Best Known (48, 63, s)-Nets in Base 3
(48, 63, 328)-Net over F3 — Constructive and digital
Digital (48, 63, 328)-net over F3, using
- 1 times m-reduction [i] based on digital (48, 64, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 16, 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, 16, 82)-net over F81, using
(48, 63, 523)-Net over F3 — Digital
Digital (48, 63, 523)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(363, 523, F3, 15) (dual of [523, 460, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(363, 748, F3, 15) (dual of [748, 685, 16]-code), using
- construction XX applied to C1 = C([725,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([725,12]) [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 = {−3,−2,…,10}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(349, 728, F3, 13) (dual of [728, 679, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(361, 728, F3, 16) (dual of [728, 667, 17]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,12}, and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(343, 728, F3, 11) (dual of [728, 685, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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(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 (see above)
- construction XX applied to C1 = C([725,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([725,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(363, 748, F3, 15) (dual of [748, 685, 16]-code), using
(48, 63, 28426)-Net in Base 3 — Upper bound on s
There is no (48, 63, 28427)-net in base 3, because
- 1 times m-reduction [i] would yield (48, 62, 28427)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 381585 287308 181039 899675 577763 > 362 [i]