MinT - Best Known (122, 122+66, s)-Nets in Base 4

Best Known (122, 122+66, s)-Nets in Base 4

(122, 122+66, 157)-Net over F₄ — Constructive and digital

Digital (122, 188, 157)-net over F₄, using

1 times m-reduction [i] based on digital (122, 189, 157)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 43, 27)-net over F₄, using
    - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
        an explicitly constructive algebraic function field [i]
  2. digital (79, 146, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 73, 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]

(122, 122+66, 208)-Net in Base 4 — Constructive

(122, 188, 208)-net in base 4, using

trace code for nets [i] based on (28, 94, 104)-net in base 16, using
- 1 times m-reduction [i] based on (28, 95, 104)-net in base 16, using
  - base change [i] based on digital (9, 76, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(122, 122+66, 433)-Net over F₄ — Digital

Digital (122, 188, 433)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(122, 122+66, 11780)-Net in Base 4 — Upper bound on s

There is no (122, 188, 11781)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 154207 799788 611474 033008 387437 251398 778497 608242 420712 039612 096349 308398 706009 966668 579528 806740 618740 323495 324416 > 4¹⁸⁸ [i]