MinT - Best Known (122, 122+63, s)-Nets in Base 4

Best Known (122, 122+63, s)-Nets in Base 4

(122, 122+63, 163)-Net over F₄ — Constructive and digital

Digital (122, 185, 163)-net over F₄, using

4¹ times duplication [i] based on digital (121, 184, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 46, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (75, 138, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 69, 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+63, 208)-Net in Base 4 — Constructive

(122, 185, 208)-net in base 4, using

3 times m-reduction [i] based on (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+63, 475)-Net over F₄ — Digital

Digital (122, 185, 475)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(122, 122+63, 15478)-Net in Base 4 — Upper bound on s

There is no (122, 185, 15479)-net in base 4, because

1 times m-reduction [i] would yield (122, 184, 15479)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 602 239625 447680 280296 790274 807764 144927 286297 569173 832479 348475 236053 880583 128335 937958 938490 819220 918478 450672 > 4¹⁸⁴ [i]