MinT - Best Known (54−32, 54, s)-Nets in Base 81

Best Known (54−32, 54, s)-Nets in Base 81

(54−32, 54, 370)-Net over F₈₁ — Constructive and digital

Digital (22, 54, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 54, 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]

(54−32, 54, 373)-Net over F₈₁ — Digital

Digital (22, 54, 373)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(81⁵⁴, 373, F₈₁, 3, 32) (dual of [(373, 3), 1065, 33]-NRT-code), using
- constructionÂ X applied to AG(3;F,1074P) âŠ‚ AG(3;F,1081P) [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(81⁴¹, 369, F₈₁, 3, 25) (dual of [(369, 3), 1066, 26]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1081P) [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370 (see above)
  3. linear OOA(81⁶, 4, F₈₁, 3, 6) (dual of [(4, 3), 6, 7]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(81⁶, 81, F₈₁, 3, 6) (dual of [(81, 3), 237, 7]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(3;237,81) [i]

(54−32, 54, 234731)-Net in Base 81 — Upper bound on s

There is no (22, 54, 234732)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 11 434177 242531 627550 863191 619238 768606 452201 479290 513617 052148 099478 436558 536353 617804 017051 842531 240961 > 81⁵⁴ [i]