MinT - Best Known (148−75, 148, s)-Nets in Base 2

Best Known (148−75, 148, s)-Nets in Base 2

(148−75, 148, 49)-Net over F₂ — Constructive and digital

Digital (73, 148, 49)-net over F₂, using

t-expansion [i] based on digital (70, 148, 49)-net over F₂, using
- net from sequence [i] based on digital (70, 48)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, and 1 place with degree 2 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(148−75, 148, 157)-Net over F₂ — Upper bound on s (digital)

There is no digital (73, 148, 158)-net over F₂, because

1 times m-reduction [i] would yield digital (73, 147, 158)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁴⁷, 158, F₂, 74) (dual of [158, 11, 75]-code), but
  - residual code [i] would yield linear OA(2⁷³, 83, F₂, 37) (dual of [83, 10, 38]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(148−75, 148, 180)-Net in Base 2 — Upper bound on s

There is no (73, 148, 181)-net in base 2, because

1 times m-reduction [i] would yield (73, 147, 181)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 196 469911 897924 635805 383756 040370 586603 866038 > 2¹⁴⁷ [i]