

More information

Unless otherwise specified, V denotes an arbitrary finite-dimensional vector space.

Unless otherwise specified, V denotes an arbitrary finite-dimensional vector space. MAT 90 // 0 points Exam Solutions Unless otherwise specified, V denotes an arbitrary finite-dimensional vector space..(0) Prove: a central arrangement A in V is essential if and only if the dual projective

More information

Vector spaces. EE 387, Notes 8, Handout #12

Vector spaces. EE 387, Notes 8, Handout #12 Vector spaces EE 387, Notes 8, Handout #12 A vector space V of vectors over a field F of scalars is a set with a binary operator + on V and a scalar-vector product satisfying these axioms: 1. (V, +) is

More information

Introduction to Association Schemes

Introduction to Association Schemes Introduction to Association Schemes Akihiro Munemasa Tohoku University June 5 6, 24 Algebraic Combinatorics Summer School, Sendai Assumed results (i) Vandermonde determinant: a a m =. a m a m m i

More information

Matroids/1. I and I 2 ,I 2 > I 1

Matroids/1. I and I 2 ,I 2 > I 1 Matroids 1 Definition A matroid is an abstraction of the notion of linear independence in a vector space. See Oxley [6], Welsh [7] for further information about matroids. A matroid is a pair (E,I ), where

More information

arxiv: v2 [cs.it] 16 Apr 2018

arxiv: v2 [cs.it] 16 Apr 2018 Rank-Metric Codes and q-polymatroids Elisa Gorla 1, Relinde Jurrius 2, Hiram H. López 3, and Alberto Ravagnani 4 1 Institut de Matématiques, Université de Neuchâtel, Switzerland 2 Faculty of Military Science,

More information

Chapter 2 Linear Transformations

Chapter 2 Linear Transformations Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more

More information

Commuting birth-and-death processes

Commuting birth-and-death processes Commuting birth-and-death processes Caroline Uhler Department of Statistics UC Berkeley (joint work with Steven N. Evans and Bernd Sturmfels) MSRI Workshop on Algebraic Statistics December 18, 2008 Birth-and-death

More information

An Introduction of Tutte Polynomial

An Introduction of Tutte Polynomial An Introduction of Tutte Polynomial Bo Lin December 12, 2013 Abstract Tutte polynomial, defined for matroids and graphs, has the important property that any multiplicative graph invariant with a deletion

More information

Chapter 6 - Orthogonality

Chapter 6 - Orthogonality Chapter 6 - Orthogonality Maggie Myers Robert A. van de Geijn The University of Texas at Austin Orthogonality Fall 2009 http://z.cs.utexas.edu/wiki/pla.wiki/ 1 Orthogonal Vectors and Subspaces http://z.cs.utexas.edu/wiki/pla.wiki/

More information

MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix.

MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix. MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix. Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis.

More information

MATROID PACKING AND COVERING WITH CIRCUITS THROUGH AN ELEMENT

MATROID PACKING AND COVERING WITH CIRCUITS THROUGH AN ELEMENT MATROID PACKING AND COVERING WITH CIRCUITS THROUGH AN ELEMENT MANOEL LEMOS AND JAMES OXLEY Abstract. In 1981, Seymour proved a conjecture of Welsh that, in a connected matroid M, the sum of the maximum

More information

ON CONTRACTING HYPERPLANE ELEMENTS FROM A 3-CONNECTED MATROID

ON CONTRACTING HYPERPLANE ELEMENTS FROM A 3-CONNECTED MATROID ON CONTRACTING HYPERPLANE ELEMENTS FROM A 3-CONNECTED MATROID RHIANNON HALL Abstract. Let K 3,n, n 3, be the simple graph obtained from K 3,n by adding three edges to a vertex part of size three. We prove

More information

Simplicial complexes, Demi-matroids, Flag of linear codes and pair of matroids

Simplicial complexes, Demi-matroids, Flag of linear codes and pair of matroids Faculty of Science and Technology Department of Mathematics and Statistics Simplicial complexes, Demi-matroids, Flag of linear codes and pair of matroids Ali Zubair MAT-3900 Master s thesis in Mathematics

