MinT - Best Known (78, 78+171, s)-Nets in Base 3

Best Known (78, 78+171, s)-Nets in Base 3

(78, 78+171, 53)-Net over F₃ — Constructive and digital

Digital (78, 249, 53)-net over F₃, using

net from sequence [i] based on digital (78, 52)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 52)-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]

(78, 78+171, 84)-Net over F₃ — Digital

Digital (78, 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]

(78, 78+171, 244)-Net over F₃ — Upper bound on s (digital)

There is no digital (78, 249, 245)-net over F₃, because

9 times m-reduction [i] would yield digital (78, 240, 245)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁰, 245, F₃, 162) (dual of [245, 5, 163]-code), but
  - 1 times code embedding in larger space [i] would yield linear OA(3²⁴¹, 246, F₃, 162) (dual of [246, 5, 163]-code), but
    - Griesmer bound [i]

(78, 78+171, 248)-Net in Base 3 — Upper bound on s

There is no (78, 249, 249)-net in base 3, because

5 times m-reduction [i] would yield (78, 244, 249)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3²⁴⁴, 249, S₃, 166), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 63561 249372 265538 529922 170457 092502 567086 808783 049445 812347 884970 295415 011324 387718 793675 882080 296992 140679 259510 242083 / 167 > 3²⁴⁴ [i]