Best Known (43−13, 43, s)-Nets in Base 3
(43−13, 43, 114)-Net over F3 — Constructive and digital
Digital (30, 43, 114)-net over F3, using
- 31 times duplication [i] based on digital (29, 42, 114)-net over F3, using
- trace code for nets [i] based on digital (1, 14, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- trace code for nets [i] based on digital (1, 14, 38)-net over F27, using
(43−13, 43, 154)-Net over F3 — Digital
Digital (30, 43, 154)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(343, 154, F3, 13) (dual of [154, 111, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(343, 251, F3, 13) (dual of [251, 208, 14]-code), using
- construction XX applied to Ce(12) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- 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(331, 243, F3, 10) (dual of [243, 212, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(31, 7, F3, 1) (dual of [7, 6, 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(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(12) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(343, 251, F3, 13) (dual of [251, 208, 14]-code), using
(43−13, 43, 3268)-Net in Base 3 — Upper bound on s
There is no (30, 43, 3269)-net in base 3, because
- 1 times m-reduction [i] would yield (30, 42, 3269)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 109 575726 801858 860481 > 342 [i]