MinT - Best Known (63, 63+128, s)-Nets in Base 3

Best Known (63, 63+128, s)-Nets in Base 3

(63, 63+128, 48)-Net over F₃ — Constructive and digital

Digital (63, 191, 48)-net over F₃, using

t-expansion [i] based on digital (45, 191, 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]

(63, 63+128, 64)-Net over F₃ — Digital

Digital (63, 191, 64)-net over F₃, using

t-expansion [i] based on digital (49, 191, 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]

(63, 63+128, 200)-Net over F₃ — Upper bound on s (digital)

There is no digital (63, 191, 201)-net over F₃, because

2 times m-reduction [i] would yield digital (63, 189, 201)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁹, 201, F₃, 126) (dual of [201, 12, 127]-code), but
  - residual code [i] would yield OA(3⁶³, 74, S₃, 42), but
    - the linear programming bound shows that MÂ â‰¥ 12 034337 282839 174538 591498 251772 066709 / 10 069525 > 3⁶³ [i]

(63, 63+128, 268)-Net in Base 3 — Upper bound on s

There is no (63, 191, 269)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 14 667483 760095 766439 388037 267493 290534 199047 189870 901161 345683 026665 086892 195396 960298 870529 > 3¹⁹¹ [i]