MinT - Best Known (27, 61, s)-Nets in Base 2

Best Known (27, 61, s)-Nets in Base 2

(27, 61, 21)-Net over F₂ — Constructive and digital

Digital (27, 61, 21)-net over F₂, using

t-expansion [i] based on digital (21, 61, 21)-net over F₂, using
- net from sequence [i] based on digital (21, 20)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
    - an explicitly constructive algebraic function field [i]

(27, 61, 24)-Net over F₂ — Digital

Digital (27, 61, 24)-net over F₂, using

t-expansion [i] based on digital (25, 61, 24)-net over F₂, using
- net from sequence [i] based on digital (25, 23)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 25 and N(F) ≥ 24, using
    - Drinfeld modules of rank 1 [i]

(27, 61, 62)-Net over F₂ — Upper bound on s (digital)

There is no digital (27, 61, 63)-net over F₂, because

6 times m-reduction [i] would yield digital (27, 55, 63)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁵⁵, 63, F₂, 28) (dual of [63, 8, 29]-code), but
  - â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]

(27, 61, 63)-Net in Base 2 — Upper bound on s

There is no (27, 61, 64)-net in base 2, because

2 times m-reduction [i] would yield (27, 59, 64)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁵⁹, 64, S₂, 32), but
  - adding a parity check bit [i] would yield OA(2⁶⁰, 65, S₂, 33), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 27 670116 110564 327424 / 17 > 2⁶⁰ [i]