Best Known (14, 22, s)-Nets in Base 3
(14, 22, 56)-Net over F3 — Constructive and digital
Digital (14, 22, 56)-net over F3, using
- trace code for nets [i] based on digital (3, 11, 28)-net over F9, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- the Hermitian function field over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
(14, 22, 66)-Net over F3 — Digital
Digital (14, 22, 66)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(322, 66, F3, 8) (dual of [66, 44, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(322, 89, F3, 8) (dual of [89, 67, 9]-code), using
- construction XX applied to C1 = C({0,1,2,4,53}), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C({0,1,2,4,5,53}) [i] based on
- linear OA(317, 80, F3, 6) (dual of [80, 63, 7]-code), using the primitive cyclic code C(A) with length 80 = 34−1, defining set A = {0,1,2,4,53}, and minimum distance d ≥ |{−1,0,…,4}|+1 = 7 (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(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(313, 80, F3, 5) (dual of [80, 67, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [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(31, 5, F3, 1) (dual of [5, 4, 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({0,1,2,4,53}), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C({0,1,2,4,5,53}) [i] based on
- discarding factors / shortening the dual code based on linear OA(322, 89, F3, 8) (dual of [89, 67, 9]-code), using
(14, 22, 462)-Net in Base 3 — Upper bound on s
There is no (14, 22, 463)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 31569 995769 > 322 [i]