MinT - Best Known (52−31, 52, s)-Nets in Base 81

Best Known (52−31, 52, s)-Nets in Base 81

(52−31, 52, 370)-Net over F₈₁ — Constructive and digital

Digital (21, 52, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 52, 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]

(52−31, 52, 372)-Net over F₈₁ — Digital

Digital (21, 52, 372)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(81⁵², 372, F₈₁, 3, 31) (dual of [(372, 3), 1064, 32]-NRT-code), using
- strength reduction [i] based on linear OOA(81⁵², 372, F₈₁, 3, 32) (dual of [(372, 3), 1064, 33]-NRT-code), using
  - constructionÂ X applied to AG(3;F,1074P) âŠ‚ AG(3;F,1079P) [i] based on
    1. linear OOA(81⁴⁸, 369, F₈₁, 3, 32) (dual of [(369, 3), 1059, 33]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1074P) [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(81⁴³, 369, F₈₁, 3, 27) (dual of [(369, 3), 1064, 28]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1079P) [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]

(52−31, 52, 247464)-Net in Base 81 — Upper bound on s

There is no (21, 52, 247465)-net in base 81, because

1 times m-reduction [i] would yield (21, 51, 247465)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 21 515237 828630 319815 775365 077975 738285 581765 583750 277573 254350 725059 681250 788243 349237 946464 798001 > 81⁵¹ [i]