MinT - Best Known (247−176, 247, s)-Nets in Base 4

Best Known (247−176, 247, s)-Nets in Base 4

(247−176, 247, 66)-Net over F₄ — Constructive and digital

Digital (71, 247, 66)-net over F₄, using

t-expansion [i] based on digital (49, 247, 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]

(247−176, 247, 105)-Net over F₄ — Digital

Digital (71, 247, 105)-net over F₄, using

t-expansion [i] based on digital (70, 247, 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]

(247−176, 247, 429)-Net over F₄ — Upper bound on s (digital)

There is no digital (71, 247, 430)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁷, 430, F₄, 176) (dual of [430, 183, 177]-code), but
- residual code [i] would yield OA(4⁷¹, 253, S₄, 44), but
  - the linear programming bound shows that MÂ â‰¥ 27385 603782 383297 661406 789870 071468 501212 578772 565955 567440 744927 980401 898659 225594 231422 713856 / 4741 255951 754751 821637 567787 914724 985401 817073 767747 > 4⁷¹ [i]

(247−176, 247, 478)-Net in Base 4 — Upper bound on s

There is no (71, 247, 479)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 57243 033892 319339 209774 415942 805419 430303 312667 292508 824913 705456 752308 327276 676371 647706 628476 644990 192568 796201 900252 341630 098970 329889 772594 754397 > 4²⁴⁷ [i]