MinT - Best Known (77−44, 77, s)-Nets in Base 64

Best Known (77−44, 77, s)-Nets in Base 64

(77−44, 77, 513)-Net over F₆₄ — Constructive and digital

Digital (33, 77, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 77, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(77−44, 77, 515)-Net over F₆₄ — Digital

Digital (33, 77, 515)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁷⁷, 515, F₆₄, 3, 44) (dual of [(515, 3), 1468, 45]-NRT-code), using
- strength reduction [i] based on linear OOA(64⁷⁷, 515, F₆₄, 3, 45) (dual of [(515, 3), 1468, 46]-NRT-code), using
  - constructionÂ X applied to AG(3;F,1490P) âŠ‚ AG(3;F,1495P) [i] based on
    1. linear OOA(64⁷³, 512, F₆₄, 3, 45) (dual of [(512, 3), 1463, 46]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1490P) [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]
    2. linear OOA(64⁶⁸, 512, F₆₄, 3, 40) (dual of [(512, 3), 1468, 41]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1495P) [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513 (see above)
    3. linear OOA(64⁴, 3, F₆₄, 3, 4) (dual of [(3, 3), 5, 5]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(64⁴, 64, F₆₄, 3, 4) (dual of [(64, 3), 188, 5]-NRT-code), using
        Reedâ€“Solomon NRT-code RS(3;188,64) [i]

(77−44, 77, 301388)-Net in Base 64 — Upper bound on s

There is no (33, 77, 301389)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 11 908914 255979 594370 665662 694522 226673 888013 524848 311606 183092 446404 708960 364447 787566 160055 510820 989595 802209 825766 667570 956217 888468 400080 > 64⁷⁷ [i]