A combinatorial view on derived codes
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1 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 Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept 16 1 / 16
2 Codes and lattices Linear code k-dim subspace of GF(q) n Arrangement of hyperplanes n-tuple of hyperplanes in GF(q) k Projective system n-tuple of points in PG(k 1, q). One-to-one correspondence between equivalence classes. Independent of choice of generator matrix, so notation: A C or P C. Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept 16 2 / 16
3 Codes and lattices Lattice: poset with join (smallest upper boud), x y and meet (greatest lower bound) x y Geometric lattice: atomic: every element is join of rank-1 elements (atoms) semimodular: rank function with r(x y) + r(x y) r(x) + r(y) no infinite chains Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept 16 3 / 16
4 Codes and lattices Projective system: atoms = points, elements = spans Hyperplane arrangement: atoms = hyperplanes, elements = intersections Example abcd ab ac ad bc bd cd a b c d Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept 16 4 / 16
5 Derived code Start with [n, k] code. Consider the projective system P C. Look at all hyperplanes spanned by k 1 points of P C. (Ignore k 1 points that span spaces of lower dimension.) Remove (multiple) copies of hyperplanes. These hyperplanes form an arrangement A. The derived code D(C) is the code such that A = A D(C). Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept 16 5 / 16
6 Derived code Example C D(C) Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept 16 6 / 16
7 Motivation The derived code was introduced in the study of the coset leader and list weight enumerator. The coset leader weight enumerator is interesting because: Determines the probability of correct decoding in coset leader decoding. Determines the average of changed symbols in steganography (information hiding). The list weight enumerator is interesting because: Determines the size of lists in list decoding. Determines the probability of correct decoding in list decoding. Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept 16 7 / 16
8 Derived code From a lattice point of view, this is how we make a derived code: Start with [n, k] code. Consider the geometric lattice L of P C. Turn it upside down: L op. Add extra elements above atom level such that L op L(D(C)) in the right way; L(D(C)) is a geometric lattice. Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept 16 8 / 16
9 Derived code Example d a f 2 c g b e abcd ab ac ad bc bd cd a b c d Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept 16 9 / 16
10 Derived code Example d a f 2 c g b e V Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept 16 9 / 16
11 Derived code Example Codes with equal geometric lattices may have different derived codes! Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept / 16
12 Results We answered two open questions: When is C = D(C)? Can we define a derived lattice, by taking the derived arrangement as general as possible? Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept / 16
13 When is C = D(C)? We need: L(C) = L(C) op = L(D(C)) and L op is a geometric lattice. Theorem The following are equivalent: C = D(C) r(x y) + r(x y) = r(x) + r(y) P C contains all points of some PG(k 1, q) C is the q-ary simplex code Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept / 16
14 Derived lattice How to create the derived lattice D(L): Start with a geometric lattice L. The atoms of D(L) are the co-atoms of L. For all subsets I of atoms: If r(l) r( I ) I in L, then I = ( I ) op in D(L). If r(l) r( I ) > I in L, then I is a new element in D(L) with r ( I ) = I. Partial ordering: I J iff I J. Rank function: r ( I ) = min{r(l) r( I ), I } Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept / 16
15 Derived lattice Example a 16 3 b c d I = {1, 2} 3 r(a) = 3 1 = no new element I = {1, 6} 3 r(0) = 3 0 = 3 3 > 2 new element I = {1, 5, 6} 3 r(0) = 3 0 = no new element Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept / 16
16 Derived lattice Steps of the proof: D(L) is a lattice L op D(L) in the right way D(L) is a geometric lattice Difficult part: semimodularity r (x y) + r (x y) r (x) + r (y) Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept / 16
17 Thank you for your attention. Relinde Jurrius (VUB/UniNe) A combinatorial view on derived codes Finite Geometries, Sept / 16
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