MinT - Best Known (58, 156, s)-Nets in Base 3

Best Known (58, 156, s)-Nets in Base 3

(58, 156, 48)-Net over F₃ — Constructive and digital

Digital (58, 156, 48)-net over F₃, using

t-expansion [i] based on digital (45, 156, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(58, 156, 64)-Net over F₃ — Digital

Digital (58, 156, 64)-net over F₃, using

t-expansion [i] based on digital (49, 156, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(58, 156, 269)-Net over F₃ — Upper bound on s (digital)

There is no digital (58, 156, 270)-net over F₃, because

2 times m-reduction [i] would yield digital (58, 154, 270)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁴, 270, F₃, 96) (dual of [270, 116, 97]-code), but
  - residual code [i] would yield OA(3⁵⁸, 173, S₃, 32), but
    - the linear programming bound shows that MÂ â‰¥ 47 314229 835710 776345 670615 418214 790563 428291 670976 086718 521800 / 9965 095822 320687 546955 148607 190123 > 3⁵⁸ [i]

(58, 156, 270)-Net in Base 3 — Upper bound on s

There is no (58, 156, 271)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 311 895013 671786 235239 586010 810735 745320 880938 003766 995775 111093 341148 574143 > 3¹⁵⁶ [i]