MinT - Best Known (42, 42+51, s)-Nets in Base 2

Best Known (42, 42+51, s)-Nets in Base 2

(42, 42+51, 33)-Net over F₂ — Constructive and digital

Digital (42, 93, 33)-net over F₂, using

t-expansion [i] based on digital (39, 93, 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]

(42, 42+51, 93)-Net over F₂ — Upper bound on s (digital)

There is no digital (42, 93, 94)-net over F₂, because

7 times m-reduction [i] would yield digital (42, 86, 94)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁸⁶, 94, F₂, 44) (dual of [94, 8, 45]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁸⁷, 95, F₂, 45) (dual of [95, 8, 46]-code), but
    - â€œDHMâ€ bound on codes from Brouwerâ€™s database [i]

(42, 42+51, 94)-Net in Base 2 — Upper bound on s

There is no (42, 93, 95)-net in base 2, because

3 times m-reduction [i] would yield (42, 90, 95)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁹⁰, 95, S₂, 48), but
  - adding a parity check bit [i] would yield OA(2⁹¹, 96, S₂, 49), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 79228 162514 264337 593543 950336 / 25 > 2⁹¹ [i]