MinT - Best Known (20−12, 20, s)-Nets in Base 32

Best Known (20−12, 20, s)-Nets in Base 32

(20−12, 20, 98)-Net over F₃₂ — Constructive and digital

Digital (8, 20, 98)-net over F₃₂, using

t-expansion [i] based on digital (7, 20, 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]

(20−12, 20, 102)-Net over F₃₂ — Digital

Digital (8, 20, 102)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32²⁰, 102, F₃₂, 12) (dual of [102, 82, 13]-code), using
- 3 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0, 0) [i] based on linear OA(32¹⁹, 98, F₃₂, 12) (dual of [98, 79, 13]-code), using
  - extended algebraic-geometric code AG_e(F,85P) [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
    - a function field by SÃ©mirat [i]

(20−12, 20, 257)-Net in Base 32 — Constructive

(8, 20, 257)-net in base 32, using

1 times m-reduction [i] based on (8, 21, 257)-net in base 32, using
- base change [i] based on (2, 15, 257)-net in base 128, using
  - 1 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
    - base change [i] based on digital (0, 14, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(20−12, 20, 10044)-Net in Base 32 — Upper bound on s

There is no (8, 20, 10045)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 268340 617311 231671 408144 848500 > 32²⁰ [i]