MinT - Best Known (30−19, 30, s)-Nets in Base 256

Best Known (30−19, 30, s)-Nets in Base 256

(30−19, 30, 516)-Net over F₂₅₆ — Constructive and digital

Digital (11, 30, 516)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 10, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (1, 20, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(30−19, 30, 578)-Net over F₂₅₆ — Digital

Digital (11, 30, 578)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256³⁰, 578, F₂₅₆, 4, 19) (dual of [(578, 4), 2282, 20]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁰, 289, F₂₅₆, 4, 9) (dual of [(289, 4), 1146, 10]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(4;F,1146P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]
  2. linear OOA(256²⁰, 289, F₂₅₆, 4, 19) (dual of [(289, 4), 1136, 20]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(4;F,1136P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289 (see above)

(30−19, 30, 935607)-Net in Base 256 — Upper bound on s

There is no (11, 30, 935608)-net in base 256, because

1 times m-reduction [i] would yield (11, 29, 935608)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 6901 774867 238766 466268 354302 886566 064319 580186 507074 077740 494104 295886 > 256²⁹ [i]