MinT - Best Known (114−71, 114, s)-Nets in Base 3

Best Known (114−71, 114, s)-Nets in Base 3

(114−71, 114, 42)-Net over F₃ — Constructive and digital

Digital (43, 114, 42)-net over F₃, using

t-expansion [i] based on digital (39, 114, 42)-net over F₃, using
- net from sequence [i] based on digital (39, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(114−71, 114, 56)-Net over F₃ — Digital

Digital (43, 114, 56)-net over F₃, using

t-expansion [i] based on digital (40, 114, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(114−71, 114, 180)-Net over F₃ — Upper bound on s (digital)

There is no digital (43, 114, 181)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹¹⁴, 181, F₃, 71) (dual of [181, 67, 72]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(3¹¹³, 143, S₃, 71), but
    - the linear programming bound shows that MÂ â‰¥ 31 672520 199297 585243 899154 531146 784746 691310 342121 601962 325858 908470 456893 / 37 514198 689317 280000 > 3¹¹³ [i]
  2. OA(3⁶⁷, 181, S₃, 38), but
    - discarding factors would yield OA(3⁶⁷, 175, S₃, 38), but
      - the linear programming bound shows that MÂ â‰¥ 387 999317 807943 805026 164281 984581 507597 966785 079804 265116 880208 641606 720469 396519 780352 / 3 893635 890543 207464 665118 176286 227992 932894 706751 830787 > 3⁶⁷ [i]

(114−71, 114, 208)-Net in Base 3 — Upper bound on s

There is no (43, 114, 209)-net in base 3, because

1 times m-reduction [i] would yield (43, 113, 209)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 850294 068741 077155 278058 126076 565308 524067 602106 137179 > 3¹¹³ [i]