MinT - Best Known (138−74, 138, s)-Nets in Base 9

Best Known (138−74, 138, s)-Nets in Base 9

(138−74, 138, 165)-Net over F₉ — Constructive and digital

Digital (64, 138, 165)-net over F₉, using

net from sequence [i] based on digital (64, 164)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 64 and N(F) ≥ 165, using
  - T₄ from the second tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(138−74, 138, 193)-Net over F₉ — Digital

Digital (64, 138, 193)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(9¹³⁸, 193, F₉, 3, 74) (dual of [(193, 3), 441, 75]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(9¹³⁵, 192, F₉, 3, 74) (dual of [(192, 3), 441, 75]-NRT-code), using
  - extended algebraic-geometric NRT-code AG_e(3;F,501P) [i] based on function field F/F₉ with g(F) = 61 and N(F) ≥ 192, using
    - Drinfeld modules of rank 1 [i]

(138−74, 138, 6613)-Net in Base 9 — Upper bound on s

There is no (64, 138, 6614)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 487065 162413 100250 667095 275096 458354 830588 747488 645745 591627 745456 754611 798263 415900 013173 412243 163679 123403 941004 611388 149391 662065 > 9¹³⁸ [i]