MinT - Best Known (19, 19+19, s)-Nets in Base 25

Best Known (19, 19+19, s)-Nets in Base 25

(19, 19+19, 152)-Net over F₂₅ — Constructive and digital

Digital (19, 38, 152)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (0, 9, 26)-net over F₂₅, using
  - net from sequence [i] based on digital (0, 25)-sequence over F₂₅, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26, using
      - the rational function field F₂₅(x) [i]
    - Niederreiter sequence [i]
2. digital (10, 29, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
      - the Hermitian function field over F₂₅ [i]

(19, 19+19, 322)-Net over F₂₅ — Digital

Digital (19, 38, 322)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25³⁸, 322, F₂₅, 19) (dual of [322, 284, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(25³⁸, 624, F₂₅, 19) (dual of [624, 586, 20]-code), using
  - the primitive BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â {2,3,…,20}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(19, 19+19, 96517)-Net in Base 25 — Upper bound on s

There is no (19, 38, 96518)-net in base 25, because

1 times m-reduction [i] would yield (19, 37, 96518)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 5294 075564 122271 293924 148761 041776 293142 589126 722065 > 25³⁷ [i]