Best Known (102, 102+34, s)-Nets in Base 3
(102, 102+34, 328)-Net over F3 — Constructive and digital
Digital (102, 136, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 34, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
(102, 102+34, 631)-Net over F3 — Digital
Digital (102, 136, 631)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3136, 631, F3, 34) (dual of [631, 495, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3136, 748, F3, 34) (dual of [748, 612, 35]-code), using
- construction XX applied to Ce(33) ⊂ Ce(30) ⊂ Ce(28) [i] based on
- linear OA(3130, 729, F3, 34) (dual of [729, 599, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3118, 729, F3, 31) (dual of [729, 611, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3112, 729, F3, 29) (dual of [729, 617, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(34, 17, F3, 2) (dual of [17, 13, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), 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(33) ⊂ Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3136, 748, F3, 34) (dual of [748, 612, 35]-code), using
(102, 102+34, 23527)-Net in Base 3 — Upper bound on s
There is no (102, 136, 23528)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 77367 919421 147233 556318 793826 842256 879234 171783 269642 776390 057169 > 3136 [i]