MinT - Best Known (74−44, 74, s)-Nets in Base 32

Best Known (74−44, 74, s)-Nets in Base 32

(74−44, 74, 142)-Net over F₃₂ — Constructive and digital

Digital (30, 74, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 23, 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, 51, 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]

(74−44, 74, 258)-Net in Base 32 — Constructive

(30, 74, 258)-net in base 32, using

3 times m-reduction [i] based on (30, 77, 258)-net in base 32, using
- base change [i] based on (8, 55, 258)-net in base 128, using
  - 1 times m-reduction [i] based on (8, 56, 258)-net in base 128, using
    - base change [i] based on digital (1, 49, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(74−44, 74, 273)-Net over F₃₂ — Digital

Digital (30, 74, 273)-net over F₃₂, using

net from sequence [i] based on digital (30, 272)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273, using
  - a curve from the manYPoints database [i]

(74−44, 74, 289)-Net in Base 32

(30, 74, 289)-net in base 32, using

3 times m-reduction [i] based on (30, 77, 289)-net in base 32, using
- base change [i] based on (8, 55, 289)-net in base 128, using
  - 1 times m-reduction [i] based on (8, 56, 289)-net in base 128, using
    - base change [i] based on digital (1, 49, 289)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
        sharp bound on the number of rational points on curves with genus 1 [i]

(74−44, 74, 33738)-Net in Base 32 — Upper bound on s

There is no (30, 74, 33739)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2405 874088 960684 442212 295646 392301 052341 942704 841153 568202 693371 310420 974012 586076 441630 263511 648612 612777 056776 > 32⁷⁴ [i]