MinT - Best Known (257−109, 257, s)-Nets in Base 4

Best Known (257−109, 257, s)-Nets in Base 4

(257−109, 257, 138)-Net over F₄ — Constructive and digital

Digital (148, 257, 138)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 75, 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]
2. digital (73, 182, 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]

(257−109, 257, 302)-Net over F₄ — Digital

Digital (148, 257, 302)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−109, 257, 4951)-Net in Base 4 — Upper bound on s

There is no (148, 257, 4952)-net in base 4, because

1 times m-reduction [i] would yield (148, 256, 4952)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13442 490283 092356 217841 359159 297300 229484 675495 496510 444627 689429 253589 840882 492939 757938 812558 605260 739943 975323 398059 389545 441842 470942 229625 119257 181430 > 4²⁵⁶ [i]