Best Known (52, 69, s)-Nets in Base 3
(52, 69, 328)-Net over F3 — Constructive and digital
Digital (52, 69, 328)-net over F3, using
- 31 times duplication [i] based on digital (51, 68, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 17, 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, 17, 82)-net over F81, using
(52, 69, 454)-Net over F3 — Digital
Digital (52, 69, 454)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(369, 454, F3, 17) (dual of [454, 385, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(369, 738, F3, 17) (dual of [738, 669, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(367, 729, F3, 17) (dual of [729, 662, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(361, 729, F3, 16) (dual of [729, 668, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(355, 729, F3, 14) (dual of [729, 674, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(16) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(369, 738, F3, 17) (dual of [738, 669, 18]-code), using
(52, 69, 21381)-Net in Base 3 — Upper bound on s
There is no (52, 69, 21382)-net in base 3, because
- 1 times m-reduction [i] would yield (52, 68, 21382)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 278 178988 497875 143259 469531 502545 > 368 [i]