MinT - Best Known (24, 24+36, s)-Nets in Base 81

Best Known (24, 24+36, s)-Nets in Base 81

(24, 24+36, 370)-Net over F₈₁ — Constructive and digital

Digital (24, 60, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 60, 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]

(24, 24+36, 374)-Net over F₈₁ — Digital

Digital (24, 60, 374)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(81⁶⁰, 374, F₈₁, 3, 36) (dual of [(374, 3), 1062, 37]-NRT-code), using
- strength reduction [i] based on linear OOA(81⁶⁰, 374, F₈₁, 3, 37) (dual of [(374, 3), 1062, 38]-NRT-code), using
  - constructionÂ X applied to AG(3;F,1069P) âŠ‚ AG(3;F,1077P) [i] based on
    1. linear OOA(81⁵³, 369, F₈₁, 3, 37) (dual of [(369, 3), 1054, 38]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1069P) [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(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 (see above)
    3. linear OOA(81⁷, 5, F₈₁, 3, 7) (dual of [(5, 3), 8, 8]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(81⁷, 81, F₈₁, 3, 7) (dual of [(81, 3), 236, 8]-NRT-code), using
        Reedâ€“Solomon NRT-code RS(3;236,81) [i]

(24, 24+36, 217089)-Net in Base 81 — Upper bound on s

There is no (24, 60, 217090)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 3 229309 387736 045260 473310 081870 764075 597577 070481 906391 169180 089620 252019 421423 478135 966549 253490 502132 491368 193601 > 81⁶⁰ [i]