MinT - Best Known (127−102, 127, s)-Nets in Base 5

Best Known (127−102, 127, s)-Nets in Base 5

(127−102, 127, 51)-Net over F₅ — Constructive and digital

Digital (25, 127, 51)-net over F₅, using

t-expansion [i] based on digital (22, 127, 51)-net over F₅, using
- net from sequence [i] based on digital (22, 50)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 22 and N(F) ≥ 51, using
    - an explicitly constructive algebraic function field [i]

(127−102, 127, 55)-Net over F₅ — Digital

Digital (25, 127, 55)-net over F₅, using

t-expansion [i] based on digital (23, 127, 55)-net over F₅, using
- net from sequence [i] based on digital (23, 54)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 23 and N(F) ≥ 55, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(127−102, 127, 128)-Net in Base 5 — Upper bound on s

There is no (25, 127, 129)-net in base 5, because

11 times m-reduction [i] would yield (25, 116, 129)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5¹¹⁶, 129, S₅, 91), but
  - the linear programming bound shows that MÂ â‰¥ 11 958351 050689 163710 066275 542623 394170 276397 455664 717239 809391 458038 589917 123317 718505 859375 / 9241 308483 > 5¹¹⁶ [i]