MinT - Best Known (19, 54, s)-Nets in Base 256

Best Known (19, 54, s)-Nets in Base 256

(19, 54, 516)-Net over F₂₅₆ — Constructive and digital

Digital (19, 54, 516)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 18, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (1, 36, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(19, 54, 578)-Net over F₂₅₆ — Digital

Digital (19, 54, 578)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁵⁴, 578, F₂₅₆, 6, 35) (dual of [(578, 6), 3414, 36]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁸, 289, F₂₅₆, 6, 17) (dual of [(289, 6), 1716, 18]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,1716P) [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(256³⁶, 289, F₂₅₆, 6, 35) (dual of [(289, 6), 1698, 36]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,1698P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289 (see above)

(19, 54, 906657)-Net in Base 256 — Upper bound on s

There is no (19, 54, 906658)-net in base 256, because

1 times m-reduction [i] would yield (19, 53, 906658)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 43 323117 903717 963518 244184 458683 804771 076245 792955 070446 005434 361683 126746 307793 911001 001898 649679 540449 079705 436730 551354 206706 > 256⁵³ [i]