MinT - Best Known (115, 115+61, s)-Nets in Base 4

Best Known (115, 115+61, s)-Nets in Base 4

(115, 115+61, 158)-Net over F₄ — Constructive and digital

Digital (115, 176, 158)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 42, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 134, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 67, 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]

(115, 115+61, 208)-Net in Base 4 — Constructive

(115, 176, 208)-net in base 4, using

trace code for nets [i] based on (27, 88, 104)-net in base 16, using
- 2 times m-reduction [i] based on (27, 90, 104)-net in base 16, using
  - base change [i] based on digital (9, 72, 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]

(115, 115+61, 428)-Net over F₄ — Digital

Digital (115, 176, 428)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(115, 115+61, 13027)-Net in Base 4 — Upper bound on s

There is no (115, 176, 13028)-net in base 4, because

1 times m-reduction [i] would yield (115, 175, 13028)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2294 938178 719607 514009 891232 367817 243992 782374 703316 753887 910722 390294 754473 144194 207243 287531 502222 506248 > 4¹⁷⁵ [i]