MinT - Best Known (48−20, 48, s)-Nets in Base 25

Best Known (48−20, 48, s)-Nets in Base 25

(48−20, 48, 208)-Net over F₂₅ — Constructive and digital

Digital (28, 48, 208)-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 (9, 29, 104)-net over F₂₅, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by GarcÃa and Quoos [i]

(48−20, 48, 1133)-Net over F₂₅ — Digital

Digital (28, 48, 1133)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(48−20, 48, 967998)-Net in Base 25 — Upper bound on s

There is no (28, 48, 967999)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 12 621777 772811 824741 678437 902473 116553 322246 171429 221306 492221 465361 > 25⁴⁸ [i]