MinT - Best Known (137−79, 137, s)-Nets in Base 5

Best Known (137−79, 137, s)-Nets in Base 5

(137−79, 137, 82)-Net over F₅ — Constructive and digital

Digital (58, 137, 82)-net over F₅, using

t-expansion [i] based on digital (48, 137, 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]

(137−79, 137, 112)-Net over F₅ — Digital

Digital (58, 137, 112)-net over F₅, using

t-expansion [i] based on digital (57, 137, 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]

(137−79, 137, 1025)-Net in Base 5 — Upper bound on s

There is no (58, 137, 1026)-net in base 5, because

1 times m-reduction [i] would yield (58, 136, 1026)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 117261 459027 092005 378116 141932 423784 238618 893268 475529 094458 270793 543412 941657 708149 428903 939625 > 5¹³⁶ [i]