MinT - Best Known (47, 47+43, s)-Nets in Base 32

Best Known (47, 47+43, s)-Nets in Base 32

(47, 47+43, 240)-Net over F₃₂ — Constructive and digital

Digital (47, 90, 240)-net over F₃₂, using

7 times m-reduction [i] based on digital (47, 97, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 36, 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, 61, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(47, 47+43, 513)-Net in Base 32 — Constructive

(47, 90, 513)-net in base 32, using

t-expansion [i] based on (46, 90, 513)-net in base 32, using
- 18 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
  - base change [i] based on digital (28, 90, 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]

(47, 47+43, 944)-Net over F₃₂ — Digital

Digital (47, 90, 944)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁹⁰, 944, F₃₂, 43) (dual of [944, 854, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁹⁰, 1042, F₃₂, 43) (dual of [1042, 952, 44]-code), using
  - constructionÂ X applied to C([0,21]) âŠ‚ C([0,18]) [i] based on
    1. linear OA(32⁸⁵, 1025, F₃₂, 43) (dual of [1025, 940, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
    2. linear OA(32⁷³, 1025, F₃₂, 37) (dual of [1025, 952, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [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]

(47, 47+43, 670027)-Net in Base 32 — Upper bound on s

There is no (47, 90, 670028)-net in base 32, because

1 times m-reduction [i] would yield (47, 89, 670028)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 90 855031 913291 640173 234780 920312 012819 379656 869035 501807 142054 121184 472023 286515 822932 659833 339287 364103 304036 865455 897287 881441 196504 > 32⁸⁹ [i]