MinT - Best Known (68−59, 68, s)-Nets in Base 4

Best Known (68−59, 68, s)-Nets in Base 4

(68−59, 68, 22)-Net over F₄ — Constructive and digital

Digital (9, 68, 22)-net over F₄, using

net from sequence [i] based on digital (9, 21)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, using
  - an explicitly constructive algebraic function field [i]

(68−59, 68, 26)-Net over F₄ — Digital

Digital (9, 68, 26)-net over F₄, using

net from sequence [i] based on digital (9, 25)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 26, using
  - Drinfeld modules of rank 1 [i]

(68−59, 68, 43)-Net over F₄ — Upper bound on s (digital)

There is no digital (9, 68, 44)-net over F₄, because

27 times m-reduction [i] would yield digital (9, 41, 44)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁴¹, 44, F₄, 32) (dual of [44, 3, 33]-code), but
  - Griesmer bound [i]

(68−59, 68, 45)-Net in Base 4 — Upper bound on s

There is no (9, 68, 46)-net in base 4, because

28 times m-reduction [i] would yield (9, 40, 46)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁴⁰, 46, S₄, 31), but
  - the linear programming bound shows that MÂ â‰¥ 976 812062 248620 373162 590208 / 703 > 4⁴⁰ [i]