Best Known (70, 70+23, s)-Nets in Base 3
(70, 70+23, 328)-Net over F3 — Constructive and digital
Digital (70, 93, 328)-net over F3, using
- 31 times duplication [i] based on digital (69, 92, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 23, 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
- trace code for nets [i] based on digital (0, 23, 82)-net over F81, using
(70, 70+23, 516)-Net over F3 — Digital
Digital (70, 93, 516)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(393, 516, F3, 23) (dual of [516, 423, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(393, 738, F3, 23) (dual of [738, 645, 24]-code), using
- construction XX applied to Ce(22) ⊂ Ce(21) ⊂ Ce(19) [i] based on
- linear OA(391, 729, F3, 23) (dual of [729, 638, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(385, 729, F3, 22) (dual of [729, 644, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(379, 729, F3, 20) (dual of [729, 650, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(30, 7, F3, 0) (dual of [7, 7, 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(22) ⊂ Ce(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(393, 738, F3, 23) (dual of [738, 645, 24]-code), using
(70, 70+23, 24002)-Net in Base 3 — Upper bound on s
There is no (70, 93, 24003)-net in base 3, because
- 1 times m-reduction [i] would yield (70, 92, 24003)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 78 556830 284290 226015 334978 703234 937098 145563 > 392 [i]