MinT - Best Known (249−168, 249, s)-Nets in Base 3

Best Known (249−168, 249, s)-Nets in Base 3

(249−168, 249, 56)-Net over F₃ — Constructive and digital

Digital (81, 249, 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]

(249−168, 249, 84)-Net over F₃ — Digital

Digital (81, 249, 84)-net over F₃, using

t-expansion [i] based on digital (71, 249, 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]

(249−168, 249, 252)-Net over F₃ — Upper bound on s (digital)

There is no digital (81, 249, 253)-net over F₃, because

6 times m-reduction [i] would yield digital (81, 243, 253)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴³, 253, F₃, 162) (dual of [253, 10, 163]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3²⁴², 249, F₃, 162) (dual of [249, 7, 163]-code), but
      - residual code [i] would yield linear OA(3⁸⁰, 86, F₃, 54) (dual of [86, 6, 55]-code), but
        residual code [i] would yield linear OA(3²⁶, 31, F₃, 18) (dual of [31, 5, 19]-code), but
        residual code [i] would yield linear OA(3⁸, 12, F₃, 6) (dual of [12, 4, 7]-code), but
        residual code [i] would yield linear OA(3², 5, F₃, 2) (dual of [5, 3, 3]-code or 5-arc in PG(1,3)), but
        improvement by Maruta on the Griesmer bound [i]
    2. OA(3¹⁰, 253, S₃, 4), but
      - discarding factors would yield OA(3¹⁰, 172, S₃, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 59169 > 3¹⁰ [i]

(249−168, 249, 339)-Net in Base 3 — Upper bound on s

There is no (81, 249, 340)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 75246 297442 451417 993679 015379 453742 988594 698914 477775 370057 969915 192069 895711 069381 314483 979806 053283 668991 771415 681985 > 3²⁴⁹ [i]