MinT - Best Known (82−45, 82, s)-Nets in Base 2

Best Known (82−45, 82, s)-Nets in Base 2

(82−45, 82, 24)-Net over F₂ — Constructive and digital

Digital (37, 82, 24)-net over F₂, using

t-expansion [i] based on digital (33, 82, 24)-net over F₂, using
- net from sequence [i] based on digital (33, 23)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 33 and N(F) ≥ 24, using
    - a function field by SÃ©mirat [i]

(82−45, 82, 30)-Net over F₂ — Digital

Digital (37, 82, 30)-net over F₂, using

t-expansion [i] based on digital (36, 82, 30)-net over F₂, using
- net from sequence [i] based on digital (36, 29)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 36 and N(F) ≥ 30, using
    - Drinfeld modules of rank 1 [i]

(82−45, 82, 82)-Net over F₂ — Upper bound on s (digital)

There is no digital (37, 82, 83)-net over F₂, because

5 times m-reduction [i] would yield digital (37, 77, 83)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁷⁷, 83, F₂, 40) (dual of [83, 6, 41]-code), but
  - Griesmer bound [i]

(82−45, 82, 83)-Net in Base 2 — Upper bound on s

There is no (37, 82, 84)-net in base 2, because

3 times m-reduction [i] would yield (37, 79, 84)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁷⁹, 84, S₂, 42), but
  - adding a parity check bit [i] would yield OA(2⁸⁰, 85, S₂, 43), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 14 507109 835375 550096 474112 / 11 > 2⁸⁰ [i]