MinT - Best Known (59−27, 59, s)-Nets in Base 2

Best Known (59−27, 59, s)-Nets in Base 2

(59−27, 59, 26)-Net over F₂ — Constructive and digital

Digital (32, 59, 26)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (8, 21, 12)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (11, 38, 14)-net over F₂, using
  - net from sequence [i] based on digital (11, 13)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 11 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]

(59−27, 59, 27)-Net over F₂ — Digital

Digital (32, 59, 27)-net over F₂, using

t-expansion [i] based on digital (31, 59, 27)-net over F₂, using
- net from sequence [i] based on digital (31, 26)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 31 and N(F) ≥ 27, using
    - Drinfeld modules of rank 1 [i]

(59−27, 59, 105)-Net over F₂ — Upper bound on s (digital)

There is no digital (32, 59, 106)-net over F₂, because

1 times m-reduction [i] would yield digital (32, 58, 106)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁵⁸, 106, F₂, 26) (dual of [106, 48, 27]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁵⁹, 107, F₂, 27) (dual of [107, 48, 28]-code), but
    - â€œLPâ€ bound on codes from Brouwerâ€™s database [i]

(59−27, 59, 106)-Net in Base 2 — Upper bound on s

There is no (32, 59, 107)-net in base 2, because

1 times m-reduction [i] would yield (32, 58, 107)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁵⁸, 107, S₂, 26), but
  - the linear programming bound shows that MÂ â‰¥ 63905 115152 312301 338462 624561 945060 065807 433728 / 207189 126759 202717 834639 422345 > 2⁵⁸ [i]