MinT - Best Known (89, 89+109, s)-Nets in Base 2

Best Known (89, 89+109, s)-Nets in Base 2

(89, 89+109, 52)-Net over F₂ — Constructive and digital

Digital (89, 198, 52)-net over F₂, using

t-expansion [i] based on digital (85, 198, 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+109, 57)-Net over F₂ — Digital

Digital (89, 198, 57)-net over F₂, using

t-expansion [i] based on digital (83, 198, 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+109, 189)-Net over F₂ — Upper bound on s (digital)

There is no digital (89, 198, 190)-net over F₂, because

19 times m-reduction [i] would yield digital (89, 179, 190)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁷⁹, 190, F₂, 90) (dual of [190, 11, 91]-code), but
  - residual code [i] would yield linear OA(2⁸⁹, 99, F₂, 45) (dual of [99, 10, 46]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(89, 89+109, 190)-Net in Base 2 — Upper bound on s

There is no (89, 198, 191)-net in base 2, because

13 times m-reduction [i] would yield (89, 185, 191)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹⁸⁵, 191, S₂, 96), but
  - adding a parity check bit [i] would yield OA(2¹⁸⁶, 192, S₂, 97), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 6277 101735 386680 763835 789423 207666 416102 355444 464034 512896 / 49 > 2¹⁸⁶ [i]