MinT - Best Known (259−226, 259, s)-Nets in Base 4

Best Known (259−226, 259, s)-Nets in Base 4

(259−226, 259, 56)-Net over F₄ — Constructive and digital

Digital (33, 259, 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]

(259−226, 259, 65)-Net over F₄ — Digital

Digital (33, 259, 65)-net over F₄, using

net from sequence [i] based on digital (33, 64)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
  - Drinfeld modules of rank 1 [i]

(259−226, 259, 117)-Net in Base 4 — Upper bound on s

There is no (33, 259, 118)-net in base 4, because

27 times m-reduction [i] would yield (33, 232, 118)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²³², 118, S₄, 2, 199), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1333 754873 792233 009013 133582 064951 424575 206725 992114 139161 016767 418066 538802 390096 486947 187588 161852 912523 713247 671286 637459 462265 320922 677248 / 25 > 4²³² [i]