Best Known (151−37, 151, s)-Nets in Base 3
(151−37, 151, 328)-Net over F3 — Constructive and digital
Digital (114, 151, 328)-net over F3, using
- 1 times m-reduction [i] based on digital (114, 152, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 38, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 38, 82)-net over F81, using
(151−37, 151, 740)-Net over F3 — Digital
Digital (114, 151, 740)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3151, 740, F3, 37) (dual of [740, 589, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3151, 763, F3, 37) (dual of [763, 612, 38]-code), using
- construction XX applied to C1 = C([722,28]), C2 = C([0,30]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([722,30]) [i] based on
- linear OA(3136, 728, F3, 35) (dual of [728, 592, 36]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−6,−5,…,28}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(3118, 728, F3, 31) (dual of [728, 610, 32]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,30], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3142, 728, F3, 37) (dual of [728, 586, 38]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−6,−5,…,30}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3112, 728, F3, 29) (dual of [728, 616, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- 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([722,28]), C2 = C([0,30]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([722,30]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3151, 763, F3, 37) (dual of [763, 612, 38]-code), using
(151−37, 151, 35718)-Net in Base 3 — Upper bound on s
There is no (114, 151, 35719)-net in base 3, because
- 1 times m-reduction [i] would yield (114, 150, 35719)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 370010 801554 298112 274052 876648 973752 575352 722183 902757 439193 313579 912045 > 3150 [i]