Best Known (136, 136+32, s)-Nets in Base 3
(136, 136+32, 688)-Net over F3 — Constructive and digital
Digital (136, 168, 688)-net over F3, using
- 4 times m-reduction [i] based on 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
(136, 136+32, 2468)-Net over F3 — Digital
Digital (136, 168, 2468)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3168, 2468, F3, 32) (dual of [2468, 2300, 33]-code), using
- 254 step Varšamov–Edel lengthening with (ri) = (4, 2, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 9 times 0, 1, 12 times 0, 1, 17 times 0, 1, 22 times 0, 1, 28 times 0, 1, 35 times 0, 1, 44 times 0, 1, 53 times 0) [i] based on linear OA(3148, 2194, F3, 32) (dual of [2194, 2046, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(30) [i] based on
- linear OA(3148, 2187, F3, 32) (dual of [2187, 2039, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3141, 2187, F3, 31) (dual of [2187, 2046, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(31) ⊂ Ce(30) [i] based on
- 254 step Varšamov–Edel lengthening with (ri) = (4, 2, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 9 times 0, 1, 12 times 0, 1, 17 times 0, 1, 22 times 0, 1, 28 times 0, 1, 35 times 0, 1, 44 times 0, 1, 53 times 0) [i] based on linear OA(3148, 2194, F3, 32) (dual of [2194, 2046, 33]-code), using
(136, 136+32, 347746)-Net in Base 3 — Upper bound on s
There is no (136, 168, 347747)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 143 345584 278029 104535 994502 614899 109094 564495 219005 746434 167726 591071 701626 224865 > 3168 [i]