MinT - Best Known (69, 245, s)-Nets in Base 4

Best Known (69, 245, s)-Nets in Base 4

(69, 245, 66)-Net over F₄ — Constructive and digital

Digital (69, 245, 66)-net over F₄, using

t-expansion [i] based on digital (49, 245, 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]

(69, 245, 99)-Net over F₄ — Digital

Digital (69, 245, 99)-net over F₄, using

t-expansion [i] based on digital (61, 245, 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]

(69, 245, 394)-Net over F₄ — Upper bound on s (digital)

There is no digital (69, 245, 395)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁵, 395, F₄, 176) (dual of [395, 150, 177]-code), but
- residual code [i] would yield OA(4⁶⁹, 218, S₄, 44), but
  - the linear programming bound shows that MÂ â‰¥ 109109 147274 768315 990931 948474 366203 066087 657712 317992 394776 862369 613323 594430 467474 878725 704856 371200 / 292745 629447 360691 756384 897898 834188 795527 737754 030331 320673 > 4⁶⁹ [i]

(69, 245, 461)-Net in Base 4 — Upper bound on s

There is no (69, 245, 462)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3521 457723 972217 855568 008537 587000 834445 019291 295768 943848 448981 473729 264577 328939 669515 108663 549852 140559 789678 545936 514710 385019 839790 572042 422328 > 4²⁴⁵ [i]