MinT - Best Known (207−91, 207, s)-Nets in Base 2

Best Known (207−91, 207, s)-Nets in Base 2

(207−91, 207, 59)-Net over F₂ — Constructive and digital

Digital (116, 207, 59)-net over F₂, using

3 times m-reduction [i] based on digital (116, 210, 59)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 62, 17)-net over F₂, using
    - net from sequence [i] based on digital (15, 16)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 15 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 148, 42)-net over F₂, using
    - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
        an explicitly constructive algebraic function field [i]

(207−91, 207, 77)-Net over F₂ — Digital

Digital (116, 207, 77)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(207−91, 207, 296)-Net in Base 2 — Upper bound on s

There is no (116, 207, 297)-net in base 2, because

1 times m-reduction [i] would yield (116, 206, 297)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁰⁶, 297, S₂, 90), but
  - 2 times code embedding in larger space [i] would yield OA(2²⁰⁸, 299, S₂, 90), but
    - adding a parity check bit [i] would yield OA(2²⁰⁹, 300, S₂, 91), but
      - the linear programming bound shows that MÂ â‰¥ 437987 836379 661851 428112 482969 460104 960459 291237 979833 080590 515802 667419 590325 215760 903409 709975 863296 / 433 753365 263741 276278 643266 060465 678125 > 2²⁰⁹ [i]