MinT - Best Known (20, 20+23, s)-Nets in Base 32

Best Known (20, 20+23, s)-Nets in Base 32

(20, 20+23, 142)-Net over F₃₂ — Constructive and digital

Digital (20, 43, 142)-net over F₃₂, using

1 times m-reduction [i] based on digital (20, 44, 142)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 13, 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, 31, 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]

(20, 20+23, 260)-Net in Base 32 — Constructive

(20, 43, 260)-net in base 32, using

32¹ times duplication [i] based on (19, 42, 260)-net in base 32, using
- base change [i] based on (12, 35, 260)-net in base 64, using
  - 1 times m-reduction [i] based on (12, 36, 260)-net in base 64, using
    - base change [i] based on digital (3, 27, 260)-net over F₂₅₆, using
      - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(20, 20+23, 278)-Net over F₃₂ — Digital

Digital (20, 43, 278)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁴³, 278, F₃₂, 23) (dual of [278, 235, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁴³, 341, F₃₂, 23) (dual of [341, 298, 24]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 341Â | 32²âˆ’1, defining interval IÂ =Â [0,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(20, 20+23, 321)-Net in Base 32

(20, 43, 321)-net in base 32, using

5 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]

(20, 20+23, 88422)-Net in Base 32 — Upper bound on s

There is no (20, 43, 88423)-net in base 32, because

1 times m-reduction [i] would yield (20, 42, 88423)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1645 621330 479252 984369 270725 339655 607679 357382 354681 277309 840624 > 32⁴² [i]