MinT - Best Known (231−147, 231, s)-Nets in Base 3

Best Known (231−147, 231, s)-Nets in Base 3

(231−147, 231, 59)-Net over F₃ — Constructive and digital

Digital (84, 231, 59)-net over F₃, using

net from sequence [i] based on digital (84, 58)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 58)-sequence over F₉, using
  - s-reduction 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]

(231−147, 231, 84)-Net over F₃ — Digital

Digital (84, 231, 84)-net over F₃, using

t-expansion [i] based on digital (71, 231, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(231−147, 231, 372)-Net over F₃ — Upper bound on s (digital)

There is no digital (84, 231, 373)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²³¹, 373, F₃, 147) (dual of [373, 142, 148]-code), but
- residual code [i] would yield linear OA(3⁸⁴, 225, F₃, 49) (dual of [225, 141, 50]-code), but
  - 1 times truncation [i] would yield linear OA(3⁸³, 224, F₃, 48) (dual of [224, 141, 49]-code), but
    - the Johnson bound shows that NÂ â‰¤ 18 650275 231410 181575 562142 123676 906294 083572 717553 183697 141026 684466 < 3¹⁴¹ [i]

(231−147, 231, 378)-Net in Base 3 — Upper bound on s

There is no (84, 231, 379)-net in base 3, because

1 times m-reduction [i] would yield (84, 230, 379)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 62 677923 630994 533209 040353 888117 946989 858436 678487 682943 203332 988099 571901 076964 091916 667642 347860 804749 954855 > 3²³⁰ [i]