MinT - Best Known (45, 45+35, s)-Nets in Base 16

Best Known (45, 45+35, s)-Nets in Base 16

(45, 45+35, 524)-Net over F₁₆ — Constructive and digital

Digital (45, 80, 524)-net over F₁₆, using

trace code for nets [i] based on digital (5, 40, 262)-net over F₂₅₆, using
- net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(45, 45+35, 654)-Net over F₁₆ — Digital

Digital (45, 80, 654)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸⁰, 654, F₁₆, 35) (dual of [654, 574, 36]-code), using
- trace code [i] based on linear OA(256⁴⁰, 327, F₂₅₆, 35) (dual of [327, 287, 36]-code), using
  - constructionÂ X applied to AG(F,284P) âŠ‚ AG(F,288P) [i] based on
    1. linear OA(256³⁷, 320, F₂₅₆, 35) (dual of [320, 283, 36]-code), using algebraic-geometric code AG(F,284P) [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]
    2. linear OA(256³³, 320, F₂₅₆, 31) (dual of [320, 287, 32]-code), using algebraic-geometric code AG(F,288P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321 (see above)
    3. linear OA(256³, 7, F₂₅₆, 3) (dual of [7, 4, 4]-code or 7-arc in PG(2,256) or 7-cap in PG(2,256)), using
      - discarding factors / shortening the dual code based on linear OA(256³, 256, F₂₅₆, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
        Reedâ€“Solomon code RS(253,256) [i]

(45, 45+35, 188559)-Net in Base 16 — Upper bound on s

There is no (45, 80, 188560)-net in base 16, because

1 times m-reduction [i] would yield (45, 79, 188560)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 133500 585415 530841 449428 515818 722130 242541 189182 437130 374548 497968 676402 006928 878829 222841 415176 > 16⁷⁹ [i]