MinT - Best Known (81, 174, s)-Nets in Base 2

Best Known (81, 174, s)-Nets in Base 2

(81, 174, 51)-Net over F₂ — Constructive and digital

Digital (81, 174, 51)-net over F₂, using

t-expansion [i] based on digital (80, 174, 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]

(81, 174, 56)-Net over F₂ — Digital

Digital (81, 174, 56)-net over F₂, using

t-expansion [i] based on digital (80, 174, 56)-net over F₂, using
- net from sequence [i] based on digital (80, 55)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 80 and N(F) ≥ 56, using
    - Drinfeld modules of rank 1 [i]

(81, 174, 172)-Net over F₂ — Upper bound on s (digital)

There is no digital (81, 174, 173)-net over F₂, because

9 times m-reduction [i] would yield digital (81, 165, 173)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁶⁵, 173, F₂, 84) (dual of [173, 8, 85]-code), but
  - residual code [i] would yield linear OA(2⁸¹, 88, F₂, 42) (dual of [88, 7, 43]-code), but
    - â€œHelâ€ bound on codes from Brouwerâ€™s database [i]

(81, 174, 174)-Net in Base 2 — Upper bound on s

There is no (81, 174, 175)-net in base 2, because

5 times m-reduction [i] would yield (81, 169, 175)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹⁶⁹, 175, S₂, 88), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 71835 728478 088540 235547 516897 670742 982128 396352 356352 / 89 > 2¹⁶⁹ [i]