MinT - Best Known (83, 83+158, s)-Nets in Base 3

Best Known (83, 83+158, s)-Nets in Base 3

(83, 83+158, 58)-Net over F₃ — Constructive and digital

Digital (83, 241, 58)-net over F₃, using

net from sequence [i] based on digital (83, 57)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 57)-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]

(83, 83+158, 84)-Net over F₃ — Digital

Digital (83, 241, 84)-net over F₃, using

t-expansion [i] based on digital (71, 241, 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]

(83, 83+158, 274)-Net over F₃ — Upper bound on s (digital)

There is no digital (83, 241, 275)-net over F₃, because

2 times m-reduction [i] would yield digital (83, 239, 275)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³⁹, 275, F₃, 156) (dual of [275, 36, 157]-code), but
  - residual code [i] would yield OA(3⁸³, 118, S₃, 52), but
    - the linear programming bound shows that MÂ â‰¥ 1996 494123 652591 132382 208614 471528 262798 278296 164094 413549 / 441117 447444 334375 > 3⁸³ [i]

(83, 83+158, 358)-Net in Base 3 — Upper bound on s

There is no (83, 241, 359)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10 967804 006896 481590 545033 137531 816440 427668 769940 550373 009677 991318 058569 745630 519048 859682 025049 634601 007459 641603 > 3²⁴¹ [i]