MinT - Best Known (108, 108+58, s)-Nets in Base 4

Best Known (108, 108+58, s)-Nets in Base 4

(108, 108+58, 152)-Net over F₄ — Constructive and digital

Digital (108, 166, 152)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 38, 22)-net over F₄, using
  - net from sequence [i] based on digital (9, 21)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, using
      - an explicitly constructive algebraic function field [i]
2. digital (70, 128, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 64, 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]

(108, 108+58, 196)-Net in Base 4 — Constructive

(108, 166, 196)-net in base 4, using

2 times m-reduction [i] based on (108, 168, 196)-net in base 4, using
- trace code for nets [i] based on (24, 84, 98)-net in base 16, using
  - 1 times m-reduction [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]

(108, 108+58, 395)-Net over F₄ — Digital

Digital (108, 166, 395)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(108, 108+58, 10847)-Net in Base 4 — Upper bound on s

There is no (108, 166, 10848)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8760 771468 275615 859190 761207 412641 858110 480914 077670 100287 529121 312214 156187 729410 686933 280090 837003 > 4¹⁶⁶ [i]