MinT - Best Known (78−34, 78, s)-Nets in Base 16

Best Known (78−34, 78, s)-Nets in Base 16

(78−34, 78, 524)-Net over F₁₆ — Constructive and digital

Digital (44, 78, 524)-net over F₁₆, using

trace code for nets [i] based on digital (5, 39, 262)-net over F₂₅₆, using
- net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(78−34, 78, 658)-Net over F₁₆ — Digital

Digital (44, 78, 658)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁷⁸, 658, F₁₆, 34) (dual of [658, 580, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁷⁸, 671, F₁₆, 34) (dual of [671, 593, 35]-code), using
  - 23 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 0, 1, 5 times 0, 1, 14 times 0) [i] based on linear OA(16⁷², 642, F₁₆, 34) (dual of [642, 570, 35]-code), using
    - trace code [i] based on linear OA(256³⁶, 321, F₂₅₆, 34) (dual of [321, 285, 35]-code), using
      - extended algebraic-geometric code AG_e(F,286P) [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]

(78−34, 78, 160182)-Net in Base 16 — Upper bound on s

There is no (44, 78, 160183)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8344 167793 147715 453510 195093 937022 166444 112542 964356 368984 547860 652439 438869 230710 483674 640766 > 16⁷⁸ [i]