MinT - Best Known (99−53, 99, s)-Nets in Base 2

Best Known (99−53, 99, s)-Nets in Base 2

(99−53, 99, 34)-Net over F₂ — Constructive and digital

Digital (46, 99, 34)-net over F₂, using

t-expansion [i] based on digital (45, 99, 34)-net over F₂, using
- net from sequence [i] based on digital (45, 33)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 1 place with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(99−53, 99, 101)-Net over F₂ — Upper bound on s (digital)

There is no digital (46, 99, 102)-net over F₂, because

5 times m-reduction [i] would yield digital (46, 94, 102)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁹⁴, 102, F₂, 48) (dual of [102, 8, 49]-code), but
  - residual code [i] would yield linear OA(2⁴⁶, 53, F₂, 24) (dual of [53, 7, 25]-code), but
    - 1 times code embedding in larger space [i] would yield linear OA(2⁴⁷, 54, F₂, 24) (dual of [54, 7, 25]-code), but
      - adding a parity check bit [i] would yield linear OA(2⁴⁸, 55, F₂, 25) (dual of [55, 7, 26]-code), but
        â€œvT4â€ bound on codes from Brouwerâ€™s database [i]

(99−53, 99, 102)-Net in Base 2 — Upper bound on s

There is no (46, 99, 103)-net in base 2, because

1 times m-reduction [i] would yield (46, 98, 103)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁹⁸, 103, S₂, 52), but
  - adding a parity check bit [i] would yield OA(2⁹⁹, 104, S₂, 53), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 20 282409 603651 670423 947251 286016 / 27 > 2⁹⁹ [i]