MinT - Best Known (79, 235, s)-Nets in Base 3

Best Known (79, 235, s)-Nets in Base 3

(79, 235, 54)-Net over F₃ — Constructive and digital

Digital (79, 235, 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, 235, 84)-Net over F₃ — Digital

Digital (79, 235, 84)-net over F₃, using

t-expansion [i] based on digital (71, 235, 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, 235, 250)-Net over F₃ — Upper bound on s (digital)

There is no digital (79, 235, 251)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²³⁵, 251, F₃, 156) (dual of [251, 16, 157]-code), but
- residual code [i] would yield OA(3⁷⁹, 94, S₃, 52), but
  - 1 times truncation [i] would yield OA(3⁷⁸, 93, S₃, 51), but
    - the linear programming bound shows that MÂ â‰¥ 2117 985825 855951 548682 979121 686352 501183 391624 / 117 447583 > 3⁷⁸ [i]

(79, 235, 336)-Net in Base 3 — Upper bound on s

There is no (79, 235, 337)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 13693 870649 905262 168769 435064 832673 972146 080806 097082 929828 607506 050649 017217 596421 376386 650946 543865 908650 373785 > 3²³⁵ [i]