Best Known (221−42, 221, s)-Nets in Base 3
(221−42, 221, 896)-Net over F3 — Constructive and digital
Digital (179, 221, 896)-net over F3, using
- 31 times duplication [i] based on digital (178, 220, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 55, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 55, 224)-net over F81, using
(221−42, 221, 3281)-Net over F3 — Digital
Digital (179, 221, 3281)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3221, 3281, F3, 42) (dual of [3281, 3060, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 3285, F3, 42) (dual of [3285, 3064, 43]-code), using
- 1 times truncation [i] based on linear OA(3222, 3286, F3, 43) (dual of [3286, 3064, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(40) [i] based on
- linear OA(3221, 3281, F3, 43) (dual of [3281, 3060, 44]-code), using an extension Ce(42) of the narrow-sense BCH-code C(I) with length 3280 | 38−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(3217, 3281, F3, 41) (dual of [3281, 3064, 42]-code), using an extension Ce(40) of the narrow-sense BCH-code C(I) with length 3280 | 38−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(31, 5, F3, 1) (dual of [5, 4, 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 Ce(42) ⊂ Ce(40) [i] based on
- 1 times truncation [i] based on linear OA(3222, 3286, F3, 43) (dual of [3286, 3064, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 3285, F3, 42) (dual of [3285, 3064, 43]-code), using
(221−42, 221, 455592)-Net in Base 3 — Upper bound on s
There is no (179, 221, 455593)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2778 524634 808355 785023 991737 859447 681597 265504 879723 686893 389898 287044 729924 283418 046711 670358 148050 025091 > 3221 [i]