MinT - Best Known (38−20, 38, s)-Nets in Base 32

Best Known (38−20, 38, s)-Nets in Base 32

(38−20, 38, 142)-Net over F₃₂ — Constructive and digital

Digital (18, 38, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 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, 27, 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]

(38−20, 38, 260)-Net in Base 32 — Constructive

(18, 38, 260)-net in base 32, using

2 times m-reduction [i] based on (18, 40, 260)-net in base 32, using
- base change [i] based on digital (3, 25, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(38−20, 38, 295)-Net over F₃₂ — Digital

Digital (18, 38, 295)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32³⁸, 295, F₃₂, 20) (dual of [295, 257, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(32³⁸, 341, F₃₂, 20) (dual of [341, 303, 21]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 341Â | 32²âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(38−20, 38, 321)-Net in Base 32

(18, 38, 321)-net in base 32, using

4 times m-reduction [i] based on (18, 42, 321)-net in base 32, using
- base change [i] based on (11, 35, 321)-net in base 64, using
  - 1 times m-reduction [i] based on (11, 36, 321)-net in base 64, using
    - base change [i] based on digital (2, 27, 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]

(38−20, 38, 76587)-Net in Base 32 — Upper bound on s

There is no (18, 38, 76588)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1569 403600 660226 816237 106279 933163 063339 507840 766468 578143 > 32³⁸ [i]