MinT - Best Known (87−43, 87, s)-Nets in Base 32

Best Known (87−43, 87, s)-Nets in Base 32

(87−43, 87, 240)-Net over F₃₂ — Constructive and digital

Digital (44, 87, 240)-net over F₃₂, using

1 times m-reduction [i] based on digital (44, 88, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 33, 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]
  2. digital (11, 55, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(87−43, 87, 513)-Net in Base 32 — Constructive

(44, 87, 513)-net in base 32, using

9 times m-reduction [i] based on (44, 96, 513)-net in base 32, using
- base change [i] based on digital (28, 80, 513)-net over F₆₄, using
  - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]

(87−43, 87, 729)-Net over F₃₂ — Digital

Digital (44, 87, 729)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸⁷, 729, F₃₂, 43) (dual of [729, 642, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁸⁷, 1041, F₃₂, 43) (dual of [1041, 954, 44]-code), using
  - constructionÂ X applied to C_e(42) âŠ‚ C_e(36) [i] based on
    1. linear OA(32⁸², 1024, F₃₂, 43) (dual of [1024, 942, 44]-code), using an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]
    2. linear OA(32⁷⁰, 1024, F₃₂, 37) (dual of [1024, 954, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    3. linear OA(32⁵, 17, F₃₂, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁵, 32, F₃₂, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
        Reedâ€“Solomon code RS(27,32) [i]

(87−43, 87, 408382)-Net in Base 32 — Upper bound on s

There is no (44, 87, 408383)-net in base 32, because

1 times m-reduction [i] would yield (44, 86, 408383)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2772 750967 094735 364069 741116 990293 531544 471155 697698 875781 875931 715365 472405 609544 749651 425755 802620 748905 136183 285205 512808 734702 > 32⁸⁶ [i]