MinT - Best Known (69, 138, s)-Nets in Base 2

Best Known (69, 138, s)-Nets in Base 2

(69, 138, 48)-Net over F₂ — Constructive and digital

Digital (69, 138, 48)-net over F₂, using

net from sequence [i] based on digital (69, 47)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using
  - an explicitly constructive algebraic function field [i]

(69, 138, 49)-Net over F₂ — Digital

Digital (69, 138, 49)-net over F₂, using

t-expansion [i] based on digital (68, 138, 49)-net over F₂, using
- net from sequence [i] based on digital (68, 48)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 68 and N(F) ≥ 49, using
    - an algebraic function field by Auer [i]

(69, 138, 152)-Net over F₂ — Upper bound on s (digital)

There is no digital (69, 138, 153)-net over F₂, because

1 times m-reduction [i] would yield digital (69, 137, 153)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹³⁷, 153, F₂, 68) (dual of [153, 16, 69]-code), but
  - residual code [i] would yield linear OA(2⁶⁹, 84, F₂, 34) (dual of [84, 15, 35]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(69, 138, 175)-Net in Base 2 — Upper bound on s

There is no (69, 138, 176)-net in base 2, because

1 times m-reduction [i] would yield (69, 137, 176)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 196792 741687 243045 673614 634077 952520 356793 > 2¹³⁷ [i]