More information

Math 121 Homework 4: Notes on Selected Problems

Math 121 Homework 4: Notes on Selected Problems Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W

More information

ON SIZE, CIRCUMFERENCE AND CIRCUIT REMOVAL IN 3 CONNECTED MATROIDS

ON SIZE, CIRCUMFERENCE AND CIRCUIT REMOVAL IN 3 CONNECTED MATROIDS ON SIZE, CIRCUMFERENCE AND CIRCUIT REMOVAL IN 3 CONNECTED MATROIDS MANOEL LEMOS AND JAMES OXLEY Abstract. This paper proves several extremal results for 3-connected matroids. In particular, it is shown

More information

Tropical aspects of algebraic matroids. Jan Draisma Universität Bern and TU Eindhoven (w/ Rudi Pendavingh and Guus Bollen TU/e)

Tropical aspects of algebraic matroids. Jan Draisma Universität Bern and TU Eindhoven (w/ Rudi Pendavingh and Guus Bollen TU/e) 1 Tropical aspects of algebraic matroids Jan Draisma Universität Bern and TU Eindhoven (w/ Rudi Pendavingh and Guus Bollen TU/e) Two advertisements 2 SIAGA: SIAM AG 19: 9 13 July 2019, Bern Algebraic matroids

More information

Second Exam. Math , Spring March 2015

Second Exam. Math , Spring March 2015 Second Exam Math 34-54, Spring 25 3 March 25. This exam has 8 questions and 2 pages. Make sure you have all pages before you begin. The eighth question is bonus (and worth less than the others). 2. This

More information

AN INTRODUCTION TO TRANSVERSAL MATROIDS

AN INTRODUCTION TO TRANSVERSAL MATROIDS AN INTRODUCTION TO TRANSVERSAL MATROIDS JOSEPH E BONIN October 26, 2010 CONTENTS 1 Prefatory Remarks 1 2 Several Perspectives on Transversal Matroids 2 21 Set systems, transversals, partial transversals,

More information

Elementary linear algebra

Elementary linear algebra Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The

More information

TOWARDS A SPLITTER THEOREM FOR INTERNALLY 4-CONNECTED BINARY MATROIDS II

TOWARDS A SPLITTER THEOREM FOR INTERNALLY 4-CONNECTED BINARY MATROIDS II TOWARDS A SPLITTER THEOREM FOR INTERNALLY 4-CONNECTED BINARY MATROIDS II CAROLYN CHUN, DILLON MAYHEW, AND JAMES OXLEY Abstract. Let M and N be internally 4-connected binary matroids such that M has a proper

More information

2018 Fall 2210Q Section 013 Midterm Exam II Solution

2018 Fall 2210Q Section 013 Midterm Exam II Solution 08 Fall 0Q Section 0 Midterm Exam II Solution True or False questions points 0 0 points) ) Let A be an n n matrix. If the equation Ax b has at least one solution for each b R n, then the solution is unique

More information

4. Images of Varieties Given a morphism f : X Y of quasi-projective varieties, a basic question might be to ask what is the image of a closed subset

4. Images of Varieties Given a morphism f : X Y of quasi-projective varieties, a basic question might be to ask what is the image of a closed subset 4. Images of Varieties Given a morphism f : X Y of quasi-projective varieties, a basic question might be to ask what is the image of a closed subset Z X. Replacing X by Z we might as well assume that Z

More information

Proof: The coding of T (x) is the left shift of the coding of x. φ(t x) n = L if T n+1 (x) L

Proof: The coding of T (x) is the left shift of the coding of x. φ(t x) n = L if T n+1 (x) L Lecture 24: Defn: Topological conjugacy: Given Z + d (resp, Zd ), actions T, S a topological conjugacy from T to S is a homeomorphism φ : M N s.t. φ T = S φ i.e., φ T n = S n φ for all n Z + d (resp, Zd

More information

GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.

GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,

More information

WHAT IS A MATROID? JAMES OXLEY

WHAT IS A MATROID? JAMES OXLEY WHAT IS A MATROID? JAMES OXLEY Abstract. Matroids were introduced by Whitney in 1935 to try to capture abstractly the essence of dependence. Whitney s definition embraces a surprising diversity of combinatorial

More information

Determinantal Probability Measures. by Russell Lyons (Indiana University)

Determinantal Probability Measures. by Russell Lyons (Indiana University) Determinantal Probability Measures by Russell Lyons (Indiana University) 1 Determinantal Measures If E is finite and H l 2 (E) is a subspace, it defines the determinantal measure T E with T = dim H P H

More information

Boolean degree 1 functions on some classical association schemes

Boolean degree 1 functions on some classical association schemes Boolean degree 1 functions on some classical association schemes Yuval Filmus, Ferdinand Ihringer February 16, 2018 Abstract We investigate Boolean degree 1 functions for several classical association

More information

RANK-WIDTH AND WELL-QUASI-ORDERING OF SKEW-SYMMETRIC OR SYMMETRIC MATRICES

RANK-WIDTH AND WELL-QUASI-ORDERING OF SKEW-SYMMETRIC OR SYMMETRIC MATRICES RANK-WIDTH AND WELL-QUASI-ORDERING OF SKEW-SYMMETRIC OR SYMMETRIC MATRICES SANG-IL OUM Abstract. We prove that every infinite sequence of skew-symmetric or symmetric matrices M, M 2,... over a fixed finite

More information

Determining a Binary Matroid from its Small Circuits

Determining a Binary Matroid from its Small Circuits Determining a Binary Matroid from its Small Circuits James Oxley Department of Mathematics Louisiana State University Louisiana, USA oxley@math.lsu.edu Charles Semple School of Mathematics and Statistics

More information

1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true?

1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true? . Let m and n be two natural numbers such that m > n. Which of the following is/are true? (i) A linear system of m equations in n variables is always consistent. (ii) A linear system of n equations in

More information

A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits

A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits Ran Raz Amir Shpilka Amir Yehudayoff Abstract We construct an explicit polynomial f(x 1,..., x n ), with coefficients in {0,

More information

TOWARDS A SPLITTER THEOREM FOR INTERNALLY 4-CONNECTED BINARY MATROIDS III

TOWARDS A SPLITTER THEOREM FOR INTERNALLY 4-CONNECTED BINARY MATROIDS III TOWARDS A SPLITTER THEOREM FOR INTERNALLY 4-CONNECTED BINARY MATROIDS III CAROLYN CHUN, DILLON MAYHEW, AND JAMES OXLEY Abstract. This paper proves a preliminary step towards a splitter theorem for internally

More information

Test 3, Linear Algebra

Test 3, Linear Algebra Test 3, Linear Algebra Dr. Adam Graham-Squire, Fall 2017 Name: I pledge that I have neither given nor received any unauthorized assistance on this exam. (signature) DIRECTIONS 1. Don t panic. 2. Show all

More information

LINEAR ALGEBRA QUESTION BANK

LINEAR ALGEBRA QUESTION BANK LINEAR ALGEBRA QUESTION BANK () ( points total) Circle True or False: TRUE / FALSE: If A is any n n matrix, and I n is the n n identity matrix, then I n A = AI n = A. TRUE / FALSE: If A, B are n n matrices,

More information

Polynomial aspects of codes, matroids and permutation groups

Polynomial aspects of codes, matroids and permutation groups Polynomial aspects of codes, matroids and permutation groups Peter J. Cameron School of Mathematical Sciences Queen Mary, University of London Mile End Road London E1 4NS UK p.j.cameron@qmul.ac.uk Contents

More information

Matroid intersection, base packing and base covering for infinite matroids

Matroid intersection, base packing and base covering for infinite matroids Matroid intersection, base packing and base covering for infinite matroids Nathan Bowler Johannes Carmesin June 25, 2014 Abstract As part of the recent developments in infinite matroid theory, there have

