MinT - Best Known (89, 89+81, s)-Nets in Base 2

Best Known (89, 89+81, s)-Nets in Base 2

(89, 89+81, 52)-Net over F₂ — Constructive and digital

Digital (89, 170, 52)-net over F₂, using

t-expansion [i] based on digital (85, 170, 52)-net over F₂, using
- net from sequence [i] based on digital (85, 51)-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, 1 place with degree 2, and 3 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(89, 89+81, 57)-Net over F₂ — Digital

Digital (89, 170, 57)-net over F₂, using

t-expansion [i] based on digital (83, 170, 57)-net over F₂, using
- net from sequence [i] based on digital (83, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 83 and N(F) ≥ 57, using
    - an algebraic function field by Auer [i]

(89, 89+81, 196)-Net over F₂ — Upper bound on s (digital)

There is no digital (89, 170, 197)-net over F₂, because

1 times m-reduction [i] would yield digital (89, 169, 197)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁶⁹, 197, F₂, 80) (dual of [197, 28, 81]-code), but
  - adding a parity check bit [i] would yield linear OA(2¹⁷⁰, 198, F₂, 81) (dual of [198, 28, 82]-code), but
    - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

(89, 89+81, 240)-Net in Base 2 — Upper bound on s

There is no (89, 170, 241)-net in base 2, because

1 times m-reduction [i] would yield (89, 169, 241)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 820 991540 653731 051583 609607 927603 130175 503483 454958 > 2¹⁶⁹ [i]