MinT - Best Known (90, 90+159, s)-Nets in Base 3

Best Known (90, 90+159, s)-Nets in Base 3

(90, 90+159, 64)-Net over F₃ — Constructive and digital

Digital (90, 249, 64)-net over F₃, using

t-expansion [i] based on digital (89, 249, 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+159, 96)-Net over F₃ — Digital

Digital (90, 249, 96)-net over F₃, using

t-expansion [i] based on digital (89, 249, 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+159, 393)-Net over F₃ — Upper bound on s (digital)

There is no digital (90, 249, 394)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁹, 394, F₃, 159) (dual of [394, 145, 160]-code), but
- residual code [i] would yield linear OA(3⁹⁰, 234, F₃, 53) (dual of [234, 144, 54]-code), but
  - 1 times truncation [i] would yield linear OA(3⁸⁹, 233, F₃, 52) (dual of [233, 144, 53]-code), but
    - the Johnson bound shows that NÂ â‰¤ 495 041572 388827 382337 917134 411719 483784 605757 670798 500294 630099 278193 < 3¹⁴⁴ [i]

(90, 90+159, 401)-Net in Base 3 — Upper bound on s

There is no (90, 249, 402)-net in base 3, because

1 times m-reduction [i] would yield (90, 248, 402)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21278 010540 459547 341254 316503 532253 901617 936478 591230 580214 131464 532363 882431 791350 799220 921257 707663 489148 452514 475737 > 3²⁴⁸ [i]