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
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.
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
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.
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.
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)
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)
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.
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.
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)
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)
Determining extended weight enumerator Determining generalized weight enumerators Determining l(j) for all J [n]
Determining extended rank weight enumerator Determining generalized rank weight enumerators Determining l(j) for all J F n q
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)
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)
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.
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)
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)
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.
Why study q-matroids? Matroids generalize: codes graphs some designs q-matroids generalize: rank metric codes q-graphs? q-designs?
Further work Equivalence between polynomials Various definetions of q-matroids Representable q-matroids Deletion and contraction
Thank you for your attention.
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}