MinT - Best Known (21, 54, s)-Nets in Base 256

Best Known (21, 54, s)-Nets in Base 256

(21, 54, 519)-Net over F₂₅₆ — Constructive and digital

Digital (21, 54, 519)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 18, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (3, 36, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(21, 54, 642)-Net over F₂₅₆ — Digital

Digital (21, 54, 642)-net over F₂₅₆, using

1 times m-reduction [i] based on digital (21, 55, 642)-net over F₂₅₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁵⁵, 642, F₂₅₆, 3, 34) (dual of [(642, 3), 1871, 35]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(256¹⁹, 321, F₂₅₆, 3, 17) (dual of [(321, 3), 944, 18]-NRT-code), using
      - extended algebraic-geometric NRT-code AG_e(3;F,945P) [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(256³⁶, 321, F₂₅₆, 3, 34) (dual of [(321, 3), 927, 35]-NRT-code), using
      - extended algebraic-geometric NRT-code AG_e(3;F,928P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321 (see above)

(21, 54, 2530999)-Net in Base 256 — Upper bound on s

There is no (21, 54, 2531000)-net in base 256, because

1 times m-reduction [i] would yield (21, 53, 2531000)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 43 322985 637090 621769 360693 285193 799803 037743 602621 208737 137155 316502 721121 371002 014538 338255 638502 955328 366683 532134 945456 261251 > 256⁵³ [i]