MinT - Best Known (126−103, 126, s)-Nets in Base 5

Best Known (126−103, 126, s)-Nets in Base 5

(126−103, 126, 51)-Net over F₅ — Constructive and digital

Digital (23, 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]

(126−103, 126, 55)-Net over F₅ — Digital

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]

(126−103, 126, 120)-Net in Base 5 — Upper bound on s

There is no (23, 126, 121)-net in base 5, because

21 times m-reduction [i] would yield (23, 105, 121)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5¹⁰⁵, 121, S₅, 82), but
  - the linear programming bound shows that MÂ â‰¥ 93 626609 426406 933398 245190 747110 019959 125490 167373 140337 758741 225115 954875 946044 921875 / 3 706166 335599 > 5¹⁰⁵ [i]