MinT - Best Known (128−78, 128, s)-Nets in Base 5

Best Known (128−78, 128, s)-Nets in Base 5

(128−78, 128, 82)-Net over F₅ — Constructive and digital

Digital (50, 128, 82)-net over F₅, using

t-expansion [i] based on digital (48, 128, 82)-net over F₅, using
- net from sequence [i] based on digital (48, 81)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 48 and N(F) ≥ 82, using
    - an explicitly constructive algebraic function field [i]

(128−78, 128, 96)-Net over F₅ — Digital

Digital (50, 128, 96)-net over F₅, using

t-expansion [i] based on digital (49, 128, 96)-net over F₅, using
- net from sequence [i] based on digital (49, 95)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 49 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(128−78, 128, 729)-Net in Base 5 — Upper bound on s

There is no (50, 128, 730)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 306254 286772 925705 849163 956271 155908 488993 671780 352739 773260 416310 820125 894685 546560 852105 > 5¹²⁸ [i]