MinT - Best Known (99, 99+49, s)-Nets in Base 4

Best Known (99, 99+49, s)-Nets in Base 4

(99, 99+49, 195)-Net over F₄ — Constructive and digital

Digital (99, 148, 195)-net over F₄, using

4¹ times duplication [i] based on digital (98, 147, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 49, 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]

(99, 99+49, 208)-Net in Base 4 — Constructive

(99, 148, 208)-net in base 4, using

2 times m-reduction [i] based on (99, 150, 208)-net in base 4, using
- trace code for nets [i] based on (24, 75, 104)-net in base 16, using
  - base change [i] based on digital (9, 60, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(99, 99+49, 437)-Net over F₄ — Digital

Digital (99, 148, 437)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(99, 99+49, 15897)-Net in Base 4 — Upper bound on s

There is no (99, 148, 15898)-net in base 4, because

1 times m-reduction [i] would yield (99, 147, 15898)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 31856 277067 248357 240047 534586 615774 145874 007422 482761 046303 101194 871053 956045 211227 571161 > 4¹⁴⁷ [i]