MinT - Best Known (159−56, 159, s)-Nets in Base 4

Best Known (159−56, 159, s)-Nets in Base 4

(159−56, 159, 151)-Net over F₄ — Constructive and digital

Digital (103, 159, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 35, 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 (68, 124, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 62, 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]

(159−56, 159, 196)-Net in Base 4 — Constructive

(103, 159, 196)-net in base 4, using

1 times m-reduction [i] based on (103, 160, 196)-net in base 4, using
- trace code for nets [i] based on (23, 80, 98)-net in base 16, using
  - base change [i] based on digital (7, 64, 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]

(159−56, 159, 370)-Net over F₄ — Digital

Digital (103, 159, 370)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(159−56, 159, 9856)-Net in Base 4 — Upper bound on s

There is no (103, 159, 9857)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 534833 572739 915221 935272 089870 840230 688123 706666 048499 081799 332223 957835 920307 719012 776685 410208 > 4¹⁵⁹ [i]