Best Known (29, 29+11, s)-Nets in Base 3
(29, 29+11, 144)-Net over F3 — Constructive and digital
Digital (29, 40, 144)-net over F3, using
- 31 times duplication [i] based on digital (28, 39, 144)-net over F3, using
- trace code for nets [i] based on digital (2, 13, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- trace code for nets [i] based on digital (2, 13, 48)-net over F27, using
(29, 29+11, 235)-Net over F3 — Digital
Digital (29, 40, 235)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(340, 235, F3, 11) (dual of [235, 195, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(340, 262, F3, 11) (dual of [262, 222, 12]-code), using
- construction XX applied to C1 = C([112,120]), C2 = C([115,122]), C3 = C1 + C2 = C([115,120]), and C∩ = C1 ∩ C2 = C([112,122]) [i] based on
- linear OA(330, 242, F3, 9) (dual of [242, 212, 10]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {112,113,…,120}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(326, 242, F3, 8) (dual of [242, 216, 9]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {115,116,…,122}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(336, 242, F3, 11) (dual of [242, 206, 12]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {112,113,…,122}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(320, 242, F3, 6) (dual of [242, 222, 7]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {115,116,…,120}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [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
- construction XX applied to C1 = C([112,120]), C2 = C([115,122]), C3 = C1 + C2 = C([115,120]), and C∩ = C1 ∩ C2 = C([112,122]) [i] based on
- discarding factors / shortening the dual code based on linear OA(340, 262, F3, 11) (dual of [262, 222, 12]-code), using
(29, 29+11, 6855)-Net in Base 3 — Upper bound on s
There is no (29, 40, 6856)-net in base 3, because
- 1 times m-reduction [i] would yield (29, 39, 6856)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4 052725 449102 254865 > 339 [i]