MinT - Best Known (128−99, 128, s)-Nets in Base 5

Best Known (128−99, 128, s)-Nets in Base 5

(128−99, 128, 51)-Net over F₅ — Constructive and digital

Digital (29, 128, 51)-net over F₅, using

t-expansion [i] based on digital (22, 128, 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]

(128−99, 128, 56)-Net over F₅ — Digital

Digital (29, 128, 56)-net over F₅, using

net from sequence [i] based on digital (29, 55)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 29 and N(F) ≥ 56, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(128−99, 128, 148)-Net in Base 5 — Upper bound on s

There is no (29, 128, 149)-net in base 5, because

1 times m-reduction [i] would yield (29, 127, 149)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5¹²⁷, 149, S₅, 98), but
  - the linear programming bound shows that MÂ â‰¥ 5193 361256 119414 558729 044685 549653 156610 735015 094949 701472 602202 230455 237241 812938 009388 744831 085205 078125 / 69869 528585 224173 > 5¹²⁷ [i]