MinT - Best Known (52, 52+126, s)-Nets in Base 3

Best Known (52, 52+126, s)-Nets in Base 3

(52, 52+126, 48)-Net over F₃ — Constructive and digital

Digital (52, 178, 48)-net over F₃, using

t-expansion [i] based on digital (45, 178, 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]

(52, 52+126, 64)-Net over F₃ — Digital

Digital (52, 178, 64)-net over F₃, using

t-expansion [i] based on digital (49, 178, 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]

(52, 52+126, 164)-Net over F₃ — Upper bound on s (digital)

There is no digital (52, 178, 165)-net over F₃, because

18 times m-reduction [i] would yield digital (52, 160, 165)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁶⁰, 165, F₃, 108) (dual of [165, 5, 109]-code), but
  - Griesmer bound [i]

(52, 52+126, 169)-Net in Base 3 — Upper bound on s

There is no (52, 178, 170)-net in base 3, because

13 times m-reduction [i] would yield (52, 165, 170)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁶⁵, 170, S₃, 113), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 143 341119 796678 074027 577337 316118 932770 382415 738332 565090 012937 927177 499182 473761 / 19 > 3¹⁶⁵ [i]