MinT - Best Known (38, 73, s)-Nets in Base 32

Best Known (38, 73, s)-Nets in Base 32

(38, 73, 224)-Net over F₃₂ — Constructive and digital

Digital (38, 73, 224)-net over F₃₂, using

1 times m-reduction [i] based on digital (38, 74, 224)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 27, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
  2. digital (11, 47, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(38, 73, 290)-Net in Base 32 — Constructive

(38, 73, 290)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (0, 17, 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. (21, 56, 257)-net in base 32, using
  - base change [i] based on digital (0, 35, 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]

(38, 73, 801)-Net over F₃₂ — Digital

Digital (38, 73, 801)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁷³, 801, F₃₂, 35) (dual of [801, 728, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁷³, 1044, F₃₂, 35) (dual of [1044, 971, 36]-code), using
  - constructionÂ X applied to C_e(34) âŠ‚ C_e(26) [i] based on
    1. linear OA(32⁶⁶, 1024, F₃₂, 35) (dual of [1024, 958, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    2. linear OA(32⁵³, 1024, F₃₂, 27) (dual of [1024, 971, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
    3. linear OA(32⁷, 20, F₃₂, 7) (dual of [20, 13, 8]-code or 20-arc in PG(6,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁷, 32, F₃₂, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
        Reedâ€“Solomon code RS(25,32) [i]

(38, 73, 548693)-Net in Base 32 — Upper bound on s

There is no (38, 73, 548694)-net in base 32, because

1 times m-reduction [i] would yield (38, 72, 548694)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 348599 113257 030210 842340 985711 313689 319108 414953 994669 275096 339553 448365 740126 654919 346109 944827 536096 349987 > 32⁷² [i]