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Configurations of Extremal Type II Lattices and Codes Scott Duke Kominers Department of Economics, Harvard University, and Harvard Business School AMS-MAA-SIAM Session on Research in Mathematics by Undergraduates Joint Mathematics Meetings

  1. Configurations of Extremal Type II Lattices Introduction Lattice Configuration Results Theorem Template If L is Type II and extremal of rank n, then the minimal-norm vectors of L generate L. Folklore: n ∈ { 8 , 24 } Venkov (1984): n ∈ { 32 } Ozeki (1986): n ∈ { 32 , 48 } Scott Duke Kominers (Harvard) January 15, 2010 4

  2. Configurations of Extremal Type II Lattices Introduction Lattice Configuration Results Theorem Template If L is Type II and extremal of rank n, then the minimal-norm vectors of L generate L. Folklore: n ∈ { 8 , 24 } Venkov (1984): n ∈ { 32 } Ozeki (1986): n ∈ { 32 , 48 } K. (2009): n ∈ { 56 , 72 , 96 } Scott Duke Kominers (Harvard) January 15, 2010 4

  3. Configurations of Extremal Type II Lattices Introduction Lattice Configuration Results Theorem Template If L is Type II and extremal of rank n, then the minimal-norm vectors of L generate L. Folklore: n ∈ { 8 , 24 } Venkov (1984): n ∈ { 32 } Ozeki (1986): n ∈ { 32 , 48 } K. (2009): n ∈ { 56 , 72 , 96 } Elkies (2010): n ∈ { 120 } Scott Duke Kominers (Harvard) January 15, 2010 4

  4. Configurations of Extremal Type II Lattices Methods Theta Functions Scott Duke Kominers (Harvard) January 15, 2010 5

  5. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function of a lattice L encodes the lengths of L ’s vectors.” Scott Duke Kominers (Harvard) January 15, 2010 5

  6. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function of a lattice L encodes the lengths of L ’s vectors.” How: “norm � x , x � ” ⇐ ⇒ “length” Scott Duke Kominers (Harvard) January 15, 2010 5

  7. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function of a lattice L encodes the lengths of L ’s vectors.” How: “norm � x , x � ” ⇐ ⇒ “length” What: ∞ � � e i πτ � x , x � = a k e i πτ ( k ) Θ L ( τ ) = x ∈ L k =1 Scott Duke Kominers (Harvard) January 15, 2010 5

  8. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L encodes the lengths of L ’s vectors.” Scott Duke Kominers (Harvard) January 15, 2010 6

  9. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L encodes the lengths of L ’s vectors.” Example: Scott Duke Kominers (Harvard) January 15, 2010 6

  10. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L encodes the lengths of L ’s vectors.” Example: Scott Duke Kominers (Harvard) January 15, 2010 6

  11. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L encodes the lengths of L ’s vectors.” Example: � e i πτ � x , x � Θ Z 2 ( τ ) = x ∈ Z 2 Scott Duke Kominers (Harvard) January 15, 2010 6

  12. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L encodes the lengths of L ’s vectors.” Example: � e i πτ � x , x � Θ Z 2 ( τ ) = x ∈ Z 2 =1 + 4 e i πτ + 4 e 2 i πτ Scott Duke Kominers (Harvard) January 15, 2010 6

  13. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L encodes the lengths of L ’s vectors.” Example: � e i πτ � x , x � Θ Z 2 ( τ ) = x ∈ Z 2 =1 + 4 e i πτ + 4 e 2 i πτ + 0 e 3 i πτ + 4 e 4 i πτ Scott Duke Kominers (Harvard) January 15, 2010 6

  14. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L encodes the lengths of L ’s vectors.” Example: � e i πτ � x , x � Θ Z 2 ( τ ) = x ∈ Z 2 =1 + 4 e i πτ + 4 e 2 i πτ + 0 e 3 i πτ + 4 e 4 i πτ + 8 e 4 i πτ + · · · Scott Duke Kominers (Harvard) January 15, 2010 6

