MinT - Best Known (232−152, 232, s)-Nets in Base 3

Best Known (232−152, 232, s)-Nets in Base 3

(232−152, 232, 55)-Net over F₃ — Constructive and digital

Digital (80, 232, 55)-net over F₃, using

net from sequence [i] based on digital (80, 54)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 54)-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]

(232−152, 232, 84)-Net over F₃ — Digital

Digital (80, 232, 84)-net over F₃, using

t-expansion [i] based on digital (71, 232, 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]

(232−152, 232, 267)-Net over F₃ — Upper bound on s (digital)

There is no digital (80, 232, 268)-net over F₃, because

2 times m-reduction [i] would yield digital (80, 230, 268)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³⁰, 268, F₃, 150) (dual of [268, 38, 151]-code), but
  - residual code [i] would yield OA(3⁸⁰, 117, S₃, 50), but
    - the linear programming bound shows that MÂ â‰¥ 43 988919 336557 262180 077487 525672 393199 697139 692324 704712 213007 / 293033 908358 491110 749536 > 3⁸⁰ [i]

(232−152, 232, 346)-Net in Base 3 — Upper bound on s

There is no (80, 232, 347)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 573 086167 618911 237614 058158 394614 502750 260862 615311 233842 295114 343684 960340 509041 908023 957724 727498 049304 676425 > 3²³² [i]