MinT - Best Known (20, 49, s)-Nets in Base 81

Best Known (20, 49, s)-Nets in Base 81

(20, 49, 370)-Net over F₈₁ — Constructive and digital

Digital (20, 49, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 49, 370)-net over F₈₁, using
- net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
    - a function field by GarcÃa and Quoos [i]

(20, 49, 372)-Net over F₈₁ — Digital

Digital (20, 49, 372)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(81⁴⁹, 372, F₈₁, 3, 29) (dual of [(372, 3), 1067, 30]-NRT-code), using
- constructionÂ X applied to AG(3;F,1077P) âŠ‚ AG(3;F,1082P) [i] based on
  1. linear OOA(81⁴⁵, 369, F₈₁, 3, 29) (dual of [(369, 3), 1062, 30]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1077P) [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
    - a function field by GarcÃa and Quoos [i]
  2. linear OOA(81⁴⁰, 369, F₈₁, 3, 24) (dual of [(369, 3), 1067, 25]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1082P) [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370 (see above)
  3. linear OOA(81⁴, 3, F₈₁, 3, 4) (dual of [(3, 3), 5, 5]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(81⁴, 81, F₈₁, 3, 4) (dual of [(81, 3), 239, 5]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(3;239,81) [i]

(20, 49, 264079)-Net in Base 81 — Upper bound on s

There is no (20, 49, 264080)-net in base 81, because

1 times m-reduction [i] would yield (20, 48, 264080)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 40 485760 848939 569619 373079 133536 352504 857407 486649 995913 833912 684047 096228 599554 980174 425601 > 81⁴⁸ [i]