MinT - Best Known (38, 62, s)-Nets in Base 32

Best Known (38, 62, s)-Nets in Base 32

(38, 62, 262)-Net over F₃₂ — Constructive and digital

Digital (38, 62, 262)-net over F₃₂, using

1 times m-reduction [i] based on digital (38, 63, 262)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (4, 12, 66)-net over F₃₂, using
    - (u,Â u+v)-construction [i] based on
      1. digital (0, 4, 33)-net over F₃₂, using
        net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
        the rational function field F₃₂(x) [i]
        Niederreiter sequence [i]
      2. digital (0, 8, 33)-net over F₃₂, using
        net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
  2. digital (7, 19, 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]
  3. digital (7, 32, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(38, 62, 516)-Net in Base 32 — Constructive

(38, 62, 516)-net in base 32, using

1 times m-reduction [i] based on (38, 63, 516)-net in base 32, using
- (u,Â u+v)-construction [i] based on
  1. (9, 21, 258)-net in base 32, using
    - base change [i] based on (3, 15, 258)-net in base 128, using
      - 1 times m-reduction [i] based on (3, 16, 258)-net in base 128, using
        base change [i] based on digital (1, 14, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
  2. (17, 42, 258)-net in base 32, using
    - base change [i] based on (10, 35, 258)-net in base 64, using
      - 1 times m-reduction [i] based on (10, 36, 258)-net in base 64, using
        base change [i] based on digital (1, 27, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(38, 62, 3483)-Net over F₃₂ — Digital

Digital (38, 62, 3483)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁶², 3483, F₃₂, 24) (dual of [3483, 3421, 25]-code), using
- 3412 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (5, 1, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 8 times 0, 1, 9 times 0, 1, 11 times 0, 1, 13 times 0, 1, 16 times 0, 1, 18 times 0, 1, 22 times 0, 1, 26 times 0, 1, 30 times 0, 1, 35 times 0, 1, 42 times 0, 1, 49 times 0, 1, 57 times 0, 1, 66 times 0, 1, 78 times 0, 1, 90 times 0, 1, 106 times 0, 1, 123 times 0, 1, 143 times 0, 1, 167 times 0, 1, 195 times 0, 1, 226 times 0, 1, 264 times 0, 1, 307 times 0, 1, 357 times 0, 1, 416 times 0, 1, 484 times 0) [i] based on linear OA(32²⁴, 33, F₃₂, 24) (dual of [33, 9, 25]-code or 33-arc in PG(23,32)), using
  - extended Reedâ€“Solomon code RS_e(9,32) [i]

(38, 62, large)-Net in Base 32 — Upper bound on s

There is no (38, 62, large)-net in base 32, because

22 times m-reduction [i] would yield (38, 40, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]