MinT - Best Known (91, 91+156, s)-Nets in Base 3

Best Known (91, 91+156, s)-Nets in Base 3

(91, 91+156, 64)-Net over F₃ — Constructive and digital

Digital (91, 247, 64)-net over F₃, using

t-expansion [i] based on digital (89, 247, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] 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]

(91, 91+156, 96)-Net over F₃ — Digital

Digital (91, 247, 96)-net over F₃, using

t-expansion [i] based on digital (89, 247, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(91, 91+156, 409)-Net over F₃ — Upper bound on s (digital)

There is no digital (91, 247, 410)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁷, 410, F₃, 156) (dual of [410, 163, 157]-code), but
- residual code [i] would yield linear OA(3⁹¹, 253, F₃, 52) (dual of [253, 162, 53]-code), but
  - the Johnson bound shows that NÂ â‰¤ 179351 448037 571926 665032 504132 411079 734421 634873 676547 341483 836618 815717 643951 < 3¹⁶² [i]

(91, 91+156, 411)-Net in Base 3 — Upper bound on s

There is no (91, 247, 412)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7884 729262 580275 682158 190576 421643 540832 993834 645085 401045 979227 361327 740869 809380 031796 085775 563395 416077 062132 262121 > 3²⁴⁷ [i]