Best Known (132−27, 132, s)-Nets in Base 3
(132−27, 132, 640)-Net over F3 — Constructive and digital
Digital (105, 132, 640)-net over F3, using
- t-expansion [i] based on digital (104, 132, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 33, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 33, 160)-net over F81, using
(132−27, 132, 1586)-Net over F3 — Digital
Digital (105, 132, 1586)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3132, 1586, F3, 27) (dual of [1586, 1454, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3132, 2209, F3, 27) (dual of [2209, 2077, 28]-code), using
- construction XX applied to Ce(27) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(3127, 2187, F3, 28) (dual of [2187, 2060, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3113, 2187, F3, 25) (dual of [2187, 2074, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3106, 2187, F3, 23) (dual of [2187, 2081, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(31, 18, F3, 1) (dual of [18, 17, 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(27) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3132, 2209, F3, 27) (dual of [2209, 2077, 28]-code), using
(132−27, 132, 182079)-Net in Base 3 — Upper bound on s
There is no (105, 132, 182080)-net in base 3, because
- 1 times m-reduction [i] would yield (105, 131, 182080)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 318 351115 993696 864181 237027 759623 416662 339929 347480 361376 463489 > 3131 [i]