MinT - Best Known (142−83, 142, s)-Nets in Base 5

Best Known (142−83, 142, s)-Nets in Base 5

(142−83, 142, 82)-Net over F₅ — Constructive and digital

Digital (59, 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]

(142−83, 142, 112)-Net over F₅ — Digital

Digital (59, 142, 112)-net over F₅, using

t-expansion [i] based on digital (57, 142, 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]

(142−83, 142, 992)-Net in Base 5 — Upper bound on s

There is no (59, 142, 993)-net in base 5, because

1 times m-reduction [i] would yield (59, 141, 993)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 366 279576 932123 104111 780332 742298 060513 413768 244013 187653 867590 688448 199673 239155 213310 837613 349477 > 5¹⁴¹ [i]