MinT - Best Known (176−116, 176, s)-Nets in Base 3

Best Known (176−116, 176, s)-Nets in Base 3

(176−116, 176, 48)-Net over F₃ — Constructive and digital

Digital (60, 176, 48)-net over F₃, using

t-expansion [i] based on digital (45, 176, 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]

(176−116, 176, 64)-Net over F₃ — Digital

Digital (60, 176, 64)-net over F₃, using

t-expansion [i] based on digital (49, 176, 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]

(176−116, 176, 208)-Net over F₃ — Upper bound on s (digital)

There is no digital (60, 176, 209)-net over F₃, because

2 times m-reduction [i] would yield digital (60, 174, 209)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁷⁴, 209, F₃, 114) (dual of [209, 35, 115]-code), but
  - residual code [i] would yield OA(3⁶⁰, 94, S₃, 38), but
    - the linear programming bound shows that MÂ â‰¥ 132 944288 351905 684103 161899 822708 298724 029601 023671 / 2537 510710 702022 341000 > 3⁶⁰ [i]

(176−116, 176, 261)-Net in Base 3 — Upper bound on s

There is no (60, 176, 262)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 077185 453534 314772 392316 484374 678165 729711 108657 076157 675543 115176 201981 177495 330653 > 3¹⁷⁶ [i]