MinT - Best Known (90, 90+157, s)-Nets in Base 3

Best Known (90, 90+157, s)-Nets in Base 3

(90, 90+157, 64)-Net over F₃ — Constructive and digital

Digital (90, 247, 64)-net over F₃, using

t-expansion [i] based on digital (89, 247, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(90, 90+157, 96)-Net over F₃ — Digital

Digital (90, 247, 96)-net over F₃, using

t-expansion [i] based on digital (89, 247, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(90, 90+157, 399)-Net over F₃ — Upper bound on s (digital)

There is no digital (90, 247, 400)-net over F₃, because

1 times m-reduction [i] would yield digital (90, 246, 400)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁶, 400, F₃, 156) (dual of [400, 154, 157]-code), but
  - residual code [i] would yield linear OA(3⁹⁰, 243, F₃, 52) (dual of [243, 153, 53]-code), but
    - the Johnson bound shows that NÂ â‰¤ 9 342763 491628 254791 739936 787780 423565 169992 498824 859636 256969 626845 605352 < 3¹⁵³ [i]

(90, 90+157, 404)-Net in Base 3 — Upper bound on s

There is no (90, 247, 405)-net in base 3, because

1 times m-reduction [i] would yield (90, 246, 405)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2508 762071 099530 452248 643034 320251 459574 993583 985751 699689 386073 561054 147982 134283 069439 218521 055199 626170 668208 597761 > 3²⁴⁶ [i]