MinT - Best Known (13, 13+22, s)-Nets in Base 256

Best Known (13, 13+22, s)-Nets in Base 256

(13, 13+22, 516)-Net over F₂₅₆ — Constructive and digital

Digital (13, 35, 516)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (1, 23, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(13, 13+22, 578)-Net over F₂₅₆ — Digital

Digital (13, 35, 578)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256³⁵, 578, F₂₅₆, 4, 22) (dual of [(578, 4), 2277, 23]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹², 289, F₂₅₆, 4, 11) (dual of [(289, 4), 1144, 12]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(4;F,1144P) [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(256²³, 289, F₂₅₆, 4, 22) (dual of [(289, 4), 1133, 23]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(4;F,1133P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289 (see above)

(13, 13+22, 885224)-Net in Base 256 — Upper bound on s

There is no (13, 35, 885225)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 1 942691 581917 227388 578204 504692 767570 064748 817931 013364 176240 392374 364668 712415 937376 > 256³⁵ [i]