More information

Math 353, Practice Midterm 1

Math 353, Practice Midterm 1 Math 353, Practice Midterm Name: This exam consists of 8 pages including this front page Ground Rules No calculator is allowed 2 Show your work for every problem unless otherwise stated Score 2 2 3 5 4

More information

Characterizations of Strongly Regular Graphs: Part II: Bose-Mesner algebras of graphs. Sung Y. Song Iowa State University

Characterizations of Strongly Regular Graphs: Part II: Bose-Mesner algebras of graphs. Sung Y. Song Iowa State University Characterizations of Strongly Regular Graphs: Part II: Bose-Mesner algebras of graphs Sung Y. Song Iowa State University sysong@iastate.edu Notation K: one of the fields R or C X: a nonempty finite set

More information

On the cycle index and the weight enumerator

On the cycle index and the weight enumerator On the cycle index and the weight enumerator Tsuyoshi Miezaki and Manabu Oura Abstract In this paper, we introduce the concept of the complete cycle index and discuss a relation with the complete weight

More information

= c. = c. c 2. We can find apply our general formula to find the inverse of the 2 2 matrix A: A 1 5 4

= c. = c. c 2. We can find apply our general formula to find the inverse of the 2 2 matrix A: A 1 5 4 . In each part, a basis B of R is given (you don t need to show B is a basis). Find he B-coordinate of the vector v. (a) B {, }, v Solution.(5 points) We have: + Therefore, the B-coordinate of v is equal

More information

Polynomials and Codes

Polynomials and Codes TU/e Eindhoven 13 September, 2012, Trieste ICTP-IPM Workshop and Conference in Combinatorics and Graph Theory. Reza and Richard: Thanks for a wonderful meeting. From now on: p is prime and q is a power

More information

Pseudoinverse & Moore-Penrose Conditions

Pseudoinverse & Moore-Penrose Conditions ECE 275AB Lecture 7 Fall 2008 V1.0 c K. Kreutz-Delgado, UC San Diego p. 1/1 Lecture 7 ECE 275A Pseudoinverse & Moore-Penrose Conditions ECE 275AB Lecture 7 Fall 2008 V1.0 c K. Kreutz-Delgado, UC San Diego

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013 18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013 Throughout this lecture k denotes an algebraically closed field. 17.1 Tangent spaces and hypersurfaces For any polynomial f k[x

More information

Tropical matrix algebra

Tropical matrix algebra Tropical matrix algebra Marianne Johnson 1 & Mark Kambites 2 NBSAN, University of York, 25th November 2009 1 University of Manchester. Supported by CICADA (EPSRC grant EP/E050441/1). 2 University of Manchester.

More information

Math 205, Summer I, Week 3a (continued): Chapter 4, Sections 5 and 6. Week 3b. Chapter 4, [Sections 7], 8 and 9

Math 205, Summer I, Week 3a (continued): Chapter 4, Sections 5 and 6. Week 3b. Chapter 4, [Sections 7], 8 and 9 Math 205, Summer I, 2016 Week 3a (continued): Chapter 4, Sections 5 and 6. Week 3b Chapter 4, [Sections 7], 8 and 9 4.5 Linear Dependence, Linear Independence 4.6 Bases and Dimension 4.7 Change of Basis,

More information

Math 21b: Linear Algebra Spring 2018

Math 21b: Linear Algebra Spring 2018 Math b: Linear Algebra Spring 08 Homework 8: Basis This homework is due on Wednesday, February 4, respectively on Thursday, February 5, 08. Which of the following sets are linear spaces? Check in each

More information

Section 2: Classes of Sets

Section 2: Classes of Sets Section 2: Classes of Sets Notation: If A, B are subsets of X, then A \ B denotes the set difference, A \ B = {x A : x B}. A B denotes the symmetric difference. A B = (A \ B) (B \ A) = (A B) \ (A B). Remarks

