MinT - Best Known (44−16, 44, s)-Nets in Base 16

Best Known (44−16, 44, s)-Nets in Base 16

(44−16, 44, 559)-Net over F₁₆ — Constructive and digital

Digital (28, 44, 559)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 12, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (16, 32, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 16, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(44−16, 44, 579)-Net in Base 16 — Constructive

(28, 44, 579)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (4, 12, 65)-net in base 16, using
  - base change [i] based on digital (0, 8, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
2. digital (16, 32, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 16, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(44−16, 44, 1466)-Net over F₁₆ — Digital

Digital (28, 44, 1466)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁴⁴, 1466, F₁₆, 16) (dual of [1466, 1422, 17]-code), using
- 878 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 0, 1, 13 times 0, 1, 37 times 0, 1, 80 times 0, 1, 126 times 0, 1, 165 times 0, 1, 202 times 0, 1, 244 times 0) [i] based on linear OA(16³⁴, 578, F₁₆, 16) (dual of [578, 544, 17]-code), using
  - trace code [i] based on linear OA(256¹⁷, 289, F₂₅₆, 16) (dual of [289, 272, 17]-code), using
    - extended algebraic-geometric code AG_e(F,272P) [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]

(44−16, 44, 1052584)-Net in Base 16 — Upper bound on s

There is no (28, 44, 1052585)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 95781 200518 017931 467436 224683 557846 858436 112896 816701 > 16⁴⁴ [i]