MinT - Best Known (13, 13+25, s)-Nets in Base 256

Best Known (13, 13+25, s)-Nets in Base 256

(13, 13+25, 515)-Net over F₂₅₆ — Constructive and digital

Digital (13, 38, 515)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 12, 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, 26, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(13, 13+25, 546)-Net over F₂₅₆ — Digital

Digital (13, 38, 546)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256³⁸, 546, F₂₅₆, 6, 25) (dual of [(546, 6), 3238, 26]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹², 257, F₂₅₆, 6, 12) (dual of [(257, 6), 1530, 13]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(6;1530,256) [i]
  2. linear OOA(256²⁶, 289, F₂₅₆, 6, 25) (dual of [(289, 6), 1708, 26]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,1708P) [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]

(13, 13+25, 552361)-Net in Base 256 — Upper bound on s

There is no (13, 38, 552362)-net in base 256, because

1 times m-reduction [i] would yield (13, 37, 552362)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 127316 021405 891900 853708 052276 857590 724207 453282 706239 579516 268820 851063 657758 050872 208296 > 256³⁷ [i]