MinT - Best Known (23, 49, s)-Nets in Base 32

Best Known (23, 49, s)-Nets in Base 32

(23, 49, 162)-Net over F₃₂ — Constructive and digital

Digital (23, 49, 162)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 16, 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, 33, 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]

(23, 49, 288)-Net in Base 32 — Constructive

(23, 49, 288)-net in base 32, using

base change [i] based on digital (9, 35, 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]

(23, 49, 314)-Net over F₃₂ — Digital

Digital (23, 49, 314)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁴⁹, 314, F₃₂, 26) (dual of [314, 265, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁴⁹, 341, F₃₂, 26) (dual of [341, 292, 27]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 341Â | 32²âˆ’1, defining interval IÂ =Â [0,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]

(23, 49, 321)-Net in Base 32

(23, 49, 321)-net in base 32, using

7 times m-reduction [i] based on (23, 56, 321)-net in base 32, using
- base change [i] based on digital (2, 35, 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]

(23, 49, 86152)-Net in Base 32 — Upper bound on s

There is no (23, 49, 86153)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 56 540236 625416 081934 264055 873481 381603 075550 846078 902797 242794 494267 487872 > 32⁴⁹ [i]