MinT - Best Known (245−93, 245, s)-Nets in Base 4

Best Known (245−93, 245, s)-Nets in Base 4

(245−93, 245, 160)-Net over F₄ — Constructive and digital

Digital (152, 245, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (33, 79, 56)-net over F₄, using
  - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
      - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (73, 166, 104)-net over F₄, using
  - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
      - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(245−93, 245, 417)-Net over F₄ — Digital

Digital (152, 245, 417)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−93, 245, 9330)-Net in Base 4 — Upper bound on s

There is no (152, 245, 9331)-net in base 4, because

1 times m-reduction [i] would yield (152, 244, 9331)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 802 448458 379992 376051 703508 409982 340165 227451 801714 965776 688913 248955 170643 466560 268894 870274 387313 862132 118104 176696 282214 948917 563397 122038 405908 > 4²⁴⁴ [i]