MinT - Best Known (51, 51+79, s)-Nets in Base 9

Best Known (51, 51+79, s)-Nets in Base 9

(51, 51+79, 81)-Net over F₉ — Constructive and digital

Digital (51, 130, 81)-net over F₉, using

t-expansion [i] based on digital (32, 130, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(51, 51+79, 82)-Net in Base 9 — Constructive

(51, 130, 82)-net in base 9, using

2 times m-reduction [i] based on (51, 132, 82)-net in base 9, using
- base change [i] based on digital (7, 88, 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]

(51, 51+79, 182)-Net over F₉ — Digital

Digital (51, 130, 182)-net over F₉, using

t-expansion [i] based on digital (50, 130, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(51, 51+79, 2734)-Net in Base 9 — Upper bound on s

There is no (51, 130, 2735)-net in base 9, because

1 times m-reduction [i] would yield (51, 129, 2735)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1256 324418 246401 980043 617362 990110 009995 739072 172315 452439 572792 895310 310205 439960 505512 958473 681082 351249 520153 601197 754185 > 9¹²⁹ [i]