MinT - Best Known (79, 79+165, s)-Nets in Base 3

Best Known (79, 79+165, s)-Nets in Base 3

(79, 79+165, 54)-Net over F₃ — Constructive and digital

Digital (79, 244, 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]

(79, 79+165, 84)-Net over F₃ — Digital

Digital (79, 244, 84)-net over F₃, using

t-expansion [i] based on digital (71, 244, 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]

(79, 79+165, 245)-Net over F₃ — Upper bound on s (digital)

There is no digital (79, 244, 246)-net over F₃, because

3 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]

(79, 79+165, 331)-Net in Base 3 — Upper bound on s

There is no (79, 244, 332)-net in base 3, because

1 times m-reduction [i] would yield (79, 243, 332)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 106 265926 687759 487720 190518 862011 182649 509129 367880 898447 038002 931077 281156 294982 810214 446366 765760 329808 210814 407145 > 3²⁴³ [i]