MinT - Best Known (167−110, 167, s)-Nets in Base 3

Best Known (167−110, 167, s)-Nets in Base 3

(167−110, 167, 48)-Net over F₃ — Constructive and digital

Digital (57, 167, 48)-net over F₃, using

t-expansion [i] based on digital (45, 167, 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]

(167−110, 167, 64)-Net over F₃ — Digital

Digital (57, 167, 64)-net over F₃, using

t-expansion [i] based on digital (49, 167, 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]

(167−110, 167, 200)-Net over F₃ — Upper bound on s (digital)

There is no digital (57, 167, 201)-net over F₃, because

2 times m-reduction [i] would yield digital (57, 165, 201)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁶⁵, 201, F₃, 108) (dual of [201, 36, 109]-code), but
  - residual code [i] would yield OA(3⁵⁷, 92, S₃, 36), but
    - the linear programming bound shows that MÂ â‰¥ 120 721890 049904 553965 455860 898229 550534 869060 685567 / 70191 477994 212742 316032 > 3⁵⁷ [i]

(167−110, 167, 249)-Net in Base 3 — Upper bound on s

There is no (57, 167, 250)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 56 771652 175477 323394 946686 425212 407217 619791 799244 552288 822758 148487 484157 390761 > 3¹⁶⁷ [i]