MinT - Best Known (75−20, 75, s)-Nets in Base 81

Best Known (75−20, 75, s)-Nets in Base 81

(75−20, 75, 53344)-Net over F₈₁ — Constructive and digital

Digital (55, 75, 53344)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (7, 17, 200)-net over F₈₁, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 6, 100)-net over F₈₁, using
      - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
        a function field by SÃ©mirat [i]
    2. digital (1, 11, 100)-net over F₈₁, using
      - net from sequence [i] based on digital (1, 99)-sequence over F₈₁ (see above)
2. digital (38, 58, 53144)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁵⁸, 53144, F₈₁, 20, 20) (dual of [(53144, 20), 1062822, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(81⁵⁸, 531440, F₈₁, 20) (dual of [531440, 531382, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(81⁵⁸, 531441, F₈₁, 20) (dual of [531441, 531383, 21]-code), using
        an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(75−20, 75, 3385480)-Net over F₈₁ — Digital

Digital (55, 75, 3385480)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−20, 75, large)-Net in Base 81 — Upper bound on s

There is no (55, 75, large)-net in base 81, because

18 times m-reduction [i] would yield (55, 57, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]