MinT - Best Known (12, 32, s)-Nets in Base 256

Best Known (12, 32, s)-Nets in Base 256

(12, 32, 516)-Net over F₂₅₆ — Constructive and digital

Digital (12, 32, 516)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (1, 21, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(12, 32, 578)-Net over F₂₅₆ — Digital

Digital (12, 32, 578)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256³², 578, F₂₅₆, 3, 20) (dual of [(578, 3), 1702, 21]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹¹, 289, F₂₅₆, 3, 10) (dual of [(289, 3), 856, 11]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,856P) [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(256²¹, 289, F₂₅₆, 3, 20) (dual of [(289, 3), 846, 21]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,846P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289 (see above)

(12, 32, 903237)-Net in Base 256 — Upper bound on s

There is no (12, 32, 903238)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 115793 078747 429530 183513 817443 037494 427735 274096 961377 156475 518040 258956 521776 > 256³² [i]