MinT - Best Known (36−20, 36, s)-Nets in Base 32

Best Known (36−20, 36, s)-Nets in Base 32

(36−20, 36, 128)-Net over F₃₂ — Constructive and digital

Digital (16, 36, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 13, 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 (3, 23, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(36−20, 36, 199)-Net over F₃₂ — Digital

Digital (16, 36, 199)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32³⁶, 199, F₃₂, 20) (dual of [199, 163, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(32³⁶, 205, F₃₂, 20) (dual of [205, 169, 21]-code), using
  - the BCH-code C(I) with length 205Â | 32⁴âˆ’1, defining interval IÂ =Â {32,61,90,…,−32}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(36−20, 36, 259)-Net in Base 32 — Constructive

(16, 36, 259)-net in base 32, using

base change [i] based on (10, 30, 259)-net in base 64, using
- 2 times m-reduction [i] based on (10, 32, 259)-net in base 64, using
  - base change [i] based on digital (2, 24, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(36−20, 36, 321)-Net in Base 32

(16, 36, 321)-net in base 32, using

base change [i] based on (10, 30, 321)-net in base 64, using
- 2 times m-reduction [i] based on (10, 32, 321)-net in base 64, using
  - base change [i] based on digital (2, 24, 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]

(36−20, 36, 38291)-Net in Base 32 — Upper bound on s

There is no (16, 36, 38292)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 532785 265855 463713 903992 453012 469360 680342 979807 254154 > 32³⁶ [i]