MinT - Best Known (77−44, 77, s)-Nets in Base 32

Best Known (77−44, 77, s)-Nets in Base 32

(77−44, 77, 162)-Net over F₃₂ — Constructive and digital

Digital (33, 77, 162)-net over F₃₂, using

2 times m-reduction [i] based on digital (33, 79, 162)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 26, 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 (7, 53, 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]

(77−44, 77, 275)-Net over F₃₂ — Digital

Digital (33, 77, 275)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁷⁷, 275, F₃₂, 2, 44) (dual of [(275, 2), 473, 45]-NRT-code), using
- constructionÂ X applied to AG(2;F,499P) âŠ‚ AG(2;F,503P) [i] based on
  1. linear OOA(32⁷⁴, 272, F₃₂, 2, 44) (dual of [(272, 2), 470, 45]-NRT-code), using algebraic-geometric NRT-code AG(2;F,499P) [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273, using
    - a curve from the manYPoints database [i]
  2. linear OOA(32⁷⁰, 272, F₃₂, 2, 40) (dual of [(272, 2), 474, 41]-NRT-code), using algebraic-geometric NRT-code AG(2;F,503P) [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273 (see above)
  3. linear OOA(32³, 3, F₃₂, 2, 3) (dual of [(3, 2), 3, 4]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(32³, 32, F₃₂, 2, 3) (dual of [(32, 2), 61, 4]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(2;61,32) [i]

(77−44, 77, 288)-Net in Base 32 — Constructive

(33, 77, 288)-net in base 32, using

7 times m-reduction [i] based on (33, 84, 288)-net in base 32, using
- base change [i] based on digital (9, 60, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(77−44, 77, 342)-Net in Base 32

(33, 77, 342)-net in base 32, using

1 times m-reduction [i] based on (33, 78, 342)-net in base 32, using
- base change [i] based on digital (20, 65, 342)-net over F₆₄, using
  - net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
      - a curve from the manYPoints database [i]

(77−44, 77, 54128)-Net in Base 32 — Upper bound on s

There is no (33, 77, 54129)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 78 818672 504996 930691 142864 660802 784652 250250 580142 419739 744683 182449 516678 050389 158327 723735 840894 102618 325989 544464 > 32⁷⁷ [i]