MinT - Best Known (81, 81+158, s)-Nets in Base 3

Best Known (81, 81+158, s)-Nets in Base 3

(81, 81+158, 56)-Net over F₃ — Constructive and digital

Digital (81, 239, 56)-net over F₃, using

net from sequence [i] based on digital (81, 55)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 55)-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]

(81, 81+158, 84)-Net over F₃ — Digital

Digital (81, 239, 84)-net over F₃, using

t-expansion [i] based on digital (71, 239, 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]

(81, 81+158, 259)-Net over F₃ — Upper bound on s (digital)

There is no digital (81, 239, 260)-net over F₃, because

2 times m-reduction [i] would yield digital (81, 237, 260)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³⁷, 260, F₃, 156) (dual of [260, 23, 157]-code), but
  - residual code [i] would yield OA(3⁸¹, 103, S₃, 52), but
    - the linear programming bound shows that MÂ â‰¥ 1648 634351 113215 149715 784496 019469 628928 501125 836083 / 3 036794 593547 > 3⁸¹ [i]

(81, 81+158, 346)-Net in Base 3 — Upper bound on s

There is no (81, 239, 347)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 153829 933840 444879 287755 393489 317228 879428 419350 718979 194356 882770 231745 906552 575459 837856 294328 582995 132851 069395 > 3²³⁹ [i]