MinT - Best Known (178−59, 178, s)-Nets in Base 4

Best Known (178−59, 178, s)-Nets in Base 4

(178−59, 178, 195)-Net over F₄ — Constructive and digital

Digital (119, 178, 195)-net over F₄, using

4¹ times duplication [i] based on digital (118, 177, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 59, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(178−59, 178, 240)-Net in Base 4 — Constructive

(119, 178, 240)-net in base 4, using

2 times m-reduction [i] based on (119, 180, 240)-net in base 4, using
- trace code for nets [i] based on (29, 90, 120)-net in base 16, using
  - base change [i] based on digital (11, 72, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(178−59, 178, 508)-Net over F₄ — Digital

Digital (119, 178, 508)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(178−59, 178, 18368)-Net in Base 4 — Upper bound on s

There is no (119, 178, 18369)-net in base 4, because

1 times m-reduction [i] would yield (119, 177, 18369)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 36699 982303 343591 346474 003776 124044 397620 835215 200214 957940 145070 811352 115047 997596 193685 571227 646544 783104 > 4¹⁷⁷ [i]