MinT - Best Known (87, 87+162, s)-Nets in Base 3

Best Known (87, 87+162, s)-Nets in Base 3

(87, 87+162, 62)-Net over F₃ — Constructive and digital

Digital (87, 249, 62)-net over F₃, using

net from sequence [i] based on digital (87, 61)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 61)-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]

(87, 87+162, 84)-Net over F₃ — Digital

Digital (87, 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]

(87, 87+162, 290)-Net over F₃ — Upper bound on s (digital)

There is no digital (87, 249, 291)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁹, 291, F₃, 162) (dual of [291, 42, 163]-code), but
- residual code [i] would yield OA(3⁸⁷, 128, S₃, 54), but
  - the linear programming bound shows that MÂ â‰¥ 15 952594 360793 907733 534555 089919 080786 131778 937341 232963 389196 637855 190777 / 46 108516 281082 645734 227925 699658 > 3⁸⁷ [i]

(87, 87+162, 378)-Net in Base 3 — Upper bound on s

There is no (87, 249, 379)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 70406 457775 798408 490332 528167 915848 634917 041348 068870 293410 511799 550644 063551 686709 630634 751802 599459 550021 675450 832599 > 3²⁴⁹ [i]