MinT - Best Known (60−37, 60, s)-Nets in Base 256

Best Known (60−37, 60, s)-Nets in Base 256

(60−37, 60, 519)-Net over F₂₅₆ — Constructive and digital

Digital (23, 60, 519)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 20, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (3, 40, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(60−37, 60, 642)-Net over F₂₅₆ — Digital

Digital (23, 60, 642)-net over F₂₅₆, using

1 times m-reduction [i] based on digital (23, 61, 642)-net over F₂₅₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁶¹, 642, F₂₅₆, 3, 38) (dual of [(642, 3), 1865, 39]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(256²¹, 321, F₂₅₆, 3, 19) (dual of [(321, 3), 942, 20]-NRT-code), using
      - extended algebraic-geometric NRT-code AG_e(3;F,943P) [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 OOA(256⁴⁰, 321, F₂₅₆, 3, 38) (dual of [(321, 3), 923, 39]-NRT-code), using
      - extended algebraic-geometric NRT-code AG_e(3;F,924P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321 (see above)

(60−37, 60, 2318802)-Net in Base 256 — Upper bound on s

There is no (23, 60, 2318803)-net in base 256, because

1 times m-reduction [i] would yield (23, 59, 2318803)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 12194 401374 837980 174778 391506 043091 864642 389418 469696 759311 210889 453365 410424 578323 463270 006099 698562 993733 448114 988560 458617 522645 811028 491221 > 256⁵⁹ [i]