MinT - Best Known (242−169, 242, s)-Nets in Base 3

Best Known (242−169, 242, s)-Nets in Base 3

(242−169, 242, 48)-Net over F₃ — Constructive and digital

Digital (73, 242, 48)-net over F₃, using

t-expansion [i] based on digital (45, 242, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(242−169, 242, 84)-Net over F₃ — Digital

Digital (73, 242, 84)-net over F₃, using

t-expansion [i] based on digital (71, 242, 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]

(242−169, 242, 229)-Net over F₃ — Upper bound on s (digital)

There is no digital (73, 242, 230)-net over F₃, because

18 times m-reduction [i] would yield digital (73, 224, 230)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²²⁴, 230, F₃, 151) (dual of [230, 6, 152]-code), but
  - the linear programming bound with additional constraints on the weights [i]

(242−169, 242, 232)-Net in Base 3 — Upper bound on s

There is no (73, 242, 233)-net in base 3, because

14 times m-reduction [i] would yield (73, 228, 233)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3²²⁸, 233, S₃, 155), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 164 062694 609488 086547 539746 812648 575659 318856 476507 740436 897453 222447 961474 036515 706307 096943 652603 236842 951947 / 26 > 3²²⁸ [i]