MinT - Best Known (85, 85+18, s)-Nets in Base 32

Best Known (85, 85+18, s)-Nets in Base 32

(85, 85+18, 932323)-Net over F₃₂ — Constructive and digital

Digital (85, 103, 932323)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (8, 17, 256)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32¹⁷, 256, F₃₂, 9, 9) (dual of [(256, 9), 2287, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(32¹⁷, 1025, F₃₂, 9) (dual of [1025, 1008, 10]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
2. digital (68, 86, 932067)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁸⁶, 932067, F₃₂, 18, 18) (dual of [(932067, 18), 16777120, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(32⁸⁶, large, F₃₂, 18) (dual of [large, large−86, 19]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(85, 85+18, 932389)-Net in Base 32 — Constructive

(85, 103, 932389)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (11, 20, 322)-net in base 32, using
  - (u,Â u+v)-construction [i] based on
    1. (1, 5, 65)-net in base 32, using
      - 1 times m-reduction [i] based on (1, 6, 65)-net in base 32, using
        base change [i] based on digital (0, 5, 65)-net over F₆₄, using
        net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
    2. (6, 15, 257)-net in base 32, using
      - 1 times m-reduction [i] based on (6, 16, 257)-net in base 32, using
        base change [i] based on digital (0, 10, 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. (65, 83, 932067)-net in base 32, using
  - net defined by OOA [i] based on OOA(32⁸³, 932067, S₃₂, 18, 18), using
    - OA 9-folding and stacking [i] based on OA(32⁸³, large, S₃₂, 18), using
      - discarding parts of the base [i] based on linear OA(64⁶⁹, large, F₆₄, 18) (dual of [large, large−69, 19]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(85, 85+18, large)-Net over F₃₂ — Digital

Digital (85, 103, large)-net over F₃₂, using

t-expansion [i] based on digital (83, 103, large)-net over F₃₂, using
- 2 times m-reduction [i] based on digital (83, 105, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(85, 85+18, large)-Net in Base 32 — Upper bound on s

There is no (85, 103, large)-net in base 32, because

16 times m-reduction [i] would yield (85, 87, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (85, 85+18, s)-Nets in Base 32

(85, 85+18, 932323)-Net over F32 — Constructive and digital

(85, 85+18, 932389)-Net in Base 32 — Constructive

(85, 85+18, large)-Net over F32 — Digital

(85, 85+18, large)-Net in Base 32 — Upper bound on s

(85, 85+18, 932323)-Net over F₃₂ — Constructive and digital

(85, 85+18, large)-Net over F₃₂ — Digital