MinT - Best Known (249−175, 249, s)-Nets in Base 3

Best Known (249−175, 249, s)-Nets in Base 3

(249−175, 249, 49)-Net over F₃ — Constructive and digital

Digital (74, 249, 49)-net over F₃, using

net from sequence [i] based on digital (74, 48)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 48)-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]

(249−175, 249, 84)-Net over F₃ — Digital

Digital (74, 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]

(249−175, 249, 232)-Net over F₃ — Upper bound on s (digital)

There is no digital (74, 249, 233)-net over F₃, because

25 times m-reduction [i] would yield digital (74, 224, 233)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²²⁴, 233, F₃, 150) (dual of [233, 9, 151]-code), but
  - residual code [i] would yield OA(3⁷⁴, 82, S₃, 50), but
    - the linear programming bound shows that MÂ â‰¥ 1133 201025 509985 412089 971278 248938 614941 / 5015 > 3⁷⁴ [i]

(249−175, 249, 235)-Net in Base 3 — Upper bound on s

There is no (74, 249, 236)-net in base 3, because

18 times m-reduction [i] would yield (74, 231, 236)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3²³¹, 236, S₃, 157), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 13289 078263 368535 010350 719491 824534 628404 827374 597126 975388 693711 018284 879396 957772 210874 852435 860862 184279 107707 / 79 > 3²³¹ [i]