MinT - Best Known (209−91, 209, s)-Nets in Base 2

Best Known (209−91, 209, s)-Nets in Base 2

(209−91, 209, 62)-Net over F₂ — Constructive and digital

Digital (118, 209, 62)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (19, 64, 20)-net over F₂, using
  - net from sequence [i] based on digital (19, 19)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
      - an explicitly constructive algebraic function field [i]
2. digital (54, 145, 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]

(209−91, 209, 79)-Net over F₂ — Digital

Digital (118, 209, 79)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(209−91, 209, 298)-Net in Base 2 — Upper bound on s

There is no (118, 209, 299)-net in base 2, because

1 times m-reduction [i] would yield (118, 208, 299)-net in base 2, but
- extracting embedded orthogonal array [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]