Best Known (229−20, 229, s)-Nets in Base 3
(229−20, 229, 838974)-Net over F3 — Constructive and digital
Digital (209, 229, 838974)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (23, 33, 114)-net over F3, using
- trace code for nets [i] based on digital (1, 11, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- trace code for nets [i] based on digital (1, 11, 38)-net over F27, using
- digital (176, 196, 838860)-net over F3, using
- net defined by OOA [i] based on linear OOA(3196, 838860, F3, 20, 20) (dual of [(838860, 20), 16777004, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3196, 8388600, F3, 20) (dual of [8388600, 8388404, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3196, large, F3, 20) (dual of [large, large−196, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(3196, large, F3, 20) (dual of [large, large−196, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3196, 8388600, F3, 20) (dual of [8388600, 8388404, 21]-code), using
- net defined by OOA [i] based on linear OOA(3196, 838860, F3, 20, 20) (dual of [(838860, 20), 16777004, 21]-NRT-code), using
- digital (23, 33, 114)-net over F3, using
(229−20, 229, 4194447)-Net over F3 — Digital
Digital (209, 229, 4194447)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3229, 4194447, F3, 2, 20) (dual of [(4194447, 2), 8388665, 21]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(333, 146, F3, 2, 10) (dual of [(146, 2), 259, 11]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(333, 146, F3, 10) (dual of [146, 113, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(333, 251, F3, 10) (dual of [251, 218, 11]-code), using
- construction XX applied to Ce(9) ⊂ Ce(7) ⊂ Ce(6) [i] based on
- linear OA(331, 243, F3, 10) (dual of [243, 212, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(326, 243, F3, 8) (dual of [243, 217, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(321, 243, F3, 7) (dual of [243, 222, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(31, 7, F3, 1) (dual of [7, 6, 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
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(9) ⊂ Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(333, 251, F3, 10) (dual of [251, 218, 11]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(333, 146, F3, 10) (dual of [146, 113, 11]-code), using
- linear OOA(3196, 4194301, F3, 2, 20) (dual of [(4194301, 2), 8388406, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3196, 8388602, F3, 20) (dual of [8388602, 8388406, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3196, large, F3, 20) (dual of [large, large−196, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(3196, large, F3, 20) (dual of [large, large−196, 21]-code), using
- OOA 2-folding [i] based on linear OA(3196, 8388602, F3, 20) (dual of [8388602, 8388406, 21]-code), using
- linear OOA(333, 146, F3, 2, 10) (dual of [(146, 2), 259, 11]-NRT-code), using
- (u, u+v)-construction [i] based on
(229−20, 229, large)-Net in Base 3 — Upper bound on s
There is no (209, 229, large)-net in base 3, because
- 18 times m-reduction [i] would yield (209, 211, large)-net in base 3, but