MinT - Best Known (100, 222, s)-Nets in Base 2

Best Known (100, 222, s)-Nets in Base 2

(100, 222, 55)-Net over F₂ — Constructive and digital

Digital (100, 222, 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]

(100, 222, 65)-Net over F₂ — Digital

Digital (100, 222, 65)-net over F₂, using

t-expansion [i] based on digital (95, 222, 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]

(100, 222, 210)-Net over F₂ — Upper bound on s (digital)

There is no digital (100, 222, 211)-net over F₂, because

18 times m-reduction [i] would yield digital (100, 204, 211)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁰⁴, 211, F₂, 104) (dual of [211, 7, 105]-code), but
  - Griesmer bound [i]

(100, 222, 211)-Net in Base 2 — Upper bound on s

There is no (100, 222, 212)-net in base 2, because

22 times m-reduction [i] would yield (100, 200, 212)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁰⁰, 212, S₂, 100), but
  - the linear programming bound shows that MÂ â‰¥ 53067 521973 608894 859497 756137 474553 785693 231666 682950 592992 641024 / 29087 > 2²⁰⁰ [i]