Best Known (42, 42+13, s)-Nets in Base 3
(42, 42+13, 328)-Net over F3 — Constructive and digital
Digital (42, 55, 328)-net over F3, using
- 1 times m-reduction [i] based on digital (42, 56, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 14, 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, 14, 82)-net over F81, using
(42, 42+13, 530)-Net over F3 — Digital
Digital (42, 55, 530)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(355, 530, F3, 13) (dual of [530, 475, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(355, 753, F3, 13) (dual of [753, 698, 14]-code), using
- construction XX applied to C1 = C([725,7]), C2 = C([1,9]), C3 = C1 + C2 = C([1,7]), and C∩ = C1 ∩ C2 = C([725,9]) [i] based on
- 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 = {−3,−2,…,7}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(336, 728, F3, 9) (dual of [728, 692, 10]-code), using the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(349, 728, F3, 13) (dual of [728, 679, 14]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,9}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(330, 728, F3, 7) (dual of [728, 698, 8]-code), using the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(35, 18, F3, 3) (dual of [18, 13, 4]-code or 18-cap in PG(4,3)), 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, using
- construction XX applied to C1 = C([725,7]), C2 = C([1,9]), C3 = C1 + C2 = C([1,7]), and C∩ = C1 ∩ C2 = C([725,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(355, 753, F3, 13) (dual of [753, 698, 14]-code), using
(42, 42+13, 29457)-Net in Base 3 — Upper bound on s
There is no (42, 55, 29458)-net in base 3, because
- 1 times m-reduction [i] would yield (42, 54, 29458)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 58 150500 174458 180176 034205 > 354 [i]