MinT - Best Known (82, 82+153, s)-Nets in Base 3

Best Known (82, 82+153, s)-Nets in Base 3

(82, 82+153, 57)-Net over F₃ — Constructive and digital

Digital (82, 235, 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+153, 84)-Net over F₃ — Digital

Digital (82, 235, 84)-net over F₃, using

t-expansion [i] based on digital (71, 235, 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+153, 275)-Net over F₃ — Upper bound on s (digital)

There is no digital (82, 235, 276)-net over F₃, because

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+153, 358)-Net in Base 3 — Upper bound on s

There is no (82, 235, 359)-net in base 3, because

1 times m-reduction [i] would yield (82, 234, 359)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5061 277742 246203 857170 651858 951630 003330 696673 041767 424135 948406 844233 005632 768921 245603 232107 991543 731261 732969 > 3²³⁴ [i]