MinT - Best Known (242−85, 242, s)-Nets in Base 4

Best Known (242−85, 242, s)-Nets in Base 4

(242−85, 242, 163)-Net over F₄ — Constructive and digital

Digital (157, 242, 163)-net over F₄, using

2 times m-reduction [i] based on digital (157, 244, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 58, 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 (99, 186, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 93, 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]

(242−85, 242, 240)-Net in Base 4 — Constructive

(157, 242, 240)-net in base 4, using

4² times duplication [i] based on (155, 240, 240)-net in base 4, using
- trace code for nets [i] based on (35, 120, 120)-net in base 16, using
  - base change [i] based on digital (11, 96, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(242−85, 242, 541)-Net over F₄ — Digital

Digital (157, 242, 541)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−85, 242, 15645)-Net in Base 4 — Upper bound on s

There is no (157, 242, 15646)-net in base 4, because

1 times m-reduction [i] would yield (157, 241, 15646)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 488111 604710 281943 487325 200658 018930 642317 270702 614532 984522 736954 805540 607404 626734 688752 695348 721486 854325 011363 150602 980845 041888 325592 961744 > 4²⁴¹ [i]