Application of hyperplane arrangements to weight enumeration

Transcription

1 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) Application to weight enumeration Joint Math Meetings / 16

2 Coding theory message channel message noise Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

3 Coding theory encoding decoding message codeword channel received word message noise Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

4 Coding theory Code Set of codewords ( vectors) of fixed length n. d(x, y) The number of places on which two vectors differ. d The minimal distance between codewords. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

5 Coding theory Code Set of codewords ( vectors) of fixed length n. d(x, y) The number of places on which two vectors differ. d The minimal distance between codewords. Linear code Linear subspace C F n q of dimension k. Generator matrix Some k n matrix G whose rows span C. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

6 Coding theory Codes are equivalent if generator matrices are the same up to left multiplication by nonsingular k k matrix over F q (i.e., same rowspace); permutation of columns; multiplication of column by element of F q. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

7 Coding theory Codes are equivalent if generator matrices are the same up to left multiplication by nonsingular k k matrix over F q (i.e., same rowspace); permutation of columns; multiplication of column by element of F q. We restrict to projective codes: they have a generator matrix where no column is zero; no column is a multiple of another column. So, all columns coordinatize a different projective point. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

8 Weight enumeration Weight The number of nonzero coordinates in a vector. For linear codes: minimum distance = minimum nonzero weight. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

9 Weight enumeration Weight The number of nonzero coordinates in a vector. For linear codes: minimum distance = minimum nonzero weight. Weight enumerator n W C (X, Y ) = A w X n w Y w w=0 where A w = number of words of weight w. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

10 Weight enumeration Extension code [n, k] code C F q m over some extension field F q m generated by the words of C. Generator matrix All extension codes of C have generator matrix G. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

11 Weight enumeration Extension code [n, k] code C F q m over some extension field F q m generated by the words of C. Generator matrix All extension codes of C have generator matrix G. Extended weight enumerator n W C (X, Y, T ) = A w (T )X n w Y w, w=0 where A w (q m ) = number of words of weight w in C F q m. Fact: the A w (T ) are polynomials of degree at most k. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

12 Weight enumeration 0 1 k k n 1 n message m generator matrix G codeword c Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

13 Weight enumeration 0 1 k k n 1 n message m generator matrix G codeword c Theorem c j = 0 m in hyperplane orthogonal to j-th column of G Weight enumeration = counting points in (intersections of) hyperplanes. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

14 Codes and hyperplane arrangements Columns of a generator matrix G of a linear [n, k] code form a linear hyperplane arrangement in F k q. Notation: (H 1,..., H n ). Theorem One-to-one correspondence between equivalence classes. Independent of choice of G, so notation: A C. Also valid over an extension field F m q. A w (T ) = number of points from vectorspace over field of T elements that are on n w hyperplanes. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

15 Codes and hyperplane arrangements Example H6 H5 Let q > 2 and C generated by G = , a 0 1 H1 H2 where a 0, 1. H4 H3 The extended weights are given by A 0 (T ) = 1 The zero word is on all hyperplanes. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

16 Codes and hyperplane arrangements Example H6 H5 Let q > 2 and C generated by G = , a 0 1 H1 H2 where a 0, 1. H4 H3 The extended weights are given by No points are on 5 hyperplanes. A 1 (T ) = 0 Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

17 Codes and hyperplane arrangements Example H6 H5 Let q > 2 and C generated by G = , a 0 1 H1 H2 where a 0, 1. H4 H3 The extended weights are given by A 2 (T ) = T 1 One projective point is on 4 hyperplanes. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

18 Codes and hyperplane arrangements Example H6 H5 Let q > 2 and C generated by G = , a 0 1 H1 H2 where a 0, 1. H4 H3 The extended weights are given by A 3 (T ) = T 1 One projective point is on 3 hyperplanes. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

19 Codes and hyperplane arrangements Example H6 H5 Let q > 2 and C generated by G = , a 0 1 H1 H2 where a 0, 1. H4 H3 The extended weights are given by A 4 (T ) = 6(T 1) Six projective points are on 2 hyperplanes. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

20 Codes and hyperplane arrangements Example H6 H5 Let q > 2 and C generated by G = , a 0 1 H1 H2 where a 0, 1. H4 H3 The extended weights are given by A 5 (T ) = (6(T + 1) )(T 1) = (6T 13)(T 1) Six lines with T + 1 points; minus the points counted before. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

21 Codes and hyperplane arrangements Example H6 H5 Let q > 2 and C generated by G = , a 0 1 H1 H2 where a 0, 1. H4 H3 The extended weights are given by A 6 (T ) = (T 1)(T 2)(T 3) The total number of projective points is T 2 + T + 1. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

22 Geometric lattice To formalize this counting, we use the geometric lattice associated to the arrangement. Notation: L. Möbius function For all x y, we have µ L (x, x) = 0 and x z y Characteristic polynomial µ L (x, z) = χ L (T ) = x L x z y µ L (z, y) = 0. µ L (ˆ0, x)t r(l) r(x) Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

23 Geometric lattice Example Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

24 Coboundary polynomial Coboundary polynomial The coboundary of a geometric lattice is defined by χ L (S, T ) = µ L (x, y)s a(x) T r(l) r(y) x L x y L where a(x) is the number of atoms smaller then x. We write: χ L (S, T ) = n i=0 S i χ i (T ), with χ i (T ) = x L a(x)=i χ [x,ˆ1](t ). Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

25 Coboundary polynomial Theorem χ i (T ) = A n i (T ) Proof: For every point in F k q m the point. there is a unique biggest element of L that contains A n i (q m ) = number of points in F k qm on exactly i hyperplanes = x L a(x)=i number of points in F k q m in x but not in any y > x Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

26 Coboundary polynomial Well-known fact: χ L (q m ) = number of points in F k qm not in the arrangement This means that: A n i (q m ) = x L a(x)=i = number of points in F k q m in ˆ0 but not in any y > ˆ0 = x L a(x)=i = χ i (q m ) number of points in F k q m χ [x,ˆ1] (qm ) So by interpolation, χ i (T ) = A n i (T ). in x but not in any y > x Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

27 Summary Codes are linear subspaces of F n q. Extending the underlying field gives extension codes C F q m, and we define the extended weight enumerator W C (X, Y, T ). By viewing the columns of G as hyperplanes, we associate an arrangement to a code. Finding the extended weight enumerator means counting points in intersections of hyperplanes. This counting can be done using the geometric lattice associated with the arrangement. The coboundary polynomial is equivalent to the extended weight enumerator. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16

28 Thank you for your attention. Relinde Jurrius (VUB) Application to weight enumeration Joint Math Meetings / 16