Best Known (159, 159+38, s)-Nets in Base 3
(159, 159+38, 688)-Net over F3 — Constructive and digital
Digital (159, 197, 688)-net over F3, using
- t-expansion [i] based on digital (157, 197, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (157, 200, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 50, 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, 50, 172)-net over F81, using
- 3 times m-reduction [i] based on digital (157, 200, 688)-net over F3, using
(159, 159+38, 2574)-Net over F3 — Digital
Digital (159, 197, 2574)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3197, 2574, F3, 38) (dual of [2574, 2377, 39]-code), using
- 359 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 16 times 0, 1, 21 times 0, 1, 26 times 0, 1, 33 times 0, 1, 41 times 0, 1, 48 times 0, 1, 56 times 0, 1, 63 times 0) [i] based on linear OA(3176, 2194, F3, 38) (dual of [2194, 2018, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(3176, 2187, F3, 38) (dual of [2187, 2011, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3169, 2187, F3, 37) (dual of [2187, 2018, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(30, 7, F3, 0) (dual of [7, 7, 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
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- 359 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 16 times 0, 1, 21 times 0, 1, 26 times 0, 1, 33 times 0, 1, 41 times 0, 1, 48 times 0, 1, 56 times 0, 1, 63 times 0) [i] based on linear OA(3176, 2194, F3, 38) (dual of [2194, 2018, 39]-code), using
(159, 159+38, 350878)-Net in Base 3 — Upper bound on s
There is no (159, 197, 350879)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 9837 856737 010453 909252 332734 952386 922125 617129 974883 988579 356764 574914 002323 009555 130923 427931 > 3197 [i]