MinT - Best Known (245−161, 245, s)-Nets in Base 3

Best Known (245−161, 245, s)-Nets in Base 3

(245−161, 245, 59)-Net over F₃ — Constructive and digital

Digital (84, 245, 59)-net over F₃, using

net from sequence [i] based on digital (84, 58)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 58)-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]

(245−161, 245, 84)-Net over F₃ — Digital

Digital (84, 245, 84)-net over F₃, using

t-expansion [i] based on digital (71, 245, 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]

(245−161, 245, 272)-Net over F₃ — Upper bound on s (digital)

There is no digital (84, 245, 273)-net over F₃, because

2 times m-reduction [i] would yield digital (84, 243, 273)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴³, 273, F₃, 159) (dual of [273, 30, 160]-code), but
  - residual code [i] would yield OA(3⁸⁴, 113, S₃, 53), but
    - the linear programming bound shows that MÂ â‰¥ 220 344254 536683 114597 471552 228690 132437 887421 343564 925242 / 18336 783241 265797 > 3⁸⁴ [i]

(245−161, 245, 362)-Net in Base 3 — Upper bound on s

There is no (84, 245, 363)-net in base 3, because

1 times m-reduction [i] would yield (84, 244, 363)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 293 343017 805224 799774 545220 869033 771984 272568 698109 585917 348600 415419 480314 441257 353260 783168 244505 477989 404641 795937 > 3²⁴⁴ [i]