MinT - Best Known (24−7, 24, s)-Nets in Base 4

Best Known (24−7, 24, s)-Nets in Base 4

(24−7, 24, 240)-Net over F₄ — Constructive and digital

Digital (17, 24, 240)-net over F₄, using

trace code for nets [i] based on digital (1, 8, 80)-net over F₆₄, using
- net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
    - a function field by SÃ©mirat [i]

(24−7, 24, 297)-Net over F₄ — Digital

Digital (17, 24, 297)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁴, 297, F₄, 7) (dual of [297, 273, 8]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(4⁶, 126, F₄, 3) (dual of [126, 120, 4]-code or 126-cap in PG(5,4)), using
    - the Glynn cap [i]
  2. linear OA(4¹⁸, 171, F₄, 7) (dual of [171, 153, 8]-code), using
    - a â€œGraâ€ code from Grasslâ€™s database [i]

(24−7, 24, 25004)-Net in Base 4 — Upper bound on s

There is no (17, 24, 25005)-net in base 4, because

1 times m-reduction [i] would yield (17, 23, 25005)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 70 371578 476396 > 4²³ [i]