MinT - Best Known (53, 53+50, s)-Nets in Base 27

Best Known (53, 53+50, s)-Nets in Base 27

(53, 53+50, 202)-Net over F₂₇ — Constructive and digital

Digital (53, 103, 202)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 35, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]
2. digital (18, 68, 108)-net over F₂₇, using
  - net from sequence [i] based on digital (18, 107)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 108, using
      - F₃ from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F₂₇ [i]

(53, 53+50, 370)-Net in Base 27 — Constructive

(53, 103, 370)-net in base 27, using

t-expansion [i] based on (43, 103, 370)-net in base 27, using
- 5 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]

(53, 53+50, 771)-Net over F₂₇ — Digital

Digital (53, 103, 771)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(53, 53+50, 308924)-Net in Base 27 — Upper bound on s

There is no (53, 103, 308925)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2694 448775 752530 951602 292862 490666 532000 109554 726954 077065 122214 918740 811885 889066 619765 089647 114730 397313 310388 545165 698665 575424 056354 707229 214483 > 27¹⁰³ [i]