MinT - Best Known (70, 70+176, s)-Nets in Base 4

Best Known (70, 70+176, s)-Nets in Base 4

(70, 70+176, 66)-Net over F₄ — Constructive and digital

Digital (70, 246, 66)-net over F₄, using

t-expansion [i] based on digital (49, 246, 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]

(70, 70+176, 105)-Net over F₄ — Digital

Digital (70, 246, 105)-net over F₄, using

net from sequence [i] based on digital (70, 104)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
  - Drinfeld modules of rank 1 [i]

(70, 70+176, 414)-Net over F₄ — Upper bound on s (digital)

There is no digital (70, 246, 415)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁶, 415, F₄, 176) (dual of [415, 169, 177]-code), but
- residual code [i] would yield OA(4⁷⁰, 238, S₄, 44), but
  - the linear programming bound shows that MÂ â‰¥ 120510 936963 900302 789124 723066 073447 088671 101178 269184 854788 285280 658913 691603 030508 274974 720000 / 77708 615139 851578 435610 585674 497674 313378 544338 862201 > 4⁷⁰ [i]

(70, 70+176, 469)-Net in Base 4 — Upper bound on s

There is no (70, 246, 470)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13226 433779 086788 460548 665333 454675 993939 782902 105936 989763 765436 849976 481646 026105 834468 817902 872884 553651 776970 288694 756860 244760 937114 968075 371120 > 4²⁴⁶ [i]