MinT - Best Known (47−25, 47, s)-Nets in Base 32

Best Known (47−25, 47, s)-Nets in Base 32

(47−25, 47, 162)-Net over F₃₂ — Constructive and digital

Digital (22, 47, 162)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 15, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (7, 32, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(47−25, 47, 261)-Net in Base 32 — Constructive

(22, 47, 261)-net in base 32, using

1 times m-reduction [i] based on (22, 48, 261)-net in base 32, using
- base change [i] based on digital (4, 30, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(47−25, 47, 302)-Net over F₃₂ — Digital

Digital (22, 47, 302)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁴⁷, 302, F₃₂, 25) (dual of [302, 255, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁴⁷, 341, F₃₂, 25) (dual of [341, 294, 26]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 341Â | 32²âˆ’1, defining interval IÂ =Â [0,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(47−25, 47, 321)-Net in Base 32

(22, 47, 321)-net in base 32, using

t-expansion [i] based on (20, 47, 321)-net in base 32, using
- 1 times m-reduction [i] based on (20, 48, 321)-net in base 32, using
  - base change [i] based on digital (2, 30, 321)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
        sharp bound on the number of rational points on curves with genus 2 [i]

(47−25, 47, 100395)-Net in Base 32 — Upper bound on s

There is no (22, 47, 100396)-net in base 32, because

1 times m-reduction [i] would yield (22, 46, 100396)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1725 463851 979469 258403 486455 526142 310988 707788 657464 053266 306549 630584 > 32⁴⁶ [i]