MinT - Best Known (174−87, 174, s)-Nets in Base 2

Best Known (174−87, 174, s)-Nets in Base 2

(174−87, 174, 52)-Net over F₂ — Constructive and digital

Digital (87, 174, 52)-net over F₂, using

t-expansion [i] based on digital (85, 174, 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]

(174−87, 174, 57)-Net over F₂ — Digital

Digital (87, 174, 57)-net over F₂, using

t-expansion [i] based on digital (83, 174, 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]

(174−87, 174, 190)-Net over F₂ — Upper bound on s (digital)

There is no digital (87, 174, 191)-net over F₂, because

1 times m-reduction [i] would yield digital (87, 173, 191)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁷³, 191, F₂, 86) (dual of [191, 18, 87]-code), but
  - residual code [i] would yield OA(2⁸⁷, 104, S₂, 43), but
    - 1 times truncation [i] would yield OA(2⁸⁶, 103, S₂, 42), but
      - the linear programming bound shows that MÂ â‰¥ 158456 325028 528675 187087 900672 / 1705 > 2⁸⁶ [i]

(174−87, 174, 216)-Net in Base 2 — Upper bound on s

There is no (87, 174, 217)-net in base 2, because

1 times m-reduction [i] would yield (87, 173, 217)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 12783 527862 841737 564418 297390 357580 858508 652897 733200 > 2¹⁷³ [i]