MinT - Best Known (125−61, 125, s)-Nets in Base 16

Best Known (125−61, 125, s)-Nets in Base 16

(125−61, 125, 516)-Net over F₁₆ — Constructive and digital

Digital (64, 125, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (64, 126, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 63, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(125−61, 125, 578)-Net over F₁₆ — Digital

Digital (64, 125, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (64, 126, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 63, 289)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(125−61, 125, 76143)-Net in Base 16 — Upper bound on s

There is no (64, 125, 76144)-net in base 16, because

1 times m-reduction [i] would yield (64, 124, 76144)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 204630 865007 016847 359285 447171 931109 391782 683024 908928 077588 029284 987607 063313 802283 441810 767597 638738 973673 280834 109273 528908 578842 836892 897741 645426 > 16¹²⁴ [i]