MinT - Best Known (50−21, 50, s)-Nets in Base 25

Best Known (50−21, 50, s)-Nets in Base 25

(50−21, 50, 251)-Net over F₂₅ — Constructive and digital

Digital (29, 50, 251)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (9, 19, 125)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25¹⁹, 125, F₂₅, 10, 10) (dual of [(125, 10), 1231, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(25¹⁹, 625, F₂₅, 10) (dual of [625, 606, 11]-code), using
      - an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
2. digital (10, 31, 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]

(50−21, 50, 1091)-Net over F₂₅ — Digital

Digital (29, 50, 1091)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(50−21, 50, 1335578)-Net in Base 25 — Upper bound on s

There is no (29, 50, 1335579)-net in base 25, because

1 times m-reduction [i] would yield (29, 49, 1335579)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 315 544917 867851 552671 693114 250829 087580 514301 682493 001549 416033 905105 > 25⁴⁹ [i]