MinT - Best Known (135, 135+77, s)-Nets in Base 4

Best Known (135, 135+77, s)-Nets in Base 4

(135, 135+77, 151)-Net over F₄ — Constructive and digital

Digital (135, 212, 151)-net over F₄, using

4¹ times duplication [i] based on digital (134, 211, 151)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 45, 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 (89, 166, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 83, 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]

(135, 135+77, 196)-Net in Base 4 — Constructive

(135, 212, 196)-net in base 4, using

4² times duplication [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]

(135, 135+77, 428)-Net over F₄ — Digital

Digital (135, 212, 428)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+77, 11002)-Net in Base 4 — Upper bound on s

There is no (135, 212, 11003)-net in base 4, because

1 times m-reduction [i] would yield (135, 211, 11003)-net in base 4, but
- 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]