MinT - Best Known (21, 21+38, s)-Nets in Base 256

Best Known (21, 21+38, s)-Nets in Base 256

(21, 21+38, 516)-Net over F₂₅₆ — Constructive and digital

Digital (21, 59, 516)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (1, 39, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(21, 21+38, 578)-Net over F₂₅₆ — Digital

Digital (21, 59, 578)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁵⁹, 578, F₂₅₆, 6, 38) (dual of [(578, 6), 3409, 39]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256²⁰, 289, F₂₅₆, 6, 19) (dual of [(289, 6), 1714, 20]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,1714P) [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(256³⁹, 289, F₂₅₆, 6, 38) (dual of [(289, 6), 1695, 39]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,1695P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289 (see above)

(21, 21+38, 935168)-Net in Base 256 — Upper bound on s

There is no (21, 59, 935169)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 12194 410576 756735 044950 078598 542583 831507 676213 525432 965350 386673 762218 250316 659398 228260 469867 901673 392592 030204 361704 259070 489701 600020 941056 > 256⁵⁹ [i]