MinT - Best Known (135−77, 135, s)-Nets in Base 5

Best Known (135−77, 135, s)-Nets in Base 5

(135−77, 135, 82)-Net over F₅ — Constructive and digital

Digital (58, 135, 82)-net over F₅, using

t-expansion [i] based on digital (48, 135, 82)-net over F₅, using
- net from sequence [i] based on digital (48, 81)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 48 and N(F) ≥ 82, using
    - an explicitly constructive algebraic function field [i]

(135−77, 135, 112)-Net over F₅ — Digital

Digital (58, 135, 112)-net over F₅, using

t-expansion [i] based on digital (57, 135, 112)-net over F₅, using
- net from sequence [i] based on digital (57, 111)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 57 and N(F) ≥ 112, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(135−77, 135, 1067)-Net in Base 5 — Upper bound on s

There is no (58, 135, 1068)-net in base 5, because

1 times m-reduction [i] would yield (58, 134, 1068)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 4664 049115 119871 473335 158708 286273 369786 212246 466862 283381 390637 350636 939604 308303 119564 839745 > 5¹³⁴ [i]