MinT - Best Known (235−105, 235, s)-Nets in Base 2

Best Known (235−105, 235, s)-Nets in Base 2

(235−105, 235, 66)-Net over F₂ — Constructive and digital

Digital (130, 235, 66)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 91, 33)-net over F₂, using
  - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (39, 144, 33)-net over F₂, using
  - net from sequence [i] based on digital (39, 32)-sequence over F₂ (see above)

(235−105, 235, 82)-Net over F₂ — Digital

Digital (130, 235, 82)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(235−105, 235, 331)-Net over F₂ — Upper bound on s (digital)

There is no digital (130, 235, 332)-net over F₂, because

1 times m-reduction [i] would yield digital (130, 234, 332)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²³⁴, 332, F₂, 104) (dual of [332, 98, 105]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(2²³³, 296, S₂, 104), but
      - the linear programming bound shows that MÂ â‰¥ 40 441027 764911 159411 999678 250893 328460 000589 698723 782371 054222 923004 931887 671319 124733 067264 / 2499 525994 551277 812375 > 2²³³ [i]
    2. linear OA(2⁹⁸, 332, F₂, 36) (dual of [332, 234, 37]-code), but
      - discarding factors / shortening the dual code would yield linear OA(2⁹⁸, 328, F₂, 36) (dual of [328, 230, 37]-code), but
        the Johnson bound shows that NÂ â‰¤ 1638 846907 564815 429866 669910 715625 391366 929534 207151 722672 384476 916602 < 2²³⁰ [i]

(235−105, 235, 385)-Net in Base 2 — Upper bound on s

There is no (130, 235, 386)-net in base 2, because

1 times m-reduction [i] would yield (130, 234, 386)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 28744 831042 914988 267685 213676 940723 688876 955444 393427 531989 967427 115936 > 2²³⁴ [i]