MinT - Best Known (142−79, 142, s)-Nets in Base 5

Best Known (142−79, 142, s)-Nets in Base 5

(142−79, 142, 82)-Net over F₅ — Constructive and digital

Digital (63, 142, 82)-net over F₅, using

t-expansion [i] based on digital (48, 142, 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]

(142−79, 142, 120)-Net over F₅ — Digital

Digital (63, 142, 120)-net over F₅, using

t-expansion [i] based on digital (61, 142, 120)-net over F₅, using
- net from sequence [i] based on digital (61, 119)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 61 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(142−79, 142, 1266)-Net in Base 5 — Upper bound on s

There is no (63, 142, 1267)-net in base 5, because

1 times m-reduction [i] would yield (63, 141, 1267)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 359 864551 849135 401863 376452 468154 270309 515910 272796 976760 888175 762096 947289 870776 604463 289996 347125 > 5¹⁴¹ [i]