MinT - Best Known (89−19, 89, s)-Nets in Base 32

Best Known (89−19, 89, s)-Nets in Base 32

(89−19, 89, 116607)-Net over F₃₂ — Constructive and digital

Digital (70, 89, 116607)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 16, 99)-net over F₃₂, using
  - generalized (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, 4, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
    3. digital (0, 9, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
2. digital (54, 73, 116508)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁷³, 116508, F₃₂, 19, 19) (dual of [(116508, 19), 2213579, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(32⁷³, 1048573, F₃₂, 19) (dual of [1048573, 1048500, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁷³, 1048576, F₃₂, 19) (dual of [1048576, 1048503, 20]-code), using
        an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(89−19, 89, 932066)-Net in Base 32 — Constructive

(70, 89, 932066)-net in base 32, using

32¹ times duplication [i] based on (69, 88, 932066)-net in base 32, using
- base change [i] based on digital (36, 55, 932066)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256⁵⁵, 932066, F₂₅₆, 19, 19) (dual of [(932066, 19), 17709199, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(256⁵⁵, 8388595, F₂₅₆, 19) (dual of [8388595, 8388540, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(256⁵⁵, large, F₂₅₆, 19) (dual of [large, large−55, 20]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(89−19, 89, 6743718)-Net over F₃₂ — Digital

Digital (70, 89, 6743718)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−19, 89, large)-Net in Base 32 — Upper bound on s

There is no (70, 89, large)-net in base 32, because

17 times m-reduction [i] would yield (70, 72, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (89−19, 89, s)-Nets in Base 32

(89−19, 89, 116607)-Net over F32 — Constructive and digital

(89−19, 89, 932066)-Net in Base 32 — Constructive

(89−19, 89, 6743718)-Net over F32 — Digital

(89−19, 89, large)-Net in Base 32 — Upper bound on s

(89−19, 89, 116607)-Net over F₃₂ — Constructive and digital

(89−19, 89, 6743718)-Net over F₃₂ — Digital