MinT - Best Known (82, 82+57, s)-Nets in Base 4

Best Known (82, 82+57, s)-Nets in Base 4

(82, 82+57, 130)-Net over F₄ — Constructive and digital

Digital (82, 139, 130)-net over F₄, using

13 times m-reduction [i] based on digital (82, 152, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 76, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(82, 82+57, 197)-Net over F₄ — Digital

Digital (82, 139, 197)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+57, 3470)-Net in Base 4 — Upper bound on s

There is no (82, 139, 3471)-net in base 4, because

1 times m-reduction [i] would yield (82, 138, 3471)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 122325 298603 872020 716490 620566 333964 680831 005105 934254 649608 626687 456283 703139 862632 > 4¹³⁸ [i]