MinT - Best Known (81, 81+144, s)-Nets in Base 3

Best Known (81, 81+144, s)-Nets in Base 3

(81, 81+144, 56)-Net over F₃ — Constructive and digital

Digital (81, 225, 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+144, 84)-Net over F₃ — Digital

Digital (81, 225, 84)-net over F₃, using

t-expansion [i] based on digital (71, 225, 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+144, 350)-Net over F₃ — Upper bound on s (digital)

There is no digital (81, 225, 351)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²²⁵, 351, F₃, 144) (dual of [351, 126, 145]-code), but
- residual code [i] would yield linear OA(3⁸¹, 206, F₃, 48) (dual of [206, 125, 49]-code), but
  - the Johnson bound shows that NÂ â‰¤ 390067 621362 525166 675859 888729 736992 372634 079787 564789 713745 < 3¹²⁵ [i]

(81, 81+144, 361)-Net in Base 3 — Upper bound on s

There is no (81, 225, 362)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 263108 319848 991262 269181 168728 573133 376674 643114 694332 841684 466799 694012 782441 867489 243016 996383 153291 374929 > 3²²⁵ [i]