MinT - Best Known (121−9, 121, s)-Nets in Base 2

Best Known (121−9, 121, s)-Nets in Base 2

(121−9, 121, 4194306)-Net over F₂ — Constructive and digital

Digital (112, 121, 4194306)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹²¹, 4194306, F₂, 9, 9) (dual of [(4194306, 9), 37748633, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹²¹, 4194306, F₂, 8, 9) (dual of [(4194306, 8), 33554327, 10]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁵, 5, F₂, 8, 4) (dual of [(5, 8), 35, 5]-NRT-code), using
      - appending 4 arbitrary columns [i] based on linear OOA(2⁵, 5, F₂, 4, 4) (dual of [(5, 4), 15, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2⁵, 5, F₂, 3, 4) (dual of [(5, 3), 10, 5]-NRT-code), using
        extended algebraic-geometric NRT-code AG_e(3;F,10P) [i] based on function field F/F₂ with g(F) = 1 and N(F) ≥ 5, using
        an explicitly constructive algebraic function field [i]
    2. linear OOA(2¹¹⁶, 4194301, F₂, 8, 9) (dual of [(4194301, 8), 33554292, 10]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OOA(2¹¹⁶, large, F₂, 2, 9), using
        OOAs constructed from BCH codes [i]

(121−9, 121, large)-Net over F₂ — Digital

Digital (112, 121, large)-net over F₂, using

2⁴ times duplication [i] based on digital (108, 117, large)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2¹¹⁷, large, F₂, 9, 9), using
  - appending kth column [i] based on linear OOA(2¹¹⁷, large, F₂, 8, 9), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹¹⁷, large, F₂, 3, 9), using
      - OOAs constructed from BCH codes [i]

(121−9, 121, large)-Net in Base 2 — Upper bound on s

There is no (112, 121, large)-net in base 2, because

7 times m-reduction [i] would yield (112, 114, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]