MinT - Best Known (84, 84+79, s)-Nets in Base 2

Best Known (84, 84+79, s)-Nets in Base 2

Digital (84, 163, 51)-net over F₂, using

t-expansion [i] based on digital (80, 163, 51)-net over F₂, using
- net from sequence [i] based on digital (80, 50)-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 2 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 (84, 163, 57)-net over F₂, using

t-expansion [i] based on digital (83, 163, 57)-net over F₂, using
- net from sequence [i] based on digital (83, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 83 and N(F) ≥ 57, using
    - an algebraic function field by Auer [i]

There is no digital (84, 163, 191)-net over F₂, because

1 times m-reduction [i] would yield digital (84, 162, 191)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁶², 191, F₂, 78) (dual of [191, 29, 79]-code), but
  - 4 times code embedding in larger space [i] would yield linear OA(2¹⁶⁶, 195, F₂, 78) (dual of [195, 29, 79]-code), but
    - adding a parity check bit [i] would yield linear OA(2¹⁶⁷, 196, F₂, 79) (dual of [196, 29, 80]-code), but
      - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

There is no (84, 163, 222)-net in base 2, because

1 times m-reduction [i] would yield (84, 162, 222)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 6 655911 793281 744533 817694 989828 307774 808151 238320 > 2¹⁶² [i]