MinT - Best Known (18−11, 18, s)-Nets in Base 49

Best Known (18−11, 18, s)-Nets in Base 49

(18−11, 18, 102)-Net over F₄₉ — Constructive and digital

Digital (7, 18, 102)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 6, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]
2. digital (1, 12, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(18−11, 18, 128)-Net over F₄₉ — Digital

Digital (7, 18, 128)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49¹⁸, 128, F₄₉, 11) (dual of [128, 110, 12]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(49⁶, 64, F₄₉, 5) (dual of [64, 58, 6]-code), using
    - extended algebraic-geometric code AG_e(F,58P) [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 OA(49¹², 64, F₄₉, 11) (dual of [64, 52, 12]-code), using
    - extended algebraic-geometric code AG_e(F,52P) [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64 (see above)

(18−11, 18, 30285)-Net in Base 49 — Upper bound on s

There is no (7, 18, 30286)-net in base 49, because

1 times m-reduction [i] would yield (7, 17, 30286)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 54123 439665 381389 504793 678625 > 49¹⁷ [i]