MinT - Best Known (25, 25+34, s)-Nets in Base 81

Best Known (25, 25+34, s)-Nets in Base 81

(25, 25+34, 370)-Net over F₈₁ — Constructive and digital

Digital (25, 59, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 59, 370)-net over F₈₁, using
- net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
    - a function field by GarcÃa and Quoos [i]

(25, 25+34, 447)-Net over F₈₁ — Digital

Digital (25, 59, 447)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁵⁹, 447, F₈₁, 34) (dual of [447, 388, 35]-code), using
- 67 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (5, 0, 0, 1, 7 times 0, 1, 18 times 0, 1, 36 times 0) [i] based on linear OA(81⁵¹, 372, F₈₁, 34) (dual of [372, 321, 35]-code), using
  - constructionÂ X applied to AG(F,334P) âŠ‚ AG(F,336P) [i] based on
    1. linear OA(81⁵⁰, 369, F₈₁, 34) (dual of [369, 319, 35]-code), using algebraic-geometric code AG(F,334P) [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]
    2. linear OA(81⁴⁸, 369, F₈₁, 32) (dual of [369, 321, 33]-code), using algebraic-geometric code AG(F,336P) [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370 (see above)
    3. linear OA(81¹, 3, F₈₁, 1) (dual of [3, 2, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(81¹, s, F₈₁, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(25, 25+34, 377060)-Net in Base 81 — Upper bound on s

There is no (25, 59, 377061)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 39867 547467 207894 517016 080095 764814 939589 504548 533531 553289 839791 339636 769894 471809 895884 802854 926080 973335 689361 > 81⁵⁹ [i]