MinT - Best Known (142−44, 142, s)-Nets in Base 2

Best Known (142−44, 142, s)-Nets in Base 2

(142−44, 142, 72)-Net over F₂ — Constructive and digital

Digital (98, 142, 72)-net over F₂, using

2¹ times duplication [i] based on digital (97, 141, 72)-net over F₂, using
- trace code for nets [i] based on digital (3, 47, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]

(142−44, 142, 84)-Net in Base 2 — Constructive

(98, 142, 84)-net in base 2, using

trace code for nets [i] based on (27, 71, 42)-net in base 4, using
- net from sequence [i] based on (27, 41)-sequence in base 4, using
  - base expansion [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(142−44, 142, 125)-Net over F₂ — Digital

Digital (98, 142, 125)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(142−44, 142, 762)-Net in Base 2 — Upper bound on s

There is no (98, 142, 763)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 5 704256 384858 969737 698632 123889 619802 008082 > 2¹⁴² [i]