MinT - Best Known (17, 17+23, s)-Nets in Base 81

Best Known (17, 17+23, s)-Nets in Base 81

(17, 17+23, 370)-Net over F₈₁ — Constructive and digital

Digital (17, 40, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 40, 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]

(17, 17+23, 371)-Net over F₈₁ — Digital

Digital (17, 40, 371)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁴⁰, 371, F₈₁, 23) (dual of [371, 331, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁴⁰, 372, F₈₁, 23) (dual of [372, 332, 24]-code), using
  - constructionÂ X applied to AG(F,345P) âŠ‚ AG(F,347P) [i] based on
    1. linear OA(81³⁹, 369, F₈₁, 23) (dual of [369, 330, 24]-code), using algebraic-geometric code AG(F,345P) [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(81³⁷, 369, F₈₁, 21) (dual of [369, 332, 22]-code), using algebraic-geometric code AG(F,347P) [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]

(17, 17+23, 358397)-Net in Base 81 — Upper bound on s

There is no (17, 40, 358398)-net in base 81, because

1 times m-reduction [i] would yield (17, 39, 358398)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 269 726719 386596 244861 286291 292594 839726 903748 736441 598112 418205 474312 542241 > 81³⁹ [i]