MinT - Best Known (61, 61+59, s)-Nets in Base 16

Best Known (61, 61+59, s)-Nets in Base 16

(61, 61+59, 516)-Net over F₁₆ — Constructive and digital

Digital (61, 120, 516)-net over F₁₆, using

trace code for nets [i] based on digital (1, 60, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(61, 61+59, 578)-Net over F₁₆ — Digital

Digital (61, 120, 578)-net over F₁₆, using

trace code for nets [i] based on digital (1, 60, 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]

(61, 61+59, 67916)-Net in Base 16 — Upper bound on s

There is no (61, 120, 67917)-net in base 16, because

1 times m-reduction [i] would yield (61, 119, 67917)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 195181 289639 302707 696820 511883 909029 423391 533859 361581 616532 299407 093309 418615 319274 142836 144664 660302 724202 564352 982563 308859 413992 054477 645696 > 16¹¹⁹ [i]