Classifying toric surface codes of dimension 7 Emily Cairncross 1 , Stephanie Ford 2 , & Eli Garcia 3 Mentor: Kelly Jabbusch University of Michigan - Dearborn REU 2019 1 Oberlin College 2 Texas A&M University 3 MIT February 1, 2020
Overview Creating a code 1 Analyzing a code 2 Monomial equivalence and lattice equivalence 3 Classification of polygons with 7 lattice points 4 Future classification for polygons with 8 lattice points 5
Creating a code k -dimensional linear code: k -dimensional subspace of F n q (where F q is a finite field of order q ) Cairncross, Ford, Garcia Toric surface codes 1 / 16
Creating a code k -dimensional linear code: k -dimensional subspace of F n q (where F q is a finite field of order q ) Toric surface code: a linear code given by a generator matrix constructed from a lattice polygon P in R 2 Cairncross, Ford, Garcia Toric surface codes 1 / 16
Creating a code k -dimensional linear code: k -dimensional subspace of F n q (where F q is a finite field of order q ) Toric surface code: a linear code given by a generator matrix constructed from a lattice polygon P in R 2 Simple example We construct a toric surface code using the following parameters: Finite field: F 5 Lattice polygon in R 2 : unit triangle Cairncross, Ford, Garcia Toric surface codes 1 / 16
Example cont. (0 , 1) (0 , 0) (1 , 0) Generator matrix ( G ): 5 ) 2 ( � Lattice points ( � e i ) Elements of ( F ∗ a j ) (0 , 0) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1 , 0) 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 (0 , 1) Cairncross, Ford, Garcia Toric surface codes 2 / 16
Example cont. (0 , 1) (0 , 0) (1 , 0) Generator matrix ( G ): 5 ) 2 ( � Lattice points ( � e i ) Elements of ( F ∗ a j ) (0 , 0) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1 , 0) 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 (0 , 1) For � e i = ( e 1 , e 2 ) and � a j = ( a 1 , a 2 ) : e i = a e 1 a j ) � 1 a e 2 G ij = ( � 2 Cairncross, Ford, Garcia Toric surface codes 2 / 16
Example cont. Generator matrix (generated by unit triangle and F 5 ): 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 G = 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Codewords: Linear combinations of rows of G : u ǫ ( F 5 ) 3 } Code = { � uG : � Cairncross, Ford, Garcia Toric surface codes 3 / 16
Example cont. Generator matrix (generated by unit triangle and F 5 ): 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 G = 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Codewords: Linear combinations of rows of G : u ǫ ( F 5 ) 3 } Code = { � uG : � Examples: (1 , 1 , 0) · G = (2 , 2 , 2 , 2 , 3 , 3 , 3 , 3 , 4 , 4 , 4 , 4 , 0 , 0 , 0 , 0) (0 , 1 , 2) · G = (3 , 0 , 2 , 4 , 4 , 1 , 3 , 0 , 0 , 2 , 4 , 1 , 1 , 3 , 0 , 2) Cairncross, Ford, Garcia Toric surface codes 3 / 16
Analyzing a code Hamming distance: number of indices at which two codewords are different Hamming distance between example codewords: 12 Cairncross, Ford, Garcia Toric surface codes 4 / 16
Analyzing a code Hamming distance: number of indices at which two codewords are different Hamming distance between example codewords: 12 Three important invariants: length of codewords n = ( q − 1) 2 n = (5 − 1) 2 = 16 Cairncross, Ford, Garcia Toric surface codes 4 / 16
Analyzing a code Hamming distance: number of indices at which two codewords are different Hamming distance between example codewords: 12 Three important invariants: length of codewords n = ( q − 1) 2 n = (5 − 1) 2 = 16 dimension of code k = #( P ) , the number of lattice points in P k = #( P ) = 3 Cairncross, Ford, Garcia Toric surface codes 4 / 16
Analyzing a code Hamming distance: number of indices at which two codewords are different Hamming distance between example codewords: 12 Three important invariants: length of codewords n = ( q − 1) 2 n = (5 − 1) 2 = 16 dimension of code k = #( P ) , the number of lattice points in P k = #( P ) = 3 minimum distance d varies (minimum Hamming distance between any two codewords) d = ( q − 1)( q − 2) = (5 − 1)(5 − 2) = 12 Cairncross, Ford, Garcia Toric surface codes 4 / 16
Motivation Previous work done by Little and Schwartz, Soprunov and Soprunova, and Yau et. al Classification of toric surface codes up to dimension k = 6 We continue this classification for dimension k = 7 Cairncross, Ford, Garcia Toric surface codes 5 / 16
