MinT - Best Known (84−48, 84, s)-Nets in Base 64

Best Known (84−48, 84, s)-Nets in Base 64

(84−48, 84, 513)-Net over F₆₄ — Constructive and digital

Digital (36, 84, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 84, 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]

(84−48, 84, 520)-Net over F₆₄ — Digital

Digital (36, 84, 520)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁸⁴, 520, F₆₄, 2, 48) (dual of [(520, 2), 956, 49]-NRT-code), using
- constructionÂ X applied to AG(2;F,975P) âŠ‚ AG(2;F,984P) [i] based on
  1. linear OOA(64⁷⁶, 512, F₆₄, 2, 48) (dual of [(512, 2), 948, 49]-NRT-code), using algebraic-geometric NRT-code AG(2;F,975P) [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(64⁶⁷, 512, F₆₄, 2, 39) (dual of [(512, 2), 957, 40]-NRT-code), using algebraic-geometric NRT-code AG(2;F,984P) [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513 (see above)
  3. linear OOA(64⁸, 8, F₆₄, 2, 8) (dual of [(8, 2), 8, 9]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(64⁸, 64, F₆₄, 2, 8) (dual of [(64, 2), 120, 9]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(2;120,64) [i]

(84−48, 84, 326314)-Net in Base 64 — Upper bound on s

There is no (36, 84, 326315)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 52 376432 392609 232385 422780 002760 930598 112065 071802 202213 167649 096545 050930 144555 014721 653456 707156 299179 716886 450721 478464 743560 827825 685180 022232 605666 > 64⁸⁴ [i]