MinT - Best Known (123−98, 123, s)-Nets in Base 3

Best Known (123−98, 123, s)-Nets in Base 3

(123−98, 123, 32)-Net over F₃ — Constructive and digital

Digital (25, 123, 32)-net over F₃, using

t-expansion [i] based on digital (21, 123, 32)-net over F₃, using
- net from sequence [i] based on digital (21, 31)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
    - an explicitly constructive algebraic function field [i]

(123−98, 123, 36)-Net over F₃ — Digital

Digital (25, 123, 36)-net over F₃, using

net from sequence [i] based on digital (25, 35)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 25 and N(F) ≥ 36, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(123−98, 123, 82)-Net over F₃ — Upper bound on s (digital)

There is no digital (25, 123, 83)-net over F₃, because

44 times m-reduction [i] would yield digital (25, 79, 83)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁷⁹, 83, F₃, 54) (dual of [83, 4, 55]-code), but
  - Griesmer bound [i]

(123−98, 123, 84)-Net in Base 3 — Upper bound on s

There is no (25, 123, 85)-net in base 3, because

46 times m-reduction [i] would yield (25, 77, 85)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁷⁷, 85, S₃, 52), but
  - the linear programming bound shows that MÂ â‰¥ 626561 627887 412368 936876 728064 858798 530639 / 100223 > 3⁷⁷ [i]