Best Known (220−45, 220, s)-Nets in Base 3
(220−45, 220, 688)-Net over F3 — Constructive and digital
Digital (175, 220, 688)-net over F3, using
- 4 times m-reduction [i] based on digital (175, 224, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 56, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 56, 172)-net over F81, using
(220−45, 220, 2232)-Net over F3 — Digital
Digital (175, 220, 2232)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3220, 2232, F3, 45) (dual of [2232, 2012, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3220, 2239, F3, 45) (dual of [2239, 2019, 46]-code), using
- 43 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0) [i] based on linear OA(3210, 2186, F3, 45) (dual of [2186, 1976, 46]-code), using
- 1 times truncation [i] based on linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using
- an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- 1 times truncation [i] based on linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using
- 43 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0) [i] based on linear OA(3210, 2186, F3, 45) (dual of [2186, 1976, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3220, 2239, F3, 45) (dual of [2239, 2019, 46]-code), using
(220−45, 220, 254279)-Net in Base 3 — Upper bound on s
There is no (175, 220, 254280)-net in base 3, because
- 1 times m-reduction [i] would yield (175, 219, 254280)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 308 723411 594777 454883 949363 845328 691887 172952 682836 639317 389945 487627 317705 652037 691872 756593 704553 429681 > 3219 [i]