MinT - Best Known (139−59, 139, s)-Nets in Base 4

Best Known (139−59, 139, s)-Nets in Base 4

(139−59, 139, 130)-Net over F₄ — Constructive and digital

Digital (80, 139, 130)-net over F₄, using

9 times m-reduction [i] based on digital (80, 148, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 74, 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]

(139−59, 139, 176)-Net over F₄ — Digital

Digital (80, 139, 176)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−59, 139, 2827)-Net in Base 4 — Upper bound on s

There is no (80, 139, 2828)-net in base 4, because

1 times m-reduction [i] would yield (80, 138, 2828)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 122323 346960 964739 703145 718383 777781 256229 312229 189272 375170 633860 230392 480265 715648 > 4¹³⁸ [i]