MinT - Best Known (61, 61+122, s)-Nets in Base 3

Best Known (61, 61+122, s)-Nets in Base 3

(61, 61+122, 48)-Net over F₃ — Constructive and digital

Digital (61, 183, 48)-net over F₃, using

t-expansion [i] based on digital (45, 183, 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]

(61, 61+122, 64)-Net over F₃ — Digital

Digital (61, 183, 64)-net over F₃, using

t-expansion [i] based on digital (49, 183, 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]

(61, 61+122, 198)-Net over F₃ — Upper bound on s (digital)

There is no digital (61, 183, 199)-net over F₃, because

2 times m-reduction [i] would yield digital (61, 181, 199)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸¹, 199, F₃, 120) (dual of [199, 18, 121]-code), but
  - residual code [i] would yield OA(3⁶¹, 78, S₃, 40), but
    - the linear programming bound shows that MÂ â‰¥ 3904 915634 579720 805714 510866 149518 645975 / 27453 221324 > 3⁶¹ [i]

(61, 61+122, 261)-Net in Base 3 — Upper bound on s

There is no (61, 183, 262)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2078 845684 576933 358067 568715 743925 965523 903994 051886 776704 306918 519316 182415 217142 216565 > 3¹⁸³ [i]