Best Known (19, 19+10, s)-Nets in Base 3
(19, 19+10, 60)-Net over F3 — Constructive and digital
Digital (19, 29, 60)-net over F3, using
- 1 times m-reduction [i] based on digital (19, 30, 60)-net over F3, using
- trace code for nets [i] based on digital (4, 15, 30)-net over F9, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 4 and N(F) ≥ 30, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- trace code for nets [i] based on digital (4, 15, 30)-net over F9, using
(19, 19+10, 82)-Net over F3 — Digital
Digital (19, 29, 82)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(329, 82, F3, 10) (dual of [82, 53, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(329, 96, F3, 10) (dual of [96, 67, 11]-code), using
- construction XX applied to C1 = C({0,1,2,4,26,53}), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C({0,1,2,4,5,26,53}) [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 = {0,1,2,4,26,53}, and minimum distance d ≥ |{−3,−2,…,4}|+1 = 9 (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(325, 80, F3, 10) (dual of [80, 55, 11]-code), using the primitive cyclic code C(A) with length 80 = 34−1, defining set A = {0,1,2,4,5,26,53}, and minimum distance d ≥ |{−3,−2,…,6}|+1 = 11 (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(33, 11, F3, 2) (dual of [11, 8, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), 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({0,1,2,4,26,53}), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C({0,1,2,4,5,26,53}) [i] based on
- discarding factors / shortening the dual code based on linear OA(329, 96, F3, 10) (dual of [96, 67, 11]-code), using
(19, 19+10, 757)-Net in Base 3 — Upper bound on s
There is no (19, 29, 758)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 68 728410 276165 > 329 [i]