MinT - Best Known (46−19, 46, s)-Nets in Base 25

Best Known (46−19, 46, s)-Nets in Base 25

(46−19, 46, 252)-Net over F₂₅ — Constructive and digital

Digital (27, 46, 252)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (8, 17, 156)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25¹⁷, 156, F₂₅, 9, 9) (dual of [(156, 9), 1387, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(25¹⁷, 625, F₂₅, 9) (dual of [625, 608, 10]-code), using
      - an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [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]

(46−19, 46, 1185)-Net over F₂₅ — Digital

Digital (27, 46, 1185)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(46−19, 46, 1687481)-Net in Base 25 — Upper bound on s

There is no (27, 46, 1687482)-net in base 25, because

1 times m-reduction [i] would yield (27, 45, 1687482)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 807 793989 070994 915643 615450 587174 750590 067831 718319 468388 131825 > 25⁴⁵ [i]