More information

Apprentice Linear Algebra, 1st day, 6/27/05

Apprentice Linear Algebra, 1st day, 6/27/05 Apprentice Linear Algebra, 1st day, 6/7/05 REU 005 Instructor: László Babai Scribe: Eric Patterson Definitions 1.1. An abelian group is a set G with the following properties: (i) ( a, b G)(!a + b G) (ii)

More information

Lecture 17: Section 4.2

Lecture 17: Section 4.2 Lecture 17: Section 4.2 Shuanglin Shao November 4, 2013 Subspaces We will discuss subspaces of vector spaces. Subspaces Definition. A subset W is a vector space V is called a subspace of V if W is itself

More information

THE NUMBER OF POINTS IN A COMBINATORIAL GEOMETRY WITH NO 8-POINT-LINE MINORS

THE NUMBER OF POINTS IN A COMBINATORIAL GEOMETRY WITH NO 8-POINT-LINE MINORS THE NUMBER OF POINTS IN A COMBINATORIAL GEOMETRY WITH NO 8-POINT-LINE MINORS JOSEPH E. BONIN AND JOSEPH P. S. KUNG ABSTRACT. We show that when n is greater than 3, the number of points in a combinatorial

More information

I. Multiple Choice Questions (Answer any eight)

I. Multiple Choice Questions (Answer any eight) Name of the student : Roll No : CS65: Linear Algebra and Random Processes Exam - Course Instructor : Prashanth L.A. Date : Sep-24, 27 Duration : 5 minutes INSTRUCTIONS: The test will be evaluated ONLY

More information

Notes on matroids and codes

Notes on matroids and codes Notes on matroids and codes Peter J. Cameron Abstract The following expository article is intended to describe a correspondence between matroids and codes. The key results are that the weight enumerator

More information

THE MINIMALLY NON-IDEAL BINARY CLUTTERS WITH A TRIANGLE 1. INTRODUCTION

THE MINIMALLY NON-IDEAL BINARY CLUTTERS WITH A TRIANGLE 1. INTRODUCTION THE MINIMALLY NON-IDEAL BINARY CLUTTERS WITH A TRIANGLE AHMAD ABDI AND BERTRAND GUENIN ABSTRACT. It is proved that the lines of the Fano plane and the odd circuits of K 5 constitute the only minimally

More information

Problem set #4. Due February 19, x 1 x 2 + x 3 + x 4 x 5 = 0 x 1 + x 3 + 2x 4 = 1 x 1 x 2 x 4 x 5 = 1.

Problem set #4. Due February 19, x 1 x 2 + x 3 + x 4 x 5 = 0 x 1 + x 3 + 2x 4 = 1 x 1 x 2 x 4 x 5 = 1. Problem set #4 Due February 19, 218 The letter V always denotes a vector space. Exercise 1. Find all solutions to 2x 1 x 2 + x 3 + x 4 x 5 = x 1 + x 3 + 2x 4 = 1 x 1 x 2 x 4 x 5 = 1. Solution. First we

More information

Chapter 3. Directions: For questions 1-11 mark each statement True or False. Justify each answer.

Chapter 3. Directions: For questions 1-11 mark each statement True or False. Justify each answer. Chapter 3 Directions: For questions 1-11 mark each statement True or False. Justify each answer. 1. (True False) Asking whether the linear system corresponding to an augmented matrix [ a 1 a 2 a 3 b ]

More information

15.1 Matching, Components, and Edge cover (Collaborate with Xin Yu)

15.1 Matching, Components, and Edge cover (Collaborate with Xin Yu) 15.1 Matching, Components, and Edge cover (Collaborate with Xin Yu) First show l = c by proving l c and c l. For a maximum matching M in G, let V be the set of vertices covered by M. Since any vertex in

More information

Lecture Summaries for Linear Algebra M51A

Lecture Summaries for Linear Algebra M51A These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture

