MinT - Best Known (50, 50+33, s)-Nets in Base 32

Best Known (50, 50+33, s)-Nets in Base 32

(50, 50+33, 300)-Net over F₃₂ — Constructive and digital

Digital (50, 83, 300)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 18, 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]
2. digital (7, 23, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
3. digital (9, 42, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(50, 50+33, 516)-Net in Base 32 — Constructive

(50, 83, 516)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (12, 28, 258)-net in base 32, using
  - base change [i] based on (4, 20, 258)-net in base 128, using
    - 4 times m-reduction [i] based on (4, 24, 258)-net in base 128, using
      - base change [i] based on digital (1, 21, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. (22, 55, 258)-net in base 32, using
  - 1 times m-reduction [i] based on (22, 56, 258)-net in base 32, using
    - base change [i] based on digital (1, 35, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(50, 50+33, 3324)-Net over F₃₂ — Digital

Digital (50, 83, 3324)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸³, 3324, F₃₂, 33) (dual of [3324, 3241, 34]-code), using
- 3240 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 7 times 0, 1, 7 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 15 times 0, 1, 16 times 0, 1, 18 times 0, 1, 21 times 0, 1, 23 times 0, 1, 26 times 0, 1, 29 times 0, 1, 33 times 0, 1, 37 times 0, 1, 42 times 0, 1, 46 times 0, 1, 52 times 0, 1, 58 times 0, 1, 65 times 0, 1, 72 times 0, 1, 81 times 0, 1, 91 times 0, 1, 101 times 0, 1, 113 times 0, 1, 126 times 0, 1, 141 times 0, 1, 157 times 0, 1, 175 times 0, 1, 196 times 0, 1, 218 times 0, 1, 243 times 0, 1, 271 times 0, 1, 303 times 0, 1, 338 times 0) [i] based on linear OA(32³³, 34, F₃₂, 33) (dual of [34, 1, 34]-code or 34-arc in PG(32,32)), using
  - dual of repetition code with length 34 [i]

(50, 50+33, large)-Net in Base 32 — Upper bound on s

There is no (50, 83, large)-net in base 32, because

31 times m-reduction [i] would yield (50, 52, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]