MinT - Best Known (80−46, 80, s)-Nets in Base 64

Best Known (80−46, 80, s)-Nets in Base 64

(80−46, 80, 513)-Net over F₆₄ — Constructive and digital

Digital (34, 80, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 80, 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]

(80−46, 80, 516)-Net over F₆₄ — Digital

Digital (34, 80, 516)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁸⁰, 516, F₆₄, 3, 46) (dual of [(516, 3), 1468, 47]-NRT-code), using
- constructionÂ X applied to AG(3;F,1489P) âŠ‚ AG(3;F,1496P) [i] based on
  1. linear OOA(64⁷⁴, 512, F₆₄, 3, 46) (dual of [(512, 3), 1462, 47]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1489P) [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, 3, 39) (dual of [(512, 3), 1469, 40]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1496P) [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513 (see above)
  3. linear OOA(64⁶, 4, F₆₄, 3, 6) (dual of [(4, 3), 6, 7]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(64⁶, 64, F₆₄, 3, 6) (dual of [(64, 3), 186, 7]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(3;186,64) [i]

(80−46, 80, 286723)-Net in Base 64 — Upper bound on s

There is no (34, 80, 286724)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 3 121815 300690 864979 368048 910911 478538 278794 712908 033800 638179 382615 396958 983662 622896 220425 748259 885165 238152 656037 178896 559048 505368 602497 165312 > 64⁸⁰ [i]