MinT - Best Known (248−169, 248, s)-Nets in Base 3

Best Known (248−169, 248, s)-Nets in Base 3

(248−169, 248, 54)-Net over F₃ — Constructive and digital

Digital (79, 248, 54)-net over F₃, using

net from sequence [i] based on digital (79, 53)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 53)-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]

(248−169, 248, 84)-Net over F₃ — Digital

Digital (79, 248, 84)-net over F₃, using

t-expansion [i] based on digital (71, 248, 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]

(248−169, 248, 245)-Net over F₃ — Upper bound on s (digital)

There is no digital (79, 248, 246)-net over F₃, because

7 times m-reduction [i] would yield digital (79, 241, 246)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴¹, 246, F₃, 162) (dual of [246, 5, 163]-code), but
  - Griesmer bound [i]

(248−169, 248, 251)-Net in Base 3 — Upper bound on s

There is no (79, 248, 252)-net in base 3, because

1 times m-reduction [i] would yield (79, 247, 252)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3²⁴⁷, 252, S₃, 168), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 1 716153 733051 169540 307898 602341 497569 311343 837142 335036 933392 894197 976205 305758 468407 429248 816168 018787 798340 006776 536241 / 169 > 3²⁴⁷ [i]