Best Known (159, 159+36, s)-Nets in Base 3
(159, 159+36, 701)-Net over F3 — Constructive and digital
Digital (159, 195, 701)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (5, 23, 13)-net over F3, using
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 4, N(F) = 12, and 1 place with degree 2 [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- digital (136, 172, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 43, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 43, 172)-net over F81, using
- digital (5, 23, 13)-net over F3, using
(159, 159+36, 3539)-Net over F3 — Digital
Digital (159, 195, 3539)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3195, 3539, F3, 36) (dual of [3539, 3344, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3195, 6566, F3, 36) (dual of [6566, 6371, 37]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(3193, 6562, F3, 37) (dual of [6562, 6369, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(3177, 6562, F3, 33) (dual of [6562, 6385, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(32, 4, F3, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
- extended Reed–Solomon code RSe(2,3) [i]
- Hamming code H(2,3) [i]
- Simplex code S(2,3) [i]
- the Tetracode [i]
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3195, 6566, F3, 36) (dual of [6566, 6371, 37]-code), using
(159, 159+36, 557057)-Net in Base 3 — Upper bound on s
There is no (159, 195, 557058)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1093 080019 676025 605618 180935 831604 077583 556749 139521 258395 181402 781295 776667 471127 538743 992501 > 3195 [i]