MinT - Best Known (211−76, 211, s)-Nets in Base 4

Best Known (211−76, 211, s)-Nets in Base 4

(211−76, 211, 152)-Net over F₄ — Constructive and digital

Digital (135, 211, 152)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 47, 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 (88, 164, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 82, 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]

(211−76, 211, 196)-Net in Base 4 — Constructive

(135, 211, 196)-net in base 4, using

4¹ times duplication [i] based on (134, 210, 196)-net in base 4, using
- t-expansion [i] based on (133, 210, 196)-net in base 4, using
  - trace code for nets [i] based on (28, 105, 98)-net in base 16, using
    - base change [i] based on digital (7, 84, 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]

(211−76, 211, 439)-Net over F₄ — Digital

Digital (135, 211, 439)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(211−76, 211, 11002)-Net in Base 4 — Upper bound on s

There is no (135, 211, 11003)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 867741 178919 992875 356980 102014 969001 305973 217680 248084 547503 954047 080139 938417 652709 795681 195123 770460 518217 563563 683790 794630 > 4²¹¹ [i]