Best Known (165−32, 165, s)-Nets in Base 3
(165−32, 165, 688)-Net over F3 — Constructive and digital
Digital (133, 165, 688)-net over F3, using
- 3 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
(165−32, 165, 2330)-Net over F3 — Digital
Digital (133, 165, 2330)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3165, 2330, F3, 32) (dual of [2330, 2165, 33]-code), using
- 119 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) [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
- 119 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) [i] based on linear OA(3148, 2194, F3, 32) (dual of [2194, 2046, 33]-code), using
(165−32, 165, 283007)-Net in Base 3 — Upper bound on s
There is no (133, 165, 283008)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5 309092 421979 854517 412896 717750 841803 748753 336449 158509 797518 697423 441051 754497 > 3165 [i]