Best Known (135, 135+32, s)-Nets in Base 3
(135, 135+32, 688)-Net over F3 — Constructive and digital
Digital (135, 167, 688)-net over F3, using
- t-expansion [i] based on digital (133, 167, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (133, 168, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 42, 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, 42, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (133, 168, 688)-net over F3, using
(135, 135+32, 2413)-Net over F3 — Digital
Digital (135, 167, 2413)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3167, 2413, F3, 32) (dual of [2413, 2246, 33]-code), using
- 200 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) [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
- 200 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) [i] based on linear OA(3148, 2194, F3, 32) (dual of [2194, 2046, 33]-code), using
(135, 135+32, 324669)-Net in Base 3 — Upper bound on s
There is no (135, 167, 324670)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 47 782254 312466 785568 059622 707067 559840 766159 223418 203858 102114 720456 800645 101985 > 3167 [i]