MinT - Best Known (18−11, 18, s)-Nets in Base 256

Best Known (18−11, 18, s)-Nets in Base 256

(18−11, 18, 516)-Net over F₂₅₆ — Constructive and digital

Digital (7, 18, 516)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 6, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (1, 12, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(18−11, 18, 578)-Net over F₂₅₆ — Digital

Digital (7, 18, 578)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256¹⁸, 578, F₂₅₆, 2, 11) (dual of [(578, 2), 1138, 12]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256⁶, 289, F₂₅₆, 2, 5) (dual of [(289, 2), 572, 6]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,572P) [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(256¹², 289, F₂₅₆, 2, 11) (dual of [(289, 2), 566, 12]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,566P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289 (see above)

(18−11, 18, 1575111)-Net in Base 256 — Upper bound on s

There is no (7, 18, 1575112)-net in base 256, because

1 times m-reduction [i] would yield (7, 17, 1575112)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 87112 332273 106741 357507 898414 977990 561051 > 256¹⁷ [i]