Best Known (107, 107+29, s)-Nets in Base 3
(107, 107+29, 640)-Net over F3 — Constructive and digital
Digital (107, 136, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 34, 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
(107, 107+29, 1303)-Net over F3 — Digital
Digital (107, 136, 1303)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3136, 1303, F3, 29) (dual of [1303, 1167, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3136, 2197, F3, 29) (dual of [2197, 2061, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(27) ⊂ Ce(25) [i] based on
- linear OA(3134, 2187, F3, 29) (dual of [2187, 2053, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(3120, 2187, F3, 26) (dual of [2187, 2067, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(28) ⊂ Ce(27) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3136, 2197, F3, 29) (dual of [2197, 2061, 30]-code), using
(107, 107+29, 120556)-Net in Base 3 — Upper bound on s
There is no (107, 136, 120557)-net in base 3, because
- 1 times m-reduction [i] would yield (107, 135, 120557)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 25786 145096 617923 322230 111943 883897 009890 730897 555942 300836 192497 > 3135 [i]