MinT - Best Known (166−82, 166, s)-Nets in Base 2

Best Known (166−82, 166, s)-Nets in Base 2

(166−82, 166, 51)-Net over F₂ — Constructive and digital

Digital (84, 166, 51)-net over F₂, using

t-expansion [i] based on digital (80, 166, 51)-net over F₂, using
- net from sequence [i] based on digital (80, 50)-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 2 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]

(166−82, 166, 57)-Net over F₂ — Digital

Digital (84, 166, 57)-net over F₂, using

t-expansion [i] based on digital (83, 166, 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]

(166−82, 166, 186)-Net over F₂ — Upper bound on s (digital)

There is no digital (84, 166, 187)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2¹⁶⁶, 187, F₂, 82) (dual of [187, 21, 83]-code), but
- residual code [i] would yield linear OA(2⁸⁴, 104, F₂, 41) (dual of [104, 20, 42]-code), but
  - â€œBroâ€ bound on codes from Brouwerâ€™s database [i]

(166−82, 166, 211)-Net in Base 2 — Upper bound on s

There is no (84, 166, 212)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 96 566697 296403 102515 495902 319018 075759 398818 418620 > 2¹⁶⁶ [i]