  15. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L encodes the lengths of L ’s vectors.” Scott Duke Kominers (Harvard) January 15, 2010 7

  16. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L encodes the lengths of L ’s vectors.” Why we care: For L Type II of rank n , the theta function Θ L is a modular form : Θ L ∈ M n / 2 . Scott Duke Kominers (Harvard) January 15, 2010 7

  17. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L encodes the lengths of L ’s vectors.” Why we care: For L Type II of rank n , the theta function Θ L is a modular form : Θ L ∈ M n / 2 . For n small, the space M n / 2 is small. Scott Duke Kominers (Harvard) January 15, 2010 7

  18. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L encodes the lengths of L ’s vectors.” Why we care: For L Type II of rank n , the theta function Θ L is a modular form : Θ L ∈ M n / 2 . For n (relatively) small, the space M n / 2 is (very) small. Scott Duke Kominers (Harvard) January 15, 2010 7

  19. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L encodes the lengths of L ’s vectors.” Why we care: For L Type II of rank n , the theta function Θ L is a modular form : Θ L ∈ M n / 2 . For n (relatively) small, the space M n / 2 is (very) small. Scott Duke Kominers (Harvard) January 15, 2010 7

  20. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L encodes the lengths of L ’s vectors.” Why we care: For L Type II of rank n , the theta function Θ L is a modular form : Θ L ∈ M n / 2 . For n (relatively) small, the space M n / 2 is (very) small. We can therefore study the function Θ L even if we cannot write down a basis for L . Scott Duke Kominers (Harvard) January 15, 2010 7

  21. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L is a modular form which encodes the lengths of L ’s vectors.” Scott Duke Kominers (Harvard) January 15, 2010 8

  22. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L is a modular form which encodes the lengths of L ’s vectors.” What we do: Scott Duke Kominers (Harvard) January 15, 2010 8

  23. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L is a modular form which encodes the lengths of L ’s vectors.” What we do: We study weighted theta functions � x ∈ L P ( x ) e i πτ � x , x � which encode norms and distributions of lattice vectors. Scott Duke Kominers (Harvard) January 15, 2010 8

  24. Configurations of Extremal Type II Lattices Methods Theta Functions Slogan: “The theta function � x ∈ L e i πτ � x , x � of a lattice L is a modular form which encodes the lengths of L ’s vectors.” What we do: We study weighted theta functions � x ∈ L P ( x ) e i πτ � x , x � which encode norms and distributions of lattice vectors. We obtain a “system of equations in vector distributions” which proves our configuration results. Scott Duke Kominers (Harvard) January 15, 2010 8

  25. Configurations of Extremal Type II Lattices Results Lattice Configuration Results Theorem Template If L is Type II and extremal of rank n, then the minimal-norm vectors of L generate L. Folklore: n ∈ { 8 , 24 } Venkov (1984): n ∈ { 32 } Ozeki (1986): n ∈ { 32 , 48 } K. (2009): n ∈ { 56 , 72 , 96 } Elkies (2010): n ∈ { 120 } Scott Duke Kominers (Harvard) January 15, 2010 9

  26. Configurations of Extremal Type II Lattices Results Lattice Configuration Results Theorem Template If L is Type II and extremal of rank n, then the minimal-norm vectors of L generate L. Folklore: n ∈ { 8 , 24 } Venkov (1984): n ∈ { 32 } Ozeki (1986): n ∈ { 32 , 48 } K. (2009): n ∈ { 56 , 72 , 96 } Elkies (2010): n ∈ { 120 } Scott Duke Kominers (Harvard) January 15, 2010 9

  27. Configurations of Extremal Type II Lattices Results Lattice Configuration Results Scott Duke Kominers (Harvard) January 15, 2010 10

