Best Known (36−12, 36, s)-Nets in Base 3
(36−12, 36, 84)-Net over F3 — Constructive and digital
Digital (24, 36, 84)-net over F3, using
- trace code for nets [i] based on digital (0, 12, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
(36−12, 36, 98)-Net over F3 — Digital
Digital (24, 36, 98)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(336, 98, F3, 12) (dual of [98, 62, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(336, 99, F3, 12) (dual of [99, 63, 13]-code), using
- construction XX applied to C1 = C({0,1,2,4,5,53}), C2 = C([0,10]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C({0,1,2,4,5,7,8,10,53}) [i] based on
- linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using the primitive cyclic code C(A) with length 80 = 34−1, defining set A = {0,1,2,4,5,53}, and minimum distance d ≥ |{−1,0,…,6}|+1 = 9 (BCH-bound) [i]
- linear OA(327, 80, F3, 11) (dual of [80, 53, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(331, 80, F3, 12) (dual of [80, 49, 13]-code), using the primitive cyclic code C(A) with length 80 = 34−1, defining set A = {0,1,2,4,5,7,8,10,53}, and minimum distance d ≥ |{−1,0,…,10}|+1 = 13 (BCH-bound) [i]
- linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(30, 4, F3, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(35, 15, F3, 3) (dual of [15, 10, 4]-code or 15-cap in PG(4,3)), using
- construction XX applied to C1 = C({0,1,2,4,5,53}), C2 = C([0,10]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C({0,1,2,4,5,7,8,10,53}) [i] based on
- discarding factors / shortening the dual code based on linear OA(336, 99, F3, 12) (dual of [99, 63, 13]-code), using
(36−12, 36, 1085)-Net in Base 3 — Upper bound on s
There is no (24, 36, 1086)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 150301 107825 075621 > 336 [i]