MinT - Best Known (110, 110+61, s)-Nets in Base 4

Best Known (110, 110+61, s)-Nets in Base 4

(110, 110+61, 151)-Net over F₄ — Constructive and digital

Digital (110, 171, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 37, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, 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]

(110, 110+61, 196)-Net in Base 4 — Constructive

(110, 171, 196)-net in base 4, using

4¹ times duplication [i] based on (109, 170, 196)-net in base 4, using
- trace code for nets [i] based on (24, 85, 98)-net in base 16, using
  - base change [i] based on digital (7, 68, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(110, 110+61, 377)-Net over F₄ — Digital

Digital (110, 171, 377)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(110, 110+61, 10335)-Net in Base 4 — Upper bound on s

There is no (110, 171, 10336)-net in base 4, because

1 times m-reduction [i] would yield (110, 170, 10336)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 245799 807566 929300 719398 948633 932852 952663 050790 415434 486871 559174 689672 159122 246188 078826 901778 926395 > 4¹⁷⁰ [i]