MinT - Best Known (81−46, 81, s)-Nets in Base 64

Best Known (81−46, 81, s)-Nets in Base 64

(81−46, 81, 513)-Net over F₆₄ — Constructive and digital

Digital (35, 81, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 81, 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]

(81−46, 81, 519)-Net over F₆₄ — Digital

Digital (35, 81, 519)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁸¹, 519, F₆₄, 2, 46) (dual of [(519, 2), 957, 47]-NRT-code), using
- constructionÂ X applied to AG(2;F,977P) âŠ‚ AG(2;F,985P) [i] based on
  1. linear OOA(64⁷⁴, 512, F₆₄, 2, 46) (dual of [(512, 2), 950, 47]-NRT-code), using algebraic-geometric NRT-code AG(2;F,977P) [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(64⁶⁶, 512, F₆₄, 2, 38) (dual of [(512, 2), 958, 39]-NRT-code), using algebraic-geometric NRT-code AG(2;F,985P) [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513 (see above)
  3. linear OOA(64⁷, 7, F₆₄, 2, 7) (dual of [(7, 2), 7, 8]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(64⁷, 64, F₆₄, 2, 7) (dual of [(64, 2), 121, 8]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(2;121,64) [i]

(81−46, 81, 343554)-Net in Base 64 — Upper bound on s

There is no (35, 81, 343555)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 199 794224 125048 918026 699424 311380 980877 263248 109887 789325 558697 997074 814969 901702 790931 016259 396955 086654 621562 544000 132709 561894 868201 926592 242176 > 64⁸¹ [i]