MinT - Best Known (23, 62, s)-Nets in Base 256

Best Known (23, 62, s)-Nets in Base 256

(23, 62, 518)-Net over F₂₅₆ — Constructive and digital

Digital (23, 62, 518)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 21, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 41, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆ (see above)

(23, 62, 642)-Net over F₂₅₆ — Digital

Digital (23, 62, 642)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁶², 642, F₂₅₆, 4, 39) (dual of [(642, 4), 2506, 40]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256²¹, 321, F₂₅₆, 4, 19) (dual of [(321, 4), 1263, 20]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(4;F,1264P) [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(256⁴¹, 321, F₂₅₆, 4, 39) (dual of [(321, 4), 1243, 40]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(4;F,1244P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321 (see above)

(23, 62, 1676450)-Net in Base 256 — Upper bound on s

There is no (23, 62, 1676451)-net in base 256, because

1 times m-reduction [i] would yield (23, 61, 1676451)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 799 171324 234751 914410 300610 271479 885819 037990 660657 410339 252427 965147 152890 425175 922537 985505 385865 248981 827613 135863 651407 908673 285494 173852 550496 > 256⁶¹ [i]