MinT - Best Known (123, 123+68, s)-Nets in Base 4

Best Known (123, 123+68, s)-Nets in Base 4

(123, 123+68, 152)-Net over F₄ — Constructive and digital

Digital (123, 191, 152)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 43, 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 (80, 148, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 74, 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]

(123, 123+68, 196)-Net in Base 4 — Constructive

(123, 191, 196)-net in base 4, using

4¹ times duplication [i] based on (122, 190, 196)-net in base 4, using
- t-expansion [i] based on (121, 190, 196)-net in base 4, using
  - trace code for nets [i] based on (26, 95, 98)-net in base 16, using
    - base change [i] based on digital (7, 76, 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]

(123, 123+68, 418)-Net over F₄ — Digital

Digital (123, 191, 418)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+68, 10849)-Net in Base 4 — Upper bound on s

There is no (123, 191, 10850)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 871032 942451 492217 945257 191408 991673 142349 874137 373744 361198 540880 337349 381873 467296 185517 575895 575500 279942 231608 > 4¹⁹¹ [i]