MinT - Best Known (79−21, 79, s)-Nets in Base 81

Best Known (79−21, 79, s)-Nets in Base 81

(79−21, 79, 53390)-Net over F₈₁ — Constructive and digital

Digital (58, 79, 53390)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (8, 18, 246)-net over F₈₁, using
  - 1 times m-reduction [i] based on digital (8, 19, 246)-net over F₈₁, using
    - generalized (u,Â u+v)-construction [i] based on
      1. digital (0, 3, 82)-net over F₈₁, using
        net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
        Niederreiter sequence [i]
      2. digital (0, 5, 82)-net over F₈₁, using
        net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
      3. digital (0, 11, 82)-net over F₈₁, using
        net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
2. digital (40, 61, 53144)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁶¹, 53144, F₈₁, 21, 21) (dual of [(53144, 21), 1115963, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(81⁶¹, 531441, F₈₁, 21) (dual of [531441, 531380, 22]-code), using
      - an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(79−21, 79, 3587016)-Net over F₈₁ — Digital

Digital (58, 79, 3587016)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(79−21, 79, large)-Net in Base 81 — Upper bound on s

There is no (58, 79, large)-net in base 81, because

19 times m-reduction [i] would yield (58, 60, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]