Best Known (35−12, 35, s)-Nets in Base 3
(35−12, 35, 64)-Net over F3 — Constructive and digital
Digital (23, 35, 64)-net over F3, using
- 1 times m-reduction [i] based on digital (23, 36, 64)-net over F3, using
- trace code for nets [i] based on digital (5, 18, 32)-net over F9, using
- net from sequence [i] based on digital (5, 31)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 5 and N(F) ≥ 32, using
- net from sequence [i] based on digital (5, 31)-sequence over F9, using
- trace code for nets [i] based on digital (5, 18, 32)-net over F9, using
(35−12, 35, 88)-Net over F3 — Digital
Digital (23, 35, 88)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(335, 88, F3, 12) (dual of [88, 53, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(335, 93, F3, 12) (dual of [93, 58, 13]-code), using
- construction XX applied to Ce(12) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- linear OA(331, 81, F3, 13) (dual of [81, 50, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(325, 81, F3, 10) (dual of [81, 56, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(321, 81, F3, 8) (dual of [81, 60, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(31, 9, F3, 1) (dual of [9, 8, 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, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(12) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(335, 93, F3, 12) (dual of [93, 58, 13]-code), using
(35−12, 35, 903)-Net in Base 3 — Upper bound on s
There is no (23, 35, 904)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 50306 461135 424817 > 335 [i]