MinT - Best Known (46−13, 46, s)-Nets in Base 81

Best Known (46−13, 46, s)-Nets in Base 81

(46−13, 46, 88737)-Net over F₈₁ — Constructive and digital

Digital (33, 46, 88737)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (3, 9, 164)-net over F₈₁, using
  - (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, 6, 82)-net over F₈₁, using
      - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
2. digital (24, 37, 88573)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81³⁷, 88573, F₈₁, 13, 13) (dual of [(88573, 13), 1151412, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(81³⁷, 531439, F₈₁, 13) (dual of [531439, 531402, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(81³⁷, 531441, F₈₁, 13) (dual of [531441, 531404, 14]-code), using
        an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(46−13, 46, 1368146)-Net over F₈₁ — Digital

Digital (33, 46, 1368146)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(46−13, 46, large)-Net in Base 81 — Upper bound on s

There is no (33, 46, large)-net in base 81, because

11 times m-reduction [i] would yield (33, 35, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]