MinT - Best Known (126−100, 126, s)-Nets in Base 5

Best Known (126−100, 126, s)-Nets in Base 5

(126−100, 126, 51)-Net over F₅ — Constructive and digital

Digital (26, 126, 51)-net over F₅, using

t-expansion [i] based on digital (22, 126, 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]

(126−100, 126, 55)-Net over F₅ — Digital

Digital (26, 126, 55)-net over F₅, using

t-expansion [i] based on digital (23, 126, 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]

(126−100, 126, 133)-Net in Base 5 — Upper bound on s

There is no (26, 126, 134)-net in base 5, because

8 times m-reduction [i] would yield (26, 118, 134)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5¹¹⁸, 134, S₅, 92), but
  - the linear programming bound shows that MÂ â‰¥ 8400 170840 072152 236399 189268 979920 771007 847581 356717 606149 153709 793608 868494 629859 924316 406250 / 238763 600747 > 5¹¹⁸ [i]