MinT - Best Known (112, 172, s)-Nets in Base 4

Best Known (112, 172, s)-Nets in Base 4

(112, 172, 157)-Net over F₄ — Constructive and digital

Digital (112, 172, 157)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 40, 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 (72, 132, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 66, 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]

(112, 172, 196)-Net in Base 4 — Constructive

(112, 172, 196)-net in base 4, using

2 times m-reduction [i] based on (112, 174, 196)-net in base 4, using
- trace code for nets [i] based on (25, 87, 98)-net in base 16, using
  - 3 times m-reduction [i] based on (25, 90, 98)-net in base 16, using
    - base change [i] based on digital (7, 72, 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]

(112, 172, 410)-Net over F₄ — Digital

Digital (112, 172, 410)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(112, 172, 11338)-Net in Base 4 — Upper bound on s

There is no (112, 172, 11339)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 35 918467 226454 393364 030621 596341 527538 544404 023254 414211 467374 261824 708041 350944 804397 334156 807505 816720 > 4¹⁷² [i]