MinT - Best Known (226−144, 226, s)-Nets in Base 3

Best Known (226−144, 226, s)-Nets in Base 3

(226−144, 226, 57)-Net over F₃ — Constructive and digital

Digital (82, 226, 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]

(226−144, 226, 84)-Net over F₃ — Digital

Digital (82, 226, 84)-net over F₃, using

t-expansion [i] based on digital (71, 226, 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]

(226−144, 226, 359)-Net over F₃ — Upper bound on s (digital)

There is no digital (82, 226, 360)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²²⁶, 360, F₃, 144) (dual of [360, 134, 145]-code), but
- residual code [i] would yield linear OA(3⁸², 215, F₃, 48) (dual of [215, 133, 49]-code), but
  - the Johnson bound shows that NÂ â‰¤ 2633 391490 584373 459265 026071 777086 821369 449862 527703 301746 510338 < 3¹³³ [i]

(226−144, 226, 367)-Net in Base 3 — Upper bound on s

There is no (82, 226, 368)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 722258 470111 389377 083320 945803 983585 316795 536341 258539 309486 170793 898948 545802 134675 490495 251720 122039 408641 > 3²²⁶ [i]