MinT - Best Known (58, 157, s)-Nets in Base 3

Best Known (58, 157, s)-Nets in Base 3

(58, 157, 48)-Net over F₃ — Constructive and digital

Digital (58, 157, 48)-net over F₃, using

t-expansion [i] based on digital (45, 157, 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, 157, 64)-Net over F₃ — Digital

Digital (58, 157, 64)-net over F₃, using

t-expansion [i] based on digital (49, 157, 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, 157, 261)-Net over F₃ — Upper bound on s (digital)

There is no digital (58, 157, 262)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁷, 262, F₃, 99) (dual of [262, 105, 100]-code), but
- residual code [i] would yield OA(3⁵⁸, 162, S₃, 33), but
  - 1 times truncation [i] would yield OA(3⁵⁷, 161, S₃, 32), but
    - the linear programming bound shows that MÂ â‰¥ 721213 722132 726616 769951 419548 792269 427090 005182 815014 600979 / 455 415925 085809 684165 452127 356779 > 3⁵⁷ [i]

(58, 157, 270)-Net in Base 3 — Upper bound on s

There is no (58, 157, 271)-net in base 3, because

1 times m-reduction [i] would yield (58, 156, 271)-net in base 3, but
- 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]