  28. Configurations of Extremal Type II Lattices Results Lattice Configuration Results Theorem Template If L is Type II and extremal of rank n with minimal norm m ( L ) , then L is generated by its vectors of norms m ( L ) and ( m ( L ) + 2) . Scott Duke Kominers (Harvard) January 15, 2010 10

  29. Configurations of Extremal Type II Lattices Results Lattice Configuration Results Theorem Template If L is Type II and extremal of rank n with minimal norm m ( L ) , then L is generated by its vectors of norms m ( L ) and ( m ( L ) + 2) . Folklore: n ∈ { 16 } Scott Duke Kominers (Harvard) January 15, 2010 10

  30. Configurations of Extremal Type II Lattices Results Lattice Configuration Results Theorem Template If L is Type II and extremal of rank n with minimal norm m ( L ) , then L is generated by its vectors of norms m ( L ) and ( m ( L ) + 2) . Folklore: n ∈ { 16 } Ozeki (1989): n ∈ { 40 } Scott Duke Kominers (Harvard) January 15, 2010 10

  31. Configurations of Extremal Type II Lattices Results Lattice Configuration Results Theorem Template If L is Type II and extremal of rank n with minimal norm m ( L ) , then L is generated by its vectors of norms m ( L ) and ( m ( L ) + 2) . Folklore: n ∈ { 16 } Ozeki (1989): n ∈ { 40 } Abel–K. (2008): n ∈ { 40 , 80 , 120 } (unified method) Scott Duke Kominers (Harvard) January 15, 2010 10

  32. Configurations of Extremal Type II Lattices Results Lattice Configuration Results Theorem Template If L is Type II and extremal of rank n with minimal norm m ( L ) , then L is generated by its vectors of norms m ( L ) and ( m ( L ) + 2) . Folklore: n ∈ { 16 } Ozeki (1989): n ∈ { 40 } Abel–K. (2008): n ∈ { 40 , 80 , 120 } (unified method) Elkies–K. (2010): Norm-( m ( L ) + 2) suffices for n ∈ { 40 , 80 } Scott Duke Kominers (Harvard) January 15, 2010 10

  33. Configurations of Extremal Type II Lattices Pause Scott Duke Kominers (Harvard) January 15, 2010 11

  34. Configurations of Extremal Type II Lattices Pause We just described configurations of lattices. Scott Duke Kominers (Harvard) January 15, 2010 11

  35. Configurations of Extremal Type II Lattices Pause We just described configurations of lattices. Recall the title slide.... Scott Duke Kominers (Harvard) January 15, 2010 11

  36. Configurations of Extremal Type II Lattices Pause Configurations of Extremal Type II Lattices and Codes Scott Duke Kominers Department of Economics, Harvard University, and Harvard Business School AMS-MAA-SIAM Session on Research in Mathematics by Undergraduates Joint Mathematics Meetings January 15, 2010 Scott Duke Kominers (Harvard) January 15, 2010 11

  37. Configurations of Extremal Type II Lattices Pause We just described configurations of lattices. Recall the title slide.... Scott Duke Kominers (Harvard) January 15, 2010 11

  38. Configurations of Extremal Type II Lattices Pause We just described configurations of lattices. Recall the title slide.... Natural Question Scott Duke Kominers (Harvard) January 15, 2010 11

  39. Configurations of Extremal Type II Lattices Pause We just described configurations of lattices. Recall the title slide.... Natural Question What about codes? Scott Duke Kominers (Harvard) January 15, 2010 11

  40. Configurations of Extremal Type II Codes Introduction Key Concepts extremal even unimodular lattice � �� � Type II Scott Duke Kominers (Harvard) January 15, 2010 12

  41. Configurations of Extremal Type II Codes Introduction Key Concepts extremal even unimodular lattice � �� � Type II extremal doubly-even self-dual code � �� � Type II Scott Duke Kominers (Harvard) January 15, 2010 12

