MinT - Best Known (22, 22+34, s)-Nets in Base 81

Best Known (22, 22+34, s)-Nets in Base 81

(22, 22+34, 370)-Net over F₈₁ — Constructive and digital

Digital (22, 56, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 56, 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]

(22, 22+34, 372)-Net over F₈₁ — Digital

Digital (22, 56, 372)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(81⁵⁶, 372, F₈₁, 4, 34) (dual of [(372, 4), 1432, 35]-NRT-code), using
- constructionÂ X applied to AG(4;F,1441P) âŠ‚ AG(4;F,1448P) [i] based on
  1. linear OOA(81⁵⁰, 369, F₈₁, 4, 34) (dual of [(369, 4), 1426, 35]-NRT-code), using algebraic-geometric NRT-code AG(4;F,1441P) [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, 4, 27) (dual of [(369, 4), 1433, 28]-NRT-code), using algebraic-geometric NRT-code AG(4;F,1448P) [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370 (see above)
  3. linear OOA(81⁶, 3, F₈₁, 4, 6) (dual of [(3, 4), 6, 7]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(81⁶, 81, F₈₁, 4, 6) (dual of [(81, 4), 318, 7]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(4;318,81) [i]

(22, 22+34, 173623)-Net in Base 81 — Upper bound on s

There is no (22, 56, 173624)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 75019 876993 830464 329557 545662 287069 236368 900793 683870 118435 087306 809281 802993 329946 034348 580859 047634 528641 > 81⁵⁶ [i]