MinT - Best Known (242−63, 242, s)-Nets in Base 4

Best Known (242−63, 242, s)-Nets in Base 4

(242−63, 242, 540)-Net over F₄ — Constructive and digital

Digital (179, 242, 540)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 32, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (147, 210, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 70, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(242−63, 242, 648)-Net in Base 4 — Constructive

(179, 242, 648)-net in base 4, using

1 times m-reduction [i] based on (179, 243, 648)-net in base 4, using
- trace code for nets [i] based on (17, 81, 216)-net in base 64, using
  - 3 times m-reduction [i] based on (17, 84, 216)-net in base 64, using
    - base change [i] based on digital (5, 72, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(242−63, 242, 1817)-Net over F₄ — Digital

Digital (179, 242, 1817)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−63, 242, 198329)-Net in Base 4 — Upper bound on s

There is no (179, 242, 198330)-net in base 4, because

1 times m-reduction [i] would yield (179, 241, 198330)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 488129 415237 991068 683627 844393 869345 562830 988736 183634 296495 623274 490901 373689 916636 986872 931190 666486 379174 672521 871101 094486 229585 478746 513232 > 4²⁴¹ [i]