MinT - Best Known (29, 29+51, s)-Nets in Base 27

Best Known (29, 29+51, s)-Nets in Base 27

(29, 29+51, 114)-Net over F₂₇ — Constructive and digital

Digital (29, 80, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 80, 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]

(29, 29+51, 172)-Net in Base 27 — Constructive

(29, 80, 172)-net in base 27, using

8 times m-reduction [i] based on (29, 88, 172)-net in base 27, using
- base change [i] based on digital (7, 66, 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]

(29, 29+51, 208)-Net over F₂₇ — Digital

Digital (29, 80, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 80, 208)-net over F₂₇, using
- net from sequence [i] based on digital (24, 207)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 24 and N(F) ≥ 208, using
    - an algebraic function field by Beelen and Pellikaan [i]

(29, 29+51, 244)-Net in Base 27

(29, 80, 244)-net in base 27, using

base change [i] based on digital (9, 60, 244)-net over F₈₁, using
- net from sequence [i] based on digital (9, 243)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 9 and N(F) ≥ 244, using
    - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₈₁ [i]

(29, 29+51, 13041)-Net in Base 27 — Upper bound on s

There is no (29, 80, 13042)-net in base 27, because

1 times m-reduction [i] would yield (29, 79, 13042)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 119694 864696 444902 290435 309625 893198 892579 421612 649244 255266 636629 895656 296371 270474 450553 709461 552227 870249 442853 > 27⁷⁹ [i]