MinT - Best Known (17, 17+19, s)-Nets in Base 32

Best Known (17, 17+19, s)-Nets in Base 32

(17, 17+19, 142)-Net over F₃₂ — Constructive and digital

Digital (17, 36, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 10, 44)-net over F₃₂, using
  - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
      - a function field by SÃ©mirat [i]
2. digital (7, 26, 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]

(17, 17+19, 260)-Net in Base 32 — Constructive

(17, 36, 260)-net in base 32, using

base change [i] based on (11, 30, 260)-net in base 64, using
- 2 times m-reduction [i] based on (11, 32, 260)-net in base 64, using
  - base change [i] based on digital (3, 24, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(17, 17+19, 283)-Net over F₃₂ — Digital

Digital (17, 36, 283)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32³⁶, 283, F₃₂, 19) (dual of [283, 247, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(32³⁶, 341, F₃₂, 19) (dual of [341, 305, 20]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 341Â | 32²âˆ’1, defining interval IÂ =Â [0,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(17, 17+19, 321)-Net in Base 32

(17, 36, 321)-net in base 32, using

4 times m-reduction [i] based on (17, 40, 321)-net in base 32, using
- base change [i] based on digital (2, 25, 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]

(17, 17+19, 95440)-Net in Base 32 — Upper bound on s

There is no (17, 36, 95441)-net in base 32, because

1 times m-reduction [i] would yield (17, 35, 95441)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 47894 179721 769390 194639 918077 530604 121573 608665 148240 > 32³⁵ [i]