Best Known (221−45, 221, s)-Nets in Base 3
(221−45, 221, 688)-Net over F3 — Constructive and digital
Digital (176, 221, 688)-net over F3, using
- t-expansion [i] based on digital (175, 221, 688)-net over F3, using
- 3 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
- 3 times m-reduction [i] based on digital (175, 224, 688)-net over F3, using
(221−45, 221, 2263)-Net over F3 — Digital
Digital (176, 221, 2263)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3221, 2263, F3, 45) (dual of [2263, 2042, 46]-code), using
- 51 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 12 times 0, 1, 17 times 0) [i] based on linear OA(3212, 2203, F3, 45) (dual of [2203, 1991, 46]-code), using
- construction X applied to C([0,22]) ⊂ C([0,21]) [i] based on
- linear OA(3211, 2188, F3, 45) (dual of [2188, 1977, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,22], and minimum distance d ≥ |{−22,−21,…,22}|+1 = 46 (BCH-bound) [i]
- linear OA(3197, 2188, F3, 43) (dual of [2188, 1991, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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
- construction X applied to C([0,22]) ⊂ C([0,21]) [i] based on
- 51 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 12 times 0, 1, 17 times 0) [i] based on linear OA(3212, 2203, F3, 45) (dual of [2203, 1991, 46]-code), using
(221−45, 221, 267301)-Net in Base 3 — Upper bound on s
There is no (176, 221, 267302)-net in base 3, because
- 1 times m-reduction [i] would yield (176, 220, 267302)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 926 211855 131274 594516 102487 934355 867851 862737 782892 882107 297526 596776 708435 173596 654492 756946 973307 781301 > 3220 [i]