MinT - Best Known (103−67, 103, s)-Nets in Base 27

Best Known (103−67, 103, s)-Nets in Base 27

(103−67, 103, 114)-Net over F₂₇ — Constructive and digital

Digital (36, 103, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 103, 114)-net over F₂₇, using
- net from sequence [i] based on digital (23, 113)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 23 and N(F) ≥ 114, using
    - an explicitly constructive algebraic function field [i]

(103−67, 103, 172)-Net in Base 27 — Constructive

(36, 103, 172)-net in base 27, using

t-expansion [i] based on (34, 103, 172)-net in base 27, using
- 5 times m-reduction [i] based on (34, 108, 172)-net in base 27, using
  - base change [i] based on digital (7, 81, 172)-net over F₈₁, using
    - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(103−67, 103, 244)-Net over F₂₇ — Digital

Digital (36, 103, 244)-net over F₂₇, using

net from sequence [i] based on digital (36, 243)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 36 and N(F) ≥ 244, using
  - Drinfeld modules of rank 1 [i]

(103−67, 103, 13428)-Net in Base 27 — Upper bound on s

There is no (36, 103, 13429)-net in base 27, because

1 times m-reduction [i] would yield (36, 102, 13429)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 99 836425 515680 528976 411651 390792 755564 002339 537073 448709 211242 204647 437389 156761 770818 886664 647201 913027 146041 158529 958994 989338 471418 040724 876899 > 27¹⁰² [i]