MinT - Best Known (62, 180, s)-Nets in Base 3

Best Known (62, 180, s)-Nets in Base 3

(62, 180, 48)-Net over F₃ — Constructive and digital

Digital (62, 180, 48)-net over F₃, using

t-expansion [i] based on digital (45, 180, 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]

(62, 180, 64)-Net over F₃ — Digital

Digital (62, 180, 64)-net over F₃, using

t-expansion [i] based on digital (49, 180, 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]

(62, 180, 216)-Net over F₃ — Upper bound on s (digital)

There is no digital (62, 180, 217)-net over F₃, because

1 times m-reduction [i] would yield digital (62, 179, 217)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁷⁹, 217, F₃, 117) (dual of [217, 38, 118]-code), but
  - residual code [i] would yield OA(3⁶², 99, S₃, 39), but
    - the linear programming bound shows that MÂ â‰¥ 8341 316354 832968 167115 756462 681271 298651 307150 960469 759639 / 19004 191492 894766 602246 400000 > 3⁶² [i]

(62, 180, 271)-Net in Base 3 — Upper bound on s

There is no (62, 180, 272)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 86 912594 997600 760306 774092 370743 679660 359491 090154 794961 919413 488492 888540 814549 518913 > 3¹⁸⁰ [i]