MinT - Best Known (215−114, 215, s)-Nets in Base 2

Best Known (215−114, 215, s)-Nets in Base 2

(215−114, 215, 55)-Net over F₂ — Constructive and digital

Digital (101, 215, 55)-net over F₂, using

t-expansion [i] based on digital (100, 215, 55)-net over F₂, using
- net from sequence [i] based on digital (100, 54)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 6 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(215−114, 215, 65)-Net over F₂ — Digital

Digital (101, 215, 65)-net over F₂, using

t-expansion [i] based on digital (95, 215, 65)-net over F₂, using
- net from sequence [i] based on digital (95, 64)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 95 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(215−114, 215, 213)-Net in Base 2 — Upper bound on s

There is no (101, 215, 214)-net in base 2, because

14 times m-reduction [i] would yield (101, 201, 214)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁰¹, 214, S₂, 100), but
  - the linear programming bound shows that MÂ â‰¥ 128760 731610 384372 798626 338535 112677 014899 081485 827624 307028 656128 / 29087 > 2²⁰¹ [i]