MinT - Best Known (122−59, 122, s)-Nets in Base 16

Best Known (122−59, 122, s)-Nets in Base 16

(122−59, 122, 518)-Net over F₁₆ — Constructive and digital

Digital (63, 122, 518)-net over F₁₆, using

trace code for nets [i] based on digital (2, 61, 259)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(122−59, 122, 642)-Net over F₁₆ — Digital

Digital (63, 122, 642)-net over F₁₆, using

trace code for nets [i] based on digital (2, 61, 321)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
    - sharp bound on the number of rational points on curves with genus 2 [i]

(122−59, 122, 82230)-Net in Base 16 — Upper bound on s

There is no (63, 122, 82231)-net in base 16, because

1 times m-reduction [i] would yield (63, 121, 82231)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 49 955035 636718 580347 865532 440679 578140 923461 368677 803546 491209 630168 607353 802110 724615 862665 818150 251668 720320 063244 437637 841790 319793 757859 407336 > 16¹²¹ [i]