MinT - Best Known (256−194, 256, s)-Nets in Base 4

Best Known (256−194, 256, s)-Nets in Base 4

(256−194, 256, 66)-Net over F₄ — Constructive and digital

Digital (62, 256, 66)-net over F₄, using

t-expansion [i] based on digital (49, 256, 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]

(256−194, 256, 99)-Net over F₄ — Digital

Digital (62, 256, 99)-net over F₄, using

t-expansion [i] based on digital (61, 256, 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]

(256−194, 256, 257)-Net over F₄ — Upper bound on s (digital)

There is no digital (62, 256, 258)-net over F₄, because

2 times m-reduction [i] would yield digital (62, 254, 258)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁴, 258, F₄, 192) (dual of [258, 4, 193]-code), but
  - Griesmer bound [i]

(256−194, 256, 401)-Net in Base 4 — Upper bound on s

There is no (62, 256, 402)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 14662 468598 848181 470999 961360 295323 575792 062590 361397 400191 784697 263244 493732 537291 382280 165500 896951 389858 551801 770050 812609 180086 161225 939116 019399 111940 > 4²⁵⁶ [i]