  42. Configurations of Extremal Type II Codes Introduction Key Concepts extremal even unimodular lattice � �� � Type II lattice of rank n ∼ “integer vector space” of rank n code of length n ∼ linear subspace of F n 2 extremal doubly-even self-dual code � �� � Type II Scott Duke Kominers (Harvard) January 15, 2010 12

  43. Configurations of Extremal Type II Codes Introduction Key Concepts extremal even unimodular lattice � �� � Type II unimodular ∼ self-dual self-dual ∼ self-dual extremal doubly-even self-dual code � �� � Type II Scott Duke Kominers (Harvard) January 15, 2010 12

  44. Configurations of Extremal Type II Codes Introduction Key Concepts extremal even unimodular lattice � �� � Type II even ∼ all vectors have even norm doubly-even ∼ 4 divides all codewords’ weights extremal doubly-even self-dual code � �� � Type II Scott Duke Kominers (Harvard) January 15, 2010 12

  45. Configurations of Extremal Type II Codes Introduction Key Concepts extremal even unimodular lattice � �� � Type II extremal doubly-even self-dual code � �� � Type II Scott Duke Kominers (Harvard) January 15, 2010 12

  46. Configurations of Extremal Type II Codes Introduction Key Concepts extremal even unimodular lattice � �� � Type II extremal ∼ shortest vector is as long as possible extremal ∼ smallest codeword is as large as possible extremal doubly-even self-dual code � �� � Type II Scott Duke Kominers (Harvard) January 15, 2010 12

  47. Configurations of Extremal Type II Codes Introduction Key Concepts extremal even unimodular lattice � �� � Type II extremal doubly-even self-dual code � �� � Type II Scott Duke Kominers (Harvard) January 15, 2010 12

  48. � Configurations of Extremal Type II Codes Introduction Key Concepts extremal even unimodular lattice � �� � Type II Construction A extremal doubly-even self-dual code � �� � Type II Scott Duke Kominers (Harvard) January 15, 2010 12

  49. Configurations of Extremal Type II Codes Methods Weight Enumerators Theta Function Slogan: “The theta function Θ L ( τ ) of a lattice L encodes the lengths of L ’s vectors.” Scott Duke Kominers (Harvard) January 15, 2010 13

  50. Configurations of Extremal Type II Codes Methods Weight Enumerators Theta Function Slogan: “The theta function Θ L ( τ ) of a lattice L encodes the lengths of L ’s vectors.” Weight Enumerator Slogan: “The weight enumerator W C ( x , y ) of a code C encodes the weights of C ’s codewords.” Scott Duke Kominers (Harvard) January 15, 2010 13

  51. Configurations of Extremal Type II Codes Methods Weight Enumerators Theta Function Slogan: “The theta function Θ L ( τ ) of a lattice L is a modular form which encodes the lengths of L ’s vectors.” Weight Enumerator Slogan: “The weight enumerator W C ( x , y ) of a code C encodes the weights of C ’s codewords.” Scott Duke Kominers (Harvard) January 15, 2010 13

  52. Configurations of Extremal Type II Codes Methods Weight Enumerators Theta Function Slogan: “The theta function Θ L ( τ ) of a lattice L is a modular form which encodes the lengths of L ’s vectors.” Weight Enumerator Slogan: “The weight enumerator W C ( x , y ) of a code C encodes the weights of C ’s codewords.” Scott Duke Kominers (Harvard) January 15, 2010 13

  53. Configurations of Extremal Type II Codes Methods Weight Enumerators Theta Function Slogan: “The theta function Θ L ( τ ) of a lattice L is a modular form which encodes the lengths of L ’s vectors.” Weight Enumerator Slogan: “The weight enumerator W C ( x , y ) of a code C is a classifiable polynomial which encodes the weights of C ’s codewords.” Scott Duke Kominers (Harvard) January 15, 2010 13

