MinT - Best Known (161−58, 161, s)-Nets in Base 4

Best Known (161−58, 161, s)-Nets in Base 4

(161−58, 161, 145)-Net over F₄ — Constructive and digital

Digital (103, 161, 145)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 33, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (70, 128, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 64, 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]

(161−58, 161, 152)-Net in Base 4 — Constructive

(103, 161, 152)-net in base 4, using

4¹ times duplication [i] based on (102, 160, 152)-net in base 4, using
- t-expansion [i] based on (101, 160, 152)-net in base 4, using
  - trace code for nets [i] based on (21, 80, 76)-net in base 16, using
    - base change [i] based on digital (5, 64, 76)-net over F₃₂, using
      - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(161−58, 161, 346)-Net over F₄ — Digital

Digital (103, 161, 346)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(161−58, 161, 8536)-Net in Base 4 — Upper bound on s

There is no (103, 161, 8537)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 563683 649795 292898 925993 312590 851047 393600 825359 092216 681551 697768 370367 110174 383285 206021 633408 > 4¹⁶¹ [i]