MinT - Best Known (254−192, 254, s)-Nets in Base 4

Best Known (254−192, 254, s)-Nets in Base 4

(254−192, 254, 66)-Net over F₄ — Constructive and digital

Digital (62, 254, 66)-net over F₄, using

t-expansion [i] based on digital (49, 254, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(254−192, 254, 99)-Net over F₄ — Digital

Digital (62, 254, 99)-net over F₄, using

t-expansion [i] based on digital (61, 254, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(254−192, 254, 257)-Net over F₄ — Upper bound on s (digital)

There is no digital (62, 254, 258)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁴, 258, F₄, 192) (dual of [258, 4, 193]-code), but
- Griesmer bound [i]

(254−192, 254, 401)-Net in Base 4 — Upper bound on s

There is no (62, 254, 402)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 854 314872 771214 102828 472497 735407 239675 452164 528034 163630 088260 716909 650604 890916 390813 337021 365482 795341 674533 194889 488209 051387 856488 802786 427782 803996 > 4²⁵⁴ [i]