MinT - Best Known (222−93, 222, s)-Nets in Base 4

Best Known (222−93, 222, s)-Nets in Base 4

(222−93, 222, 131)-Net over F₄ — Constructive and digital

Digital (129, 222, 131)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 56, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 166, 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]

(222−93, 222, 276)-Net over F₄ — Digital

Digital (129, 222, 276)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−93, 222, 4646)-Net in Base 4 — Upper bound on s

There is no (129, 222, 4647)-net in base 4, because

1 times m-reduction [i] would yield (129, 221, 4647)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 440418 898573 057465 810777 289203 059261 261542 031638 669586 933026 835360 304718 323300 870528 402709 866651 737369 057781 502848 977410 427480 838208 > 4²²¹ [i]