MinT - Best Known (104−60, 104, s)-Nets in Base 27

Best Known (104−60, 104, s)-Nets in Base 27

(104−60, 104, 164)-Net over F₂₇ — Constructive and digital

Digital (44, 104, 164)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 37, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]
2. digital (7, 67, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(104−60, 104, 281)-Net over F₂₇ — Digital

Digital (44, 104, 281)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(27¹⁰⁴, 281, F₂₇, 2, 60) (dual of [(281, 2), 458, 61]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(27¹⁰², 280, F₂₇, 2, 60) (dual of [(280, 2), 458, 61]-NRT-code), using
  - extended algebraic-geometric NRT-code AG_e(2;F,499P) [i] based on function field F/F₂₇ with g(F) = 42 and N(F) ≥ 280, using
    - Drinfeld modules of rank 1 [i]

(104−60, 104, 370)-Net in Base 27 — Constructive

(44, 104, 370)-net in base 27, using

t-expansion [i] based on (43, 104, 370)-net in base 27, using
- 4 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(104−60, 104, 42434)-Net in Base 27 — Upper bound on s

There is no (44, 104, 42435)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 72789 960439 275324 240728 977508 352078 918920 747302 996216 718677 571248 848967 414233 305884 003209 465817 154899 097773 687036 335783 000047 389605 409228 143658 244317 > 27¹⁰⁴ [i]