MinT - Best Known (109, 133, s)-Nets in Base 5

Best Known (109, 133, s)-Nets in Base 5

(109, 133, 1323)-Net over F₅ — Constructive and digital

Digital (109, 133, 1323)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (6, 18, 21)-net over F₅, using
  - net from sequence [i] based on digital (6, 20)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 6 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (91, 115, 1302)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹¹⁵, 1302, F₅, 24, 24) (dual of [(1302, 24), 31133, 25]-NRT-code), using
    - OA 12-folding and stacking [i] based on linear OA(5¹¹⁵, 15624, F₅, 24) (dual of [15624, 15509, 25]-code), using
      - discarding factors / shortening the dual code based on linear OA(5¹¹⁵, 15625, F₅, 24) (dual of [15625, 15510, 25]-code), using
        an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(109, 133, 25969)-Net over F₅ — Digital

Digital (109, 133, 25969)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 133, large)-Net in Base 5 — Upper bound on s

There is no (109, 133, large)-net in base 5, because

22 times m-reduction [i] would yield (109, 111, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]