MinT - Best Known (234−151, 234, s)-Nets in Base 3

Best Known (234−151, 234, s)-Nets in Base 3

(234−151, 234, 58)-Net over F₃ — Constructive and digital

Digital (83, 234, 58)-net over F₃, using

net from sequence [i] based on digital (83, 57)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 57)-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]

(234−151, 234, 84)-Net over F₃ — Digital

Digital (83, 234, 84)-net over F₃, using

t-expansion [i] based on digital (71, 234, 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]

(234−151, 234, 352)-Net over F₃ — Upper bound on s (digital)

There is no digital (83, 234, 353)-net over F₃, because

1 times m-reduction [i] would yield digital (83, 233, 353)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³³, 353, F₃, 150) (dual of [353, 120, 151]-code), but
  - residual code [i] would yield linear OA(3⁸³, 202, F₃, 50) (dual of [202, 119, 51]-code), but
    - the Johnson bound shows that NÂ â‰¤ 550 972990 432084 950775 442566 323188 109097 170341 593919 868682 < 3¹¹⁹ [i]

(234−151, 234, 366)-Net in Base 3 — Upper bound on s

There is no (83, 234, 367)-net in base 3, because

1 times m-reduction [i] would yield (83, 233, 367)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1558 956485 145448 812240 807506 573498 970399 143262 012191 673752 589242 249383 450853 264441 065985 275680 467562 764830 254891 > 3²³³ [i]