MinT - Best Known (105−19, 105, s)-Nets in Base 5

Best Known (105−19, 105, s)-Nets in Base 5

(105−19, 105, 1756)-Net over F₅ — Constructive and digital

Digital (86, 105, 1756)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (5, 14, 20)-net over F₅, using
  - net from sequence [i] based on digital (5, 19)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 5 and N(F) ≥ 20, using
      - an explicitly constructive algebraic function field [i]
2. digital (72, 91, 1736)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁹¹, 1736, F₅, 19, 19) (dual of [(1736, 19), 32893, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(5⁹¹, 15625, F₅, 19) (dual of [15625, 15534, 20]-code), using
      - an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(105−19, 105, 22572)-Net over F₅ — Digital

Digital (86, 105, 22572)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(105−19, 105, large)-Net in Base 5 — Upper bound on s

There is no (86, 105, large)-net in base 5, because

17 times m-reduction [i] would yield (86, 88, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]