More information

Ma/CS 6b Class 23: Eigenvalues in Regular Graphs

Ma/CS 6b Class 23: Eigenvalues in Regular Graphs Ma/CS 6b Class 3: Eigenvalues in Regular Graphs By Adam Sheffer Recall: The Spectrum of a Graph Consider a graph G = V, E and let A be the adjacency matrix of G. The eigenvalues of G are the eigenvalues

More information

arxiv: v1 [math.co] 19 Oct 2018

arxiv: v1 [math.co] 19 Oct 2018 On the structure of spikes arxiv:1810.08416v1 [math.co] 19 Oct 2018 Abstract Vahid Ghorbani, Ghodratollah Azadi and Habib Azanchiler Department of mathematics, University of Urmia, Iran Spikes are an important

More information

MATROIDS DENSER THAN A PROJECTIVE GEOMETRY

MATROIDS DENSER THAN A PROJECTIVE GEOMETRY MATROIDS DENSER THAN A PROJECTIVE GEOMETRY PETER NELSON Abstract. The growth-rate function for a minor-closed class M of matroids is the function h where, for each non-negative integer r, h(r) is the maximum

More information

Submodular Functions, Optimization, and Applications to Machine Learning

Submodular Functions, Optimization, and Applications to Machine Learning Submodular Functions, Optimization, and Applications to Machine Learning Spring Quarter, Lecture http://www.ee.washington.edu/people/faculty/bilmes/classes/eeb_spring_0/ Prof. Jeff Bilmes University of

More information

Semimatroids and their Tutte polynomials

Semimatroids and their Tutte polynomials Semimatroids and their Tutte polynomials Federico Ardila Abstract We define and study semimatroids, a class of objects which abstracts the dependence properties of an affine hyperplane arrangement. We

More information

Counting bases of representable matroids

Counting bases of representable matroids Counting bases of representable matroids Michael Snook School of Mathematics, Statistics and Operations Research Victoria University of Wellington Wellington, New Zealand michael.snook@msor.vuw.ac.nz Submitted:

More information

Linear Systems. Math A Bianca Santoro. September 23, 2016

Linear Systems. Math A Bianca Santoro. September 23, 2016 Linear Systems Math A4600 - Bianca Santoro September 3, 06 Goal: Understand how to solve Ax = b. Toy Model: Let s study the following system There are two nice ways of thinking about this system: x + y

More information

LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE

LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds

More information

Linear Algebra Highlights

Linear Algebra Highlights Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to

More information

Submodular Functions, Optimization, and Applications to Machine Learning

Submodular Functions, Optimization, and Applications to Machine Learning Submodular Functions, Optimization, and Applications to Machine Learning Spring Quarter, Lecture http://www.ee.washington.edu/people/faculty/bilmes/classes/ee_spring_0/ Prof. Jeff Bilmes University of

More information

Fall 2016 MATH*1160 Final Exam

Fall 2016 MATH*1160 Final Exam Fall 2016 MATH*1160 Final Exam Last name: (PRINT) First name: Student #: Instructor: M. R. Garvie Dec 16, 2016 INSTRUCTIONS: 1. The exam is 2 hours long. Do NOT start until instructed. You may use blank

More information

NAME MATH 304 Examination 2 Page 1

NAME MATH 304 Examination 2 Page 1 NAME MATH 4 Examination 2 Page. [8 points (a) Find the following determinant. However, use only properties of determinants, without calculating directly (that is without expanding along a column or row

More information

New binary self-dual codes of lengths 50 to 60

New binary self-dual codes of lengths 50 to 60 Designs, Codes and Cryptography manuscript No. (will be inserted by the editor) New binary self-dual codes of lengths 50 to 60 Nikolay Yankov Moon Ho Lee Received: date / Accepted: date Abstract Using

More information

The definition of a vector space (V, +, )

The definition of a vector space (V, +, ) The definition of a vector space (V, +, ) 1. For any u and v in V, u + v is also in V. 2. For any u and v in V, u + v = v + u. 3. For any u, v, w in V, u + ( v + w) = ( u + v) + w. 4. There is an element

