MinT - Best Known (48−31, 48, s)-Nets in Base 49

Best Known (48−31, 48, s)-Nets in Base 49

(48−31, 48, 102)-Net over F₄₉ — Constructive and digital

Digital (17, 48, 102)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 16, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]
2. digital (1, 32, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(48−31, 48, 128)-Net over F₄₉ — Digital

Digital (17, 48, 128)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49⁴⁸, 128, F₄₉, 2, 31) (dual of [(128, 2), 208, 32]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49¹⁶, 64, F₄₉, 2, 15) (dual of [(64, 2), 112, 16]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,112P) [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64, using
      - sharp bound on the number of rational points on curves with genus 1 [i]
  2. linear OOA(49³², 64, F₄₉, 2, 31) (dual of [(64, 2), 96, 32]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,96P) [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64 (see above)

(48−31, 48, 26445)-Net in Base 49 — Upper bound on s

There is no (17, 48, 26446)-net in base 49, because

1 times m-reduction [i] would yield (17, 47, 26446)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 27 499182 155182 844262 884420 142416 388543 965937 191814 672423 813633 738354 303476 450401 > 49⁴⁷ [i]