MinT - Best Known (162−102, 162, s)-Nets in Base 3

Best Known (162−102, 162, s)-Nets in Base 3

(162−102, 162, 48)-Net over F₃ — Constructive and digital

Digital (60, 162, 48)-net over F₃, using

t-expansion [i] based on digital (45, 162, 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]

(162−102, 162, 64)-Net over F₃ — Digital

Digital (60, 162, 64)-net over F₃, using

t-expansion [i] based on digital (49, 162, 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]

(162−102, 162, 264)-Net over F₃ — Upper bound on s (digital)

There is no digital (60, 162, 265)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁶², 265, F₃, 102) (dual of [265, 103, 103]-code), but
- residual code [i] would yield OA(3⁶⁰, 162, S₃, 34), but
  - the linear programming bound shows that MÂ â‰¥ 68980 367968 436452 870293 317186 511729 045056 682286 749727 903703 040000 000000 / 1 592376 446885 204670 935611 188490 612206 288097 > 3⁶⁰ [i]

(162−102, 162, 277)-Net in Base 3 — Upper bound on s

There is no (60, 162, 278)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 199990 777482 418360 372802 079588 524257 542719 672539 594850 333437 163804 225937 791937 > 3¹⁶² [i]