Best Known (155−27, 155, s)-Nets in Base 3
(155−27, 155, 702)-Net over F3 — Constructive and digital
Digital (128, 155, 702)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (6, 19, 14)-net over F3, using
- net from sequence [i] based on digital (6, 13)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 6 and N(F) ≥ 14, using
- net from sequence [i] based on digital (6, 13)-sequence over F3, using
- digital (109, 136, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 34, 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, 34, 172)-net over F81, using
- digital (6, 19, 14)-net over F3, using
(155−27, 155, 4399)-Net over F3 — Digital
Digital (128, 155, 4399)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3155, 4399, F3, 27) (dual of [4399, 4244, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3155, 6598, F3, 27) (dual of [6598, 6443, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(3145, 6562, F3, 27) (dual of [6562, 6417, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3113, 6562, F3, 21) (dual of [6562, 6449, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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(33, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3155, 6598, F3, 27) (dual of [6598, 6443, 28]-code), using
(155−27, 155, 1271814)-Net in Base 3 — Upper bound on s
There is no (128, 155, 1271815)-net in base 3, because
- 1 times m-reduction [i] would yield (128, 154, 1271815)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 29 969284 413300 820825 229859 682551 068816 016247 940753 164976 166053 773632 081207 > 3154 [i]