MinT - Best Known (47−30, 47, s)-Nets in Base 256

Best Known (47−30, 47, s)-Nets in Base 256

(47−30, 47, 516)-Net over F₂₅₆ — Constructive and digital

Digital (17, 47, 516)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 16, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (1, 31, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(47−30, 47, 578)-Net over F₂₅₆ — Digital

Digital (17, 47, 578)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁴⁷, 578, F₂₅₆, 5, 30) (dual of [(578, 5), 2843, 31]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁶, 289, F₂₅₆, 5, 15) (dual of [(289, 5), 1429, 16]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(5;F,1429P) [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(256³¹, 289, F₂₅₆, 5, 30) (dual of [(289, 5), 1414, 31]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(5;F,1414P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289 (see above)

(47−30, 47, 885200)-Net in Base 256 — Upper bound on s

There is no (17, 47, 885201)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 153915 898121 251920 067152 832242 397291 316675 637353 825740 810555 736786 937467 072551 456810 371714 684966 060923 039295 334576 > 256⁴⁷ [i]