Best Known (33−10, 33, s)-Nets in Base 3
(33−10, 33, 114)-Net over F3 — Constructive and digital
Digital (23, 33, 114)-net over F3, using
- trace code for nets [i] based on digital (1, 11, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
(33−10, 33, 146)-Net over F3 — Digital
Digital (23, 33, 146)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(333, 146, F3, 10) (dual of [146, 113, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(333, 251, F3, 10) (dual of [251, 218, 11]-code), using
- construction XX applied to Ce(9) ⊂ Ce(7) ⊂ Ce(6) [i] based on
- linear OA(331, 243, F3, 10) (dual of [243, 212, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(326, 243, F3, 8) (dual of [243, 217, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(321, 243, F3, 7) (dual of [243, 222, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- 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
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(9) ⊂ Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(333, 251, F3, 10) (dual of [251, 218, 11]-code), using
(33−10, 33, 1831)-Net in Base 3 — Upper bound on s
There is no (23, 33, 1832)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5570 804635 983825 > 333 [i]