MinT - Best Known (10, 10+16, s)-Nets in Base 49

Best Known (10, 10+16, s)-Nets in Base 49

(10, 10+16, 102)-Net over F₄₉ — Constructive and digital

Digital (10, 26, 102)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]
2. digital (1, 17, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(10, 10+16, 128)-Net over F₄₉ — Digital

Digital (10, 26, 128)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49²⁶, 128, F₄₉, 2, 16) (dual of [(128, 2), 230, 17]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49⁹, 64, F₄₉, 2, 8) (dual of [(64, 2), 119, 9]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,119P) [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(49¹⁷, 64, F₄₉, 2, 16) (dual of [(64, 2), 111, 17]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,111P) [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64 (see above)

(10, 10+16, 24407)-Net in Base 49 — Upper bound on s

There is no (10, 26, 24408)-net in base 49, because

the generalized Rao bound for nets shows that 49^mÂ â‰¥ 88 143198 552520 565329 433477 864925 644556 063745 > 49²⁶ [i]