Best Known (24−7, 24, s)-Nets in Base 3
(24−7, 24, 114)-Net over F3 — Constructive and digital
Digital (17, 24, 114)-net over F3, using
- trace code for nets [i] based on digital (1, 8, 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
(24−7, 24, 200)-Net over F3 — Digital
Digital (17, 24, 200)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(324, 200, F3, 7) (dual of [200, 176, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(324, 248, F3, 7) (dual of [248, 224, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([1,3]) [i] based on
- linear OA(321, 244, F3, 7) (dual of [244, 223, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 244 | 310−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(320, 244, F3, 3) (dual of [244, 224, 4]-code or 244-cap in PG(19,3)), using the narrow-sense BCH-code C(I) with length 244 | 310−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(33, 4, F3, 3) (dual of [4, 1, 4]-code or 4-arc in PG(2,3) or 4-cap in PG(2,3)), using
- dual of repetition code with length 4 [i]
- oval in PG(2, 3) [i]
- construction X applied to C([0,3]) ⊂ C([1,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(324, 248, F3, 7) (dual of [248, 224, 8]-code), using
(24−7, 24, 4130)-Net in Base 3 — Upper bound on s
There is no (17, 24, 4131)-net in base 3, because
- 1 times m-reduction [i] would yield (17, 23, 4131)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 94165 618987 > 323 [i]