MinT - Best Known (48, 86, s)-Nets in Base 16

Best Known (48, 86, s)-Nets in Base 16

(48, 86, 524)-Net over F₁₆ — Constructive and digital

Digital (48, 86, 524)-net over F₁₆, using

trace code for nets [i] based on digital (5, 43, 262)-net over F₂₅₆, using
- net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(48, 86, 646)-Net over F₁₆ — Digital

Digital (48, 86, 646)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸⁶, 646, F₁₆, 38) (dual of [646, 560, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁸⁶, 654, F₁₆, 38) (dual of [654, 568, 39]-code), using
  - trace code [i] based on linear OA(256⁴³, 327, F₂₅₆, 38) (dual of [327, 284, 39]-code), using
    - constructionÂ X applied to AG(F,281P) âŠ‚ AG(F,285P) [i] based on
      1. linear OA(256⁴⁰, 320, F₂₅₆, 38) (dual of [320, 280, 39]-code), using algebraic-geometric code AG(F,281P) [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(256³⁶, 320, F₂₅₆, 34) (dual of [320, 284, 35]-code), using algebraic-geometric code AG(F,285P) [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]

(48, 86, 149046)-Net in Base 16 — Upper bound on s

There is no (48, 86, 149047)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 35 838448 467559 411394 105049 293570 188958 387733 144020 477856 172544 997921 735982 795938 967680 232553 670051 642196 > 16⁸⁶ [i]