MinT - Best Known (83, 83+153, s)-Nets in Base 3

Best Known (83, 83+153, s)-Nets in Base 3

(83, 83+153, 58)-Net over F₃ — Constructive and digital

Digital (83, 236, 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]

(83, 83+153, 84)-Net over F₃ — Digital

Digital (83, 236, 84)-net over F₃, using

t-expansion [i] based on digital (71, 236, 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]

(83, 83+153, 287)-Net over F₃ — Upper bound on s (digital)

There is no digital (83, 236, 288)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²³⁶, 288, F₃, 153) (dual of [288, 52, 154]-code), but
- residual code [i] would yield OA(3⁸³, 134, S₃, 51), but
  - the linear programming bound shows that MÂ â‰¥ 625760 414962 993510 058854 039475 945995 304619 276439 915262 016003 058873 437945 073915 147163 250093 / 135 349071 693745 981183 738057 225151 833299 159482 850560 > 3⁸³ [i]

(83, 83+153, 364)-Net in Base 3 — Upper bound on s

There is no (83, 236, 365)-net in base 3, because

1 times m-reduction [i] would yield (83, 235, 365)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 14692 780942 913007 242776 209922 727567 269339 707932 172735 246017 282966 641334 615032 615246 976982 552253 375540 617361 378161 > 3²³⁵ [i]