MinT - Best Known (243−101, 243, s)-Nets in Base 4

Best Known (243−101, 243, s)-Nets in Base 4

(243−101, 243, 137)-Net over F₄ — Constructive and digital

Digital (142, 243, 137)-net over F₄, using

7 times m-reduction [i] based on digital (142, 250, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 69, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 181, 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]

(243−101, 243, 307)-Net over F₄ — Digital

Digital (142, 243, 307)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−101, 243, 5286)-Net in Base 4 — Upper bound on s

There is no (142, 243, 5287)-net in base 4, because

1 times m-reduction [i] would yield (142, 242, 5287)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 50 278683 024971 080281 535688 127911 922117 471912 954871 118086 318946 197505 514458 916685 635626 757545 705488 384572 447663 509027 686958 973199 487157 217849 443640 > 4²⁴² [i]