MinT - Best Known (138−67, 138, s)-Nets in Base 5

Best Known (138−67, 138, s)-Nets in Base 5

(138−67, 138, 94)-Net over F₅ — Constructive and digital

Digital (71, 138, 94)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (7, 40, 22)-net over F₅, using
  - net from sequence [i] based on digital (7, 21)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 7 and N(F) ≥ 22, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 98, 72)-net over F₅, using
  - net from sequence [i] based on digital (31, 71)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
      - an explicitly constructive algebraic function field [i]

(138−67, 138, 150)-Net over F₅ — Digital

Digital (71, 138, 150)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−67, 138, 2600)-Net in Base 5 — Upper bound on s

There is no (71, 138, 2601)-net in base 5, because

1 times m-reduction [i] would yield (71, 137, 2601)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 577418 435522 450952 429597 033989 984304 045409 392987 977368 426584 718543 208347 627885 807322 093108 274725 > 5¹³⁷ [i]