Best Known (186, 186+36, s)-Nets in Base 3
(186, 186+36, 1480)-Net over F3 — Constructive and digital
Digital (186, 222, 1480)-net over F3, using
- t-expansion [i] based on digital (184, 222, 1480)-net over F3, using
- 2 times m-reduction [i] based on digital (184, 224, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 56, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 56, 370)-net over F81, using
- 2 times m-reduction [i] based on digital (184, 224, 1480)-net over F3, using
(186, 186+36, 9854)-Net over F3 — Digital
Digital (186, 222, 9854)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3222, 9854, F3, 2, 36) (dual of [(9854, 2), 19486, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3222, 19708, F3, 36) (dual of [19708, 19486, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3222, 19709, F3, 36) (dual of [19709, 19487, 37]-code), using
- construction XX applied to Ce(36) ⊂ Ce(33) ⊂ Ce(31) [i] based on
- linear OA(3217, 19683, F3, 37) (dual of [19683, 19466, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3190, 19683, F3, 32) (dual of [19683, 19493, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(31, 22, F3, 1) (dual of [22, 21, 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
- linear OA(31, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(36) ⊂ Ce(33) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3222, 19709, F3, 36) (dual of [19709, 19487, 37]-code), using
- OOA 2-folding [i] based on linear OA(3222, 19708, F3, 36) (dual of [19708, 19486, 37]-code), using
(186, 186+36, 2894629)-Net in Base 3 — Upper bound on s
There is no (186, 222, 2894630)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 8335 299729 577507 987970 618914 694137 470034 042372 646506 318878 708698 664002 839986 751062 718700 270476 510484 216397 > 3222 [i]