MinT - Best Known (63−42, 63, s)-Nets in Base 4

Best Known (63−42, 63, s)-Nets in Base 4

(63−42, 63, 34)-Net over F₄ — Constructive and digital

Digital (21, 63, 34)-net over F₄, using

net from sequence [i] based on digital (21, 33)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
  - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(63−42, 63, 44)-Net over F₄ — Digital

Digital (21, 63, 44)-net over F₄, using

net from sequence [i] based on digital (21, 43)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 44, using
  - a curve from the manYPoints database [i]

(63−42, 63, 164)-Net in Base 4 — Upper bound on s

There is no (21, 63, 165)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁶³, 165, S₄, 42), but
- the linear programming bound shows that MÂ â‰¥ 34 077190 942729 726882 024839 335902 035522 717143 016283 935337 510998 524998 580004 361097 586089 334532 997120 / 383476 318008 814835 283839 224897 225684 554347 388608 827379 925139 > 4⁶³ [i]