MinT - Best Known (30, 68, s)-Nets in Base 32

Best Known (30, 68, s)-Nets in Base 32

(30, 68, 162)-Net over F₃₂ — Constructive and digital

Digital (30, 68, 162)-net over F₃₂, using

2 times m-reduction [i] based on digital (30, 70, 162)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 23, 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, 47, 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]

(30, 68, 277)-Net over F₃₂ — Digital

Digital (30, 68, 277)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(30, 68, 288)-Net in Base 32 — Constructive

(30, 68, 288)-net in base 32, using

4 times m-reduction [i] based on (30, 72, 288)-net in base 32, using
- base change [i] based on (18, 60, 288)-net in base 64, using
  - 3 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
    - base change [i] based on digital (9, 54, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(30, 68, 321)-Net in Base 32

(30, 68, 321)-net in base 32, using

t-expansion [i] based on (29, 68, 321)-net in base 32, using
- 4 times m-reduction [i] based on (29, 72, 321)-net in base 32, using
  - base change [i] based on digital (2, 45, 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]

(30, 68, 62321)-Net in Base 32 — Upper bound on s

There is no (30, 68, 62322)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 239940 339179 076216 906739 423069 626496 819317 682108 438710 646560 893904 958811 773003 018132 327667 012800 810928 > 32⁶⁸ [i]