MinT - Best Known (127−67, 127, s)-Nets in Base 5

Best Known (127−67, 127, s)-Nets in Base 5

(127−67, 127, 82)-Net over F₅ — Constructive and digital

Digital (60, 127, 82)-net over F₅, using

t-expansion [i] based on digital (48, 127, 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]

(127−67, 127, 112)-Net over F₅ — Digital

Digital (60, 127, 112)-net over F₅, using

t-expansion [i] based on digital (57, 127, 112)-net over F₅, using
- net from sequence [i] based on digital (57, 111)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 57 and N(F) ≥ 112, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(127−67, 127, 1510)-Net in Base 5 — Upper bound on s

There is no (60, 127, 1511)-net in base 5, because

1 times m-reduction [i] would yield (60, 126, 1511)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 11801 451045 738267 942398 807732 488756 596164 683282 110654 606435 118724 196618 161493 955709 744925 > 5¹²⁶ [i]