MinT - Best Known (62, 62+122, s)-Nets in Base 3

Best Known (62, 62+122, s)-Nets in Base 3

(62, 62+122, 48)-Net over F₃ — Constructive and digital

Digital (62, 184, 48)-net over F₃, using

t-expansion [i] based on digital (45, 184, 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, 62+122, 64)-Net over F₃ — Digital

Digital (62, 184, 64)-net over F₃, using

t-expansion [i] based on digital (49, 184, 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, 62+122, 203)-Net over F₃ — Upper bound on s (digital)

There is no digital (62, 184, 204)-net over F₃, because

2 times m-reduction [i] would yield digital (62, 182, 204)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸², 204, F₃, 120) (dual of [204, 22, 121]-code), but
  - residual code [i] would yield OA(3⁶², 83, S₃, 40), but
    - the linear programming bound shows that MÂ â‰¥ 463 942350 761962 378112 632058 617109 511207 / 1197 196966 > 3⁶² [i]

(62, 62+122, 267)-Net in Base 3 — Upper bound on s

There is no (62, 184, 268)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6589 723799 408262 773152 422961 757289 343095 150698 318332 475608 893182 454611 039518 335271 583481 > 3¹⁸⁴ [i]