MinT - Best Known (23−13, 23, s)-Nets in Base 32

Best Known (23−13, 23, s)-Nets in Base 32

(23−13, 23, 108)-Net over F₃₂ — Constructive and digital

Digital (10, 23, 108)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 7, 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 (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]

(23−13, 23, 138)-Net over F₃₂ — Digital

Digital (10, 23, 138)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32²³, 138, F₃₂, 13) (dual of [138, 115, 14]-code), using
- 37 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 9 times 0, 1, 26 times 0) [i] based on linear OA(32²⁰, 98, F₃₂, 13) (dual of [98, 78, 14]-code), using
  - extended algebraic-geometric code AG_e(F,84P) [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
    - a function field by SÃ©mirat [i]

(23−13, 23, 258)-Net in Base 32 — Constructive

(10, 23, 258)-net in base 32, using

1 times m-reduction [i] based on (10, 24, 258)-net in base 32, using
- base change [i] based on digital (1, 15, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(23−13, 23, 289)-Net in Base 32

(10, 23, 289)-net in base 32, using

1 times m-reduction [i] based on (10, 24, 289)-net in base 32, using
- base change [i] based on digital (1, 15, 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]

(23−13, 23, 31893)-Net in Base 32 — Upper bound on s

There is no (10, 23, 31894)-net in base 32, because

1 times m-reduction [i] would yield (10, 22, 31894)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1298 110482 089498 352779 619987 257080 > 32²² [i]