MinT - Best Known (70, 70+168, s)-Nets in Base 4

Best Known (70, 70+168, s)-Nets in Base 4

(70, 70+168, 66)-Net over F₄ — Constructive and digital

Digital (70, 238, 66)-net over F₄, using

t-expansion [i] based on digital (49, 238, 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+168, 105)-Net over F₄ — Digital

Digital (70, 238, 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+168, 453)-Net over F₄ — Upper bound on s (digital)

There is no digital (70, 238, 454)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²³⁸, 454, F₄, 168) (dual of [454, 216, 169]-code), but
- residual code [i] would yield OA(4⁷⁰, 285, S₄, 42), but
  - the linear programming bound shows that MÂ â‰¥ 2 520198 658998 621179 007148 715142 691743 667539 102383 827900 254205 426285 806015 463738 326023 123475 890176 / 1 791905 782922 287300 464200 415513 167226 158849 588749 853743 > 4⁷⁰ [i]

(70, 70+168, 476)-Net in Base 4 — Upper bound on s

There is no (70, 238, 477)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 200683 720008 909449 789281 832803 390323 557012 821384 902907 924305 752762 624815 423774 985484 585143 747944 135288 719816 492763 797794 371215 709941 027093 389971 > 4²³⁸ [i]