More information

Dr. Abdulla Eid. Section 4.2 Subspaces. Dr. Abdulla Eid. MATHS 211: Linear Algebra. College of Science

Dr. Abdulla Eid. Section 4.2 Subspaces. Dr. Abdulla Eid. MATHS 211: Linear Algebra. College of Science Section 4.2 Subspaces College of Science MATHS 211: Linear Algebra (University of Bahrain) Subspaces 1 / 42 Goal: 1 Define subspaces. 2 Subspace test. 3 Linear Combination of elements. 4 Subspace generated

More information

Lecture: Linear algebra. 4. Solutions of linear equation systems The fundamental theorem of linear algebra

Lecture: Linear algebra. 4. Solutions of linear equation systems The fundamental theorem of linear algebra Lecture: Linear algebra. 1. Subspaces. 2. Orthogonal complement. 3. The four fundamental subspaces 4. Solutions of linear equation systems The fundamental theorem of linear algebra 5. Determining the fundamental

More information

Reconstruction and Higher Dimensional Geometry

Reconstruction and Higher Dimensional Geometry Reconstruction and Higher Dimensional Geometry Hongyu He Department of Mathematics Louisiana State University email: hongyu@math.lsu.edu Abstract Tutte proved that, if two graphs, both with more than two

More information

A Course in Combinatorics

A Course in Combinatorics A Course in Combinatorics J. H. van Lint Technical Universüy of Eindhoven and R. M. Wilson California Institute of Technology H CAMBRIDGE UNIVERSITY PRESS CONTENTS Preface xi 1. Graphs 1 Terminology of

More information

Triply even codes binary codes, lattices and framed vertex operator algebras

Triply even codes binary codes, lattices and framed vertex operator algebras Triply even codes binary codes, lattices and framed vertex operator algebras Akihiro Munemasa 1 1 Graduate School of Information Sciences Tohoku University (joint work with Koichi Betsumiya, Masaaki Harada

More information

SOME DESIGNS AND CODES FROM L 2 (q) Communicated by Alireza Abdollahi

SOME DESIGNS AND CODES FROM L 2 (q) Communicated by Alireza Abdollahi Transactions on Combinatorics ISSN (print): 2251-8657, ISSN (on-line): 2251-8665 Vol. 3 No. 1 (2014), pp. 15-28. c 2014 University of Isfahan www.combinatorics.ir www.ui.ac.ir SOME DESIGNS AND CODES FROM

More information

B ɛ (P ) B. n N } R. Q P

B ɛ (P ) B. n N } R. Q P 8. Limits Definition 8.1. Let P R n be a point. The open ball of radius ɛ > 0 about P is the set B ɛ (P ) = { Q R n P Q < ɛ }. The closed ball of radius ɛ > 0 about P is the set { Q R n P Q ɛ }. Definition

More information

Optimization on the Grassmann manifold: a case study

Optimization on the Grassmann manifold: a case study Optimization on the Grassmann manifold: a case study Konstantin Usevich and Ivan Markovsky Department ELEC, Vrije Universiteit Brussel 28 March 2013 32nd Benelux Meeting on Systems and Control, Houffalize,

More information

Review for Exam 2 Solutions

Review for Exam 2 Solutions Review for Exam 2 Solutions Note: All vector spaces are real vector spaces. Definition 4.4 will be provided on the exam as it appears in the textbook.. Determine if the following sets V together with operations

More information

Simplicial and Cellular Spanning Trees, II: Applications

Simplicial and Cellular Spanning Trees, II: Applications Simplicial and Cellular Spanning Trees, II: Applications Art Duval (University of Texas at El Paso) Caroline Klivans (Brown University) Jeremy Martin (University of Kansas) University of California, Davis

More information

Instructions. 2. Four possible answers are provided for each question and only one of these is correct.

Instructions. 2. Four possible answers are provided for each question and only one of these is correct. Instructions 1. This question paper has forty multiple choice questions. 2. Four possible answers are provided for each question and only one of these is correct. 3. Marking scheme: Each correct answer

More information