MinT - Best Known (107−62, 107, s)-Nets in Base 27

Best Known (107−62, 107, s)-Nets in Base 27

(107−62, 107, 164)-Net over F₂₇ — Constructive and digital

Digital (45, 107, 164)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 38, 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, 69, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(107−62, 107, 281)-Net over F₂₇ — Digital

Digital (45, 107, 281)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(27¹⁰⁷, 281, F₂₇, 2, 62) (dual of [(281, 2), 455, 63]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(27¹⁰⁷, 282, F₂₇, 2, 62) (dual of [(282, 2), 457, 63]-NRT-code), using
  - constructionÂ X applied to AG(2;F,495P) âŠ‚ AG(2;F,499P) [i] based on
    1. linear OOA(27¹⁰⁴, 279, F₂₇, 2, 62) (dual of [(279, 2), 454, 63]-NRT-code), using algebraic-geometric NRT-code AG(2;F,495P) [i] based on function field F/F₂₇ with g(F) = 42 and N(F) ≥ 280, using
      - Drinfeld modules of rank 1 [i]
    2. linear OOA(27¹⁰⁰, 279, F₂₇, 2, 58) (dual of [(279, 2), 458, 59]-NRT-code), using algebraic-geometric NRT-code AG(2;F,499P) [i] based on function field F/F₂₇ with g(F) = 42 and N(F) ≥ 280 (see above)
    3. linear OOA(27³, 3, F₂₇, 2, 3) (dual of [(3, 2), 3, 4]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(27³, 27, F₂₇, 2, 3) (dual of [(27, 2), 51, 4]-NRT-code), using
        Reedâ€“Solomon NRT-code RS(2;51,27) [i]

(107−62, 107, 370)-Net in Base 27 — Constructive

(45, 107, 370)-net in base 27, using

t-expansion [i] based on (43, 107, 370)-net in base 27, using
- 1 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]

(107−62, 107, 41629)-Net in Base 27 — Upper bound on s

There is no (45, 107, 41630)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1432 141303 983423 726876 281653 429580 838361 422225 929364 084500 367956 283804 363579 681406 649023 609506 039179 724994 537808 827828 103188 568584 982083 399855 124870 006809 > 27¹⁰⁷ [i]