MinT - Best Known (182, 182+37, s)-Nets in Base 3

Best Known (182, 182+37, s)-Nets in Base 3

(182, 182+37, 1480)-Net over F₃ — Constructive and digital

Digital (182, 219, 1480)-net over F₃, using

t-expansion [i] based on digital (181, 219, 1480)-net over F₃, using
- 1 times m-reduction [i] based on digital (181, 220, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 55, 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]

(182, 182+37, 7476)-Net over F₃ — Digital

Digital (182, 219, 7476)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²¹⁹, 7476, F₃, 2, 37) (dual of [(7476, 2), 14733, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²¹⁹, 9847, F₃, 2, 37) (dual of [(9847, 2), 19475, 38]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3²¹⁹, 19694, F₃, 37) (dual of [19694, 19475, 38]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(3²¹⁸, 19693, F₃, 37) (dual of [19693, 19475, 38]-code), using
      - constructionÂ X applied to C_e(36) âŠ‚ C_e(34) [i] based on
        linear OA(3²¹⁷, 19683, F₃, 37) (dual of [19683, 19466, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
        linear OA(3²⁰⁸, 19683, F₃, 35) (dual of [19683, 19475, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
        linear OA(3¹, 10, F₃, 1) (dual of [10, 9, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(182, 182+37, 2267593)-Net in Base 3 — Upper bound on s

There is no (182, 219, 2267594)-net in base 3, because

1 times m-reduction [i] would yield (182, 218, 2267594)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 102 904559 070599 182498 128589 699797 719978 135286 472087 685668 192206 328165 953420 396916 703657 356691 367516 907045 > 3²¹⁸ [i]