MinT - Best Known (51, 51+138, s)-Nets in Base 3

Best Known (51, 51+138, s)-Nets in Base 3

(51, 51+138, 48)-Net over F₃ — Constructive and digital

Digital (51, 189, 48)-net over F₃, using

t-expansion [i] based on digital (45, 189, 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]

(51, 51+138, 64)-Net over F₃ — Digital

Digital (51, 189, 64)-net over F₃, using

t-expansion [i] based on digital (49, 189, 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]

(51, 51+138, 162)-Net over F₃ — Upper bound on s (digital)

There is no digital (51, 189, 163)-net over F₃, because

30 times m-reduction [i] would yield digital (51, 159, 163)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁹, 163, F₃, 108) (dual of [163, 4, 109]-code), but
  - Griesmer bound [i]

(51, 51+138, 166)-Net in Base 3 — Upper bound on s

There is no (51, 189, 167)-net in base 3, because

27 times m-reduction [i] would yield (51, 162, 167)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁶², 167, S₃, 111), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 15 926791 088519 786003 064148 590679 881418 931379 526481 396121 112548 658575 277686 941529 / 56 > 3¹⁶² [i]