MinT - Best Known (203−89, 203, s)-Nets in Base 2

Best Known (203−89, 203, s)-Nets in Base 2

(203−89, 203, 59)-Net over F₂ — Constructive and digital

Digital (114, 203, 59)-net over F₂, using

1 times m-reduction [i] based on digital (114, 204, 59)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 60, 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, 144, 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]

(203−89, 203, 76)-Net over F₂ — Digital

Digital (114, 203, 76)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(203−89, 203, 296)-Net in Base 2 — Upper bound on s

There is no (114, 203, 297)-net in base 2, because

1 times m-reduction [i] would yield (114, 202, 297)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁰², 297, S₂, 88), but
  - 2 times code embedding in larger space [i] would yield OA(2²⁰⁴, 299, S₂, 88), but
    - adding a parity check bit [i] would yield OA(2²⁰⁵, 300, S₂, 89), but
      - the linear programming bound shows that MÂ â‰¥ 416927 463188 475542 795463 940731 898983 181308 104641 213119 598295 609827 594303 154474 074874 094714 331435 368448 / 5584 401256 756961 297780 513782 298869 514125 > 2²⁰⁵ [i]