Best Known (167, 167+34, s)-Nets in Base 3
(167, 167+34, 1480)-Net over F3 — Constructive and digital
Digital (167, 201, 1480)-net over F3, using
- 31 times duplication [i] based on digital (166