  54. Configurations of Extremal Type II Codes Methods Weight Enumerators Theta Function Slogan: “The theta function Θ L ( τ ) of a lattice L is a modular form which encodes the lengths of L ’s vectors.” Weight Enumerator Slogan: “The weight enumerator W C ( x , y ) of a code C is a classifiable polynomial which encodes the weights of C ’s codewords.” Scott Duke Kominers (Harvard) January 15, 2010 13

  55. Configurations of Extremal Type II Codes Methods Weight Enumerators Theta Function Slogan: “The weighted theta function Θ L , P ( τ ) of L is a modular form which encodes the distributions of L ’s vectors.” Weight Enumerator Slogan: “The harmonic weight enumerator W C , Q ( x , y ) of C is a classifiable polynomial which encodes the distributions of C ’s codewords.” Scott Duke Kominers (Harvard) January 15, 2010 13

  56. Configurations of Extremal Type II Codes Methods Weight Enumerators Theta Function Slogan: “The weighted theta function Θ L , P ( τ ) of L is a modular form which encodes the distributions of L ’s vectors.” Weight Enumerator Slogan: “The harmonic weight enumerator W C , Q ( x , y ) of C is a classifiable polynomial which encodes the distributions of C ’s codewords.” Scott Duke Kominers (Harvard) January 15, 2010 13

  57. Configurations of Extremal Type II Codes Results Code Configuration Results Scott Duke Kominers (Harvard) January 15, 2010 14

  58. Configurations of Extremal Type II Codes Results Code Configuration Results Theorem Template If C is Type II and extremal of length n, then the minimal-weight codewords of C generate C. Scott Duke Kominers (Harvard) January 15, 2010 14

  59. Configurations of Extremal Type II Codes Results Code Configuration Results Theorem Template If C is Type II and extremal of length n, then the minimal-weight codewords of C generate C. Folklore(?): n ∈ { 8 , 24 } Scott Duke Kominers (Harvard) January 15, 2010 14

  60. Configurations of Extremal Type II Codes Results Code Configuration Results Theorem Template If C is Type II and extremal of length n, then the minimal-weight codewords of C generate C. Folklore(?): n ∈ { 8 , 24 } K. (2009): n ∈ { 32 , 48 , 56 , 72 , 96 } Scott Duke Kominers (Harvard) January 15, 2010 14

  61. Configurations of Extremal Type II Codes Results Code Configuration Results Theorem Template If C is Type II and extremal of length n, then the minimal-weight codewords of C generate C. Folklore(?): n ∈ { 8 , 24 } K. (2009): n ∈ { 32 , 48 , 56 , 72 , 96 } Likely: Analog of slightly weaker result for n ∈ { 40 , 80 , 120 } Scott Duke Kominers (Harvard) January 15, 2010 14

  62. Configurations of Extremal Type II Lattices and Codes Conclusion Scott Duke Kominers (Harvard) January 15, 2010 15

  63. Configurations of Extremal Type II Lattices and Codes Conclusion Lattices Codes Scott Duke Kominers (Harvard) January 15, 2010 15

  64. � Configurations of Extremal Type II Lattices and Codes Conclusion Lattices common Codes Scott Duke Kominers (Harvard) January 15, 2010 15

  65. � � Configurations of Extremal Type II Lattices and Codes Conclusion Lattices common uncommon Codes Scott Duke Kominers (Harvard) January 15, 2010 15

  66. Configurations of Extremal Type II Lattices and Codes Acknowledgments Scott Duke Kominers (Harvard) January 15, 2010 16

  67. Configurations of Extremal Type II Lattices and Codes Acknowledgments Prof. Noam D. Elkies Scott Duke Kominers (Harvard) January 15, 2010 16

  68. Configurations of Extremal Type II Lattices and Codes Acknowledgments Prof. Noam D. Elkies Mrs. Susan Schwartz Wildstrom Scott Duke Kominers (Harvard) January 15, 2010 16

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