MinT - Best Known (238−159, 238, s)-Nets in Base 3

Best Known (238−159, 238, s)-Nets in Base 3

(238−159, 238, 54)-Net over F₃ — Constructive and digital

Digital (79, 238, 54)-net over F₃, using

net from sequence [i] based on digital (79, 53)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 53)-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]

(238−159, 238, 84)-Net over F₃ — Digital

Digital (79, 238, 84)-net over F₃, using

t-expansion [i] based on digital (71, 238, 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]

(238−159, 238, 249)-Net over F₃ — Upper bound on s (digital)

There is no digital (79, 238, 250)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²³⁸, 250, F₃, 159) (dual of [250, 12, 160]-code), but
- residual code [i] would yield OA(3⁷⁹, 90, S₃, 53), but
  - 1 times truncation [i] would yield OA(3⁷⁸, 89, S₃, 52), but
    - the linear programming bound shows that MÂ â‰¥ 298 630446 198644 459587 118144 486403 305957 001183 / 16 135903 > 3⁷⁸ [i]

(238−159, 238, 335)-Net in Base 3 — Upper bound on s

There is no (79, 238, 336)-net in base 3, because

1 times m-reduction [i] would yield (79, 237, 336)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 138011 006303 335228 766492 843975 611808 447575 001049 078787 382329 397035 033721 239601 777383 613655 646842 834011 308000 232385 > 3²³⁷ [i]