MinT - Best Known (88−56, 88, s)-Nets in Base 3

Best Known (88−56, 88, s)-Nets in Base 3

(88−56, 88, 38)-Net over F₃ — Constructive and digital

Digital (32, 88, 38)-net over F₃, using

net from sequence [i] based on digital (32, 37)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
  - an explicitly constructive algebraic function field [i]

(88−56, 88, 42)-Net over F₃ — Digital

Digital (32, 88, 42)-net over F₃, using

t-expansion [i] based on digital (29, 88, 42)-net over F₃, using
- net from sequence [i] based on digital (29, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 29 and N(F) ≥ 42, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(88−56, 88, 113)-Net in Base 3 — Upper bound on s

There is no (32, 88, 114)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁸⁸, 114, S₃, 56), but
- the linear programming bound shows that MÂ â‰¥ 55244 064096 672202 650192 010466 118056 001703 652594 733528 976683 / 40173 727271 768090 > 3⁸⁸ [i]