Monomial Equivalence Definition Let G 1 and G 2 be the generator matrices for linear codes C 1 and C 2 with dimension k and length n . We call C 1 and C 2 monomially equivalent if there exists an invertible n × n diagonal matrix ∆ and an n × n permutation matrix Π such that G 1 = G 2 ∆Π . Cairncross, Ford, Garcia Toric surface codes 6 / 16
Lattice equivalence Definition Let P 1 and P 2 be lattice convex polytopes in R m . We call P 1 and P 2 lattice equivalent if there exists a unimodular affine transformation T : R m → R m defined by T ( � x ) = M � x + λ where M ∈ SL( m , Z ) and λ ∈ Z m such that T ( P 1 ) = P 2 . Cairncross, Ford, Garcia Toric surface codes 7 / 16
Lattice equivalence Definition Let P 1 and P 2 be lattice convex polytopes in R m . We call P 1 and P 2 lattice equivalent if there exists a unimodular affine transformation T : R m → R m defined by T ( � x ) = M � x + λ where M ∈ SL( m , Z ) and λ ∈ Z m such that T ( P 1 ) = P 2 . Valid transformations: shear, translation, rotation by a multiple of 90 ◦ Scaling is not an affine transformation Cairncross, Ford, Garcia Toric surface codes 7 / 16
Lattice equivalence Definition Let P 1 and P 2 be lattice convex polytopes in R m . We call P 1 and P 2 lattice equivalent if there exists a unimodular affine transformation T : R m → R m defined by T ( � x ) = M � x + λ where M ∈ SL( m , Z ) and λ ∈ Z m such that T ( P 1 ) = P 2 . Valid transformations: shear, translation, rotation by a multiple of 90 ◦ Scaling is not an affine transformation Lattice equivalence ⇒ monomial equivalence Cairncross, Ford, Garcia Toric surface codes 7 / 16
Lattice equivalence Lattice equivalent: Cairncross, Ford, Garcia Toric surface codes 8 / 16
Lattice equivalence Lattice equivalent: Lattice inequivalent: Cairncross, Ford, Garcia Toric surface codes 8 / 16
Lattice equivalence classes for k = 7 For P ( i ) k , k refers to the number of lattice points while i is the number assigned to the equivalence class. Cairncross, Ford, Garcia Toric surface codes 9 / 16
Lattice equivalence classes for k = 7 Cairncross, Ford, Garcia Toric surface codes 9 / 16
Lattice equivalence classes for k = 7 Cairncross, Ford, Garcia Toric surface codes 9 / 16
Classification of k = 7 polygons Theorem: C.F.G. 2019 Every toric surface code generated by a polygon with k = 7 lattice points is monomially equivalent to a code given by one of the polygons in the preceding slides. Cairncross, Ford, Garcia Toric surface codes 10 / 16
Classification of k = 7 polygons Theorem: C.F.G. 2019 Every toric surface code generated by a polygon with k = 7 lattice points is monomially equivalent to a code given by one of the polygons in the preceding slides. Sketch of the proof Cairncross, Ford, Garcia Toric surface codes 10 / 16
Classification of k = 7 polygons Theorem: C.F.G. 2019 Every toric surface code generated by a polygon with k = 7 lattice points is monomially equivalent to a code given by one of the polygons in the preceding slides. Sketch of the proof Goal: prove that we have all polygons with 7 lattice points Each P 7 polygon has at least one P 6 polygon as a subset Cairncross, Ford, Garcia Toric surface codes 10 / 16
Classification of k = 7 polygons Theorem: C.F.G. 2019 Every toric surface code generated by a polygon with k = 7 lattice points is monomially equivalent to a code given by one of the polygons in the preceding slides. Sketch of the proof Goal: prove that we have all polygons with 7 lattice points Each P 7 polygon has at least one P 6 polygon as a subset Take each P 6 and find all possible P 7 by adding lattice points Cairncross, Ford, Garcia Toric surface codes 10 / 16
Illustration of the proof Figure: Illustration for P (2) 6 . Cairncross, Ford, Garcia Toric surface codes 11 / 16
Classification of k = 7 codes Theorem: C.F.G. 2019 The toric surface codes C P ( i ) 7 , 1 ≤ i ≤ 22 , are pairwise monomially inequivalent over F q for sufficiently large q . Cairncross, Ford, Garcia Toric surface codes 12 / 16
Classification of k = 7 codes Theorem: C.F.G. 2019 The toric surface codes C P ( i ) 7 , 1 ≤ i ≤ 22 , are pairwise monomially inequivalent over F q for sufficiently large q . Sketch of the proof Cairncross, Ford, Garcia Toric surface codes 12 / 16
Classification of k = 7 codes Theorem: C.F.G. 2019 The toric surface codes C P ( i ) 7 , 1 ≤ i ≤ 22 , are pairwise monomially inequivalent over F q for sufficiently large q . Sketch of the proof Goal: prove that no pair of the 22 codes are monomially equivalent Cairncross, Ford, Garcia Toric surface codes 12 / 16
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