MinT - Best Known (90, 111, s)-Nets in Base 5

Best Known (90, 111, s)-Nets in Base 5

(90, 111, 1580)-Net over F₅ — Constructive and digital

Digital (90, 111, 1580)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 14, 18)-net over F₅, using
  - net from sequence [i] based on digital (4, 17)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 4 and N(F) ≥ 18, using
      - an explicitly constructive algebraic function field [i]
2. digital (76, 97, 1562)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁹⁷, 1562, F₅, 21, 21) (dual of [(1562, 21), 32705, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(5⁹⁷, 15621, F₅, 21) (dual of [15621, 15524, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁹⁷, 15625, F₅, 21) (dual of [15625, 15528, 22]-code), using
        an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(90, 111, 15733)-Net over F₅ — Digital

Digital (90, 111, 15733)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(90, 111, large)-Net in Base 5 — Upper bound on s

There is no (90, 111, large)-net in base 5, because

19 times m-reduction [i] would yield (90, 92, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]