Best Known (34, 34+14, s)-Nets in Base 3
(34, 34+14, 144)-Net over F3 — Constructive and digital
Digital (34, 48, 144)-net over F3, using
- trace code for nets [i] based on digital (2, 16, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
(34, 34+14, 186)-Net over F3 — Digital
Digital (34, 48, 186)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(348, 186, F3, 14) (dual of [186, 138, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(348, 251, F3, 14) (dual of [251, 203, 15]-code), using
- construction XX applied to Ce(13) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- linear OA(346, 243, F3, 14) (dual of [243, 197, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(341, 243, F3, 13) (dual of [243, 202, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(336, 243, F3, 11) (dual of [243, 207, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(30, 6, F3, 0) (dual of [6, 6, 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(13) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(348, 251, F3, 14) (dual of [251, 203, 15]-code), using
(34, 34+14, 3152)-Net in Base 3 — Upper bound on s
There is no (34, 48, 3153)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 79818 314107 522258 793931 > 348 [i]