MinT - Best Known (98, 189, s)-Nets in Base 2

Best Known (98, 189, s)-Nets in Base 2

Digital (98, 189, 54)-net over F₂, using

t-expansion [i] based on digital (95, 189, 54)-net over F₂, using
- net from sequence [i] based on digital (95, 53)-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 5 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]

Digital (98, 189, 65)-net over F₂, using

t-expansion [i] based on digital (95, 189, 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]

There is no digital (98, 189, 233)-net over F₂, because

1 times m-reduction [i] would yield digital (98, 188, 233)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁸⁸, 233, F₂, 90) (dual of [233, 45, 91]-code), but
  - residual code [i] would yield OA(2⁹⁸, 142, S₂, 45), but
    - 1 times truncation [i] would yield OA(2⁹⁷, 141, S₂, 44), but
      - the linear programming bound shows that MÂ â‰¥ 33406 063517 297373 977917 219831 876457 216369 229824 / 189199 322419 332537 > 2⁹⁷ [i]

There is no (98, 189, 258)-net in base 2, because

1 times m-reduction [i] would yield (98, 188, 258)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 404 156488 291774 329724 601337 587786 626691 539236 067650 717376 > 2¹⁸⁸ [i]