Best Known (60−17, 60, s)-Nets in Base 3
(60−17, 60, 156)-Net over F3 — Constructive and digital
Digital (43, 60, 156)-net over F3, using
- trace code for nets [i] based on digital (3, 20, 52)-net over F27, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
(60−17, 60, 230)-Net over F3 — Digital
Digital (43, 60, 230)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(360, 230, F3, 17) (dual of [230, 170, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(360, 262, F3, 17) (dual of [262, 202, 18]-code), using
- construction XX applied to C1 = C([106,120]), C2 = C([109,122]), C3 = C1 + C2 = C([109,120]), and C∩ = C1 ∩ C2 = C([106,122]) [i] based on
- linear OA(350, 242, F3, 15) (dual of [242, 192, 16]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {106,107,…,120}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(346, 242, F3, 14) (dual of [242, 196, 15]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {109,110,…,122}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(356, 242, F3, 17) (dual of [242, 186, 18]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {106,107,…,122}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(340, 242, F3, 12) (dual of [242, 202, 13]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {109,110,…,120}, and designed minimum distance d ≥ |I|+1 = 13 [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([106,120]), C2 = C([109,122]), C3 = C1 + C2 = C([109,120]), and C∩ = C1 ∩ C2 = C([106,122]) [i] based on
- discarding factors / shortening the dual code based on linear OA(360, 262, F3, 17) (dual of [262, 202, 18]-code), using
(60−17, 60, 6207)-Net in Base 3 — Upper bound on s
There is no (43, 60, 6208)-net in base 3, because
- 1 times m-reduction [i] would yield (43, 59, 6208)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 14142 490143 304320 017768 075265 > 359 [i]