Best Known (49, 64, s)-Nets in Base 3
(49, 64, 400)-Net over F3 — Constructive and digital
Digital (49, 64, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 16, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
(49, 64, 570)-Net over F3 — Digital
Digital (49, 64, 570)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(364, 570, F3, 15) (dual of [570, 506, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(364, 747, F3, 15) (dual of [747, 683, 16]-code), using
- construction XX applied to C1 = C([351,364]), C2 = C([354,365]), C3 = C1 + C2 = C([354,364]), and C∩ = C1 ∩ C2 = C([351,365]) [i] based on
- linear OA(355, 728, F3, 14) (dual of [728, 673, 15]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {351,352,…,364}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(349, 728, F3, 12) (dual of [728, 679, 13]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {354,355,…,365}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(361, 728, F3, 15) (dual of [728, 667, 16]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {351,352,…,365}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(343, 728, F3, 11) (dual of [728, 685, 12]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {354,355,…,364}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([351,364]), C2 = C([354,365]), C3 = C1 + C2 = C([354,364]), and C∩ = C1 ∩ C2 = C([351,365]) [i] based on
- discarding factors / shortening the dual code based on linear OA(364, 747, F3, 15) (dual of [747, 683, 16]-code), using
(49, 64, 33257)-Net in Base 3 — Upper bound on s
There is no (49, 64, 33258)-net in base 3, because
- 1 times m-reduction [i] would yield (49, 63, 33258)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 144580 608935 576415 004798 164425 > 363 [i]