MinT - Best Known (66, 81, s)-Nets in Base 32

Best Known (66, 81, s)-Nets in Base 32

(66, 81, 1198437)-Net over F₃₂ — Constructive and digital

Digital (66, 81, 1198437)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 10, 66)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
        the rational function field F₃₂(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 7, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
2. digital (56, 71, 1198371)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁷¹, 1198371, F₃₂, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(32⁷¹, 8388598, F₃₂, 15) (dual of [8388598, 8388527, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁷¹, large, F₃₂, 15) (dual of [large, large−71, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(66, 81, 1198628)-Net in Base 32 — Constructive

(66, 81, 1198628)-net in base 32, using

32¹ times duplication [i] based on (65, 80, 1198628)-net in base 32, using
- base change [i] based on digital (35, 50, 1198628)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 7, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
    2. digital (28, 43, 1198371)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256⁴³, 1198371, F₂₅₆, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
        OOA 7-folding and stacking with additional row [i] based on linear OA(256⁴³, 8388598, F₂₅₆, 15) (dual of [8388598, 8388555, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴³, large, F₂₅₆, 15) (dual of [large, large−43, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(66, 81, large)-Net over F₃₂ — Digital

Digital (66, 81, large)-net over F₃₂, using

t-expansion [i] based on digital (64, 81, large)-net over F₃₂, using
- Gilbertâ€“VarÅ¡amov bound [i]

(66, 81, large)-Net in Base 32 — Upper bound on s

There is no (66, 81, large)-net in base 32, because

13 times m-reduction [i] would yield (66, 68, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]