MinT - Best Known (63, 63+81, s)-Nets in Base 5

Best Known (63, 63+81, s)-Nets in Base 5

(63, 63+81, 82)-Net over F₅ — Constructive and digital

Digital (63, 144, 82)-net over F₅, using

t-expansion [i] based on digital (48, 144, 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]

(63, 63+81, 120)-Net over F₅ — Digital

Digital (63, 144, 120)-net over F₅, using

t-expansion [i] based on digital (61, 144, 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]

(63, 63+81, 1213)-Net in Base 5 — Upper bound on s

There is no (63, 144, 1214)-net in base 5, because

1 times m-reduction [i] would yield (63, 143, 1214)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 8977 164707 810151 587993 941896 059117 742467 160696 077837 961440 525895 708120 839858 578748 332137 240140 787585 > 5¹⁴³ [i]