MinT - Best Known (12, 12+23, s)-Nets in Base 256

Best Known (12, 12+23, s)-Nets in Base 256

(12, 12+23, 515)-Net over F₂₅₆ — Constructive and digital

Digital (12, 35, 515)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 11, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
      - the rational function field F₂₅₆(x) [i]
    - Niederreiter sequence [i]
2. digital (1, 24, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(12, 12+23, 546)-Net over F₂₅₆ — Digital

Digital (12, 35, 546)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256³⁵, 546, F₂₅₆, 6, 23) (dual of [(546, 6), 3241, 24]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹¹, 257, F₂₅₆, 6, 11) (dual of [(257, 6), 1531, 12]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(6;1531,256) [i]
  2. linear OOA(256²⁴, 289, F₂₅₆, 6, 23) (dual of [(289, 6), 1710, 24]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,1710P) [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]

(12, 12+23, 534712)-Net in Base 256 — Upper bound on s

There is no (12, 35, 534713)-net in base 256, because

1 times m-reduction [i] would yield (12, 34, 534713)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 7588 562129 657279 668909 004196 006701 591458 245281 296661 843442 165982 812866 907811 123216 > 256³⁴ [i]