MinT - Best Known (187−93, 187, s)-Nets in Base 2

Best Known (187−93, 187, s)-Nets in Base 2

(187−93, 187, 53)-Net over F₂ — Constructive and digital

Digital (94, 187, 53)-net over F₂, using

t-expansion [i] based on digital (90, 187, 53)-net over F₂, using
- net from sequence [i] based on digital (90, 52)-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 4 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]

(187−93, 187, 60)-Net over F₂ — Digital

Digital (94, 187, 60)-net over F₂, using

t-expansion [i] based on digital (92, 187, 60)-net over F₂, using
- net from sequence [i] based on digital (92, 59)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 92 and N(F) ≥ 60, using
    - Drinfeld modules of rank 1 [i]

(187−93, 187, 202)-Net over F₂ — Upper bound on s (digital)

There is no digital (94, 187, 203)-net over F₂, because

1 times m-reduction [i] would yield digital (94, 186, 203)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁸⁶, 203, F₂, 92) (dual of [203, 17, 93]-code), but
  - residual code [i] would yield OA(2⁹⁴, 110, S₂, 46), but
    - the linear programming bound shows that MÂ â‰¥ 70 671520 962723 789133 441203 699712 / 3451 > 2⁹⁴ [i]

(187−93, 187, 234)-Net in Base 2 — Upper bound on s

There is no (94, 187, 235)-net in base 2, because

1 times m-reduction [i] would yield (94, 186, 235)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 102 645883 626922 117628 423624 296451 763750 489363 702684 247328 > 2¹⁸⁶ [i]