MinT - Best Known (82, 82+155, s)-Nets in Base 3

Best Known (82, 82+155, s)-Nets in Base 3

(82, 82+155, 57)-Net over F₃ — Constructive and digital

Digital (82, 237, 57)-net over F₃, using

net from sequence [i] based on digital (82, 56)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 56)-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]

(82, 82+155, 84)-Net over F₃ — Digital

Digital (82, 237, 84)-net over F₃, using

t-expansion [i] based on digital (71, 237, 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]

(82, 82+155, 275)-Net over F₃ — Upper bound on s (digital)

There is no digital (82, 237, 276)-net over F₃, because

2 times m-reduction [i] would yield digital (82, 235, 276)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³⁵, 276, F₃, 153) (dual of [276, 41, 154]-code), but
  - residual code [i] would yield OA(3⁸², 122, S₃, 51), but
    - the linear programming bound shows that MÂ â‰¥ 2 857259 335023 875460 783166 319642 601001 954259 886029 570784 083010 116237 359893 / 2097 368232 043591 449722 787486 700000 > 3⁸² [i]

(82, 82+155, 356)-Net in Base 3 — Upper bound on s

There is no (82, 237, 357)-net in base 3, because

1 times m-reduction [i] would yield (82, 236, 357)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 45995 564824 302654 381015 873461 797985 065360 558966 581717 062619 503103 968648 774921 882970 542074 382309 847488 489581 973691 > 3²³⁶ [i]