MinT - Best Known (231−117, 231, s)-Nets in Base 2

Best Known (231−117, 231, s)-Nets in Base 2

(231−117, 231, 57)-Net over F₂ — Constructive and digital

Digital (114, 231, 57)-net over F₂, using

t-expansion [i] based on digital (110, 231, 57)-net over F₂, using
- net from sequence [i] based on digital (110, 56)-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 8 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]

(231−117, 231, 73)-Net over F₂ — Digital

Digital (114, 231, 73)-net over F₂, using

net from sequence [i] based on digital (114, 72)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 114 and N(F) ≥ 73, using
  - Drinfeld modules of rank 1 [i]

(231−117, 231, 239)-Net over F₂ — Upper bound on s (digital)

There is no digital (114, 231, 240)-net over F₂, because

1 times m-reduction [i] would yield digital (114, 230, 240)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²³⁰, 240, F₂, 116) (dual of [240, 10, 117]-code), but
  - residual code [i] would yield linear OA(2¹¹⁴, 123, F₂, 58) (dual of [123, 9, 59]-code), but
    - residual code [i] would yield linear OA(2⁵⁶, 64, F₂, 29) (dual of [64, 8, 30]-code), but
      - 1 times truncation [i] would yield linear OA(2⁵⁵, 63, F₂, 28) (dual of [63, 8, 29]-code), but
        â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]

(231−117, 231, 240)-Net in Base 2 — Upper bound on s

There is no (114, 231, 241)-net in base 2, because

9 times m-reduction [i] would yield (114, 222, 241)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²²², 241, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 65 939284 597237 046425 151657 508194 701172 717895 120115 873148 022756 434178 473984 / 7 611275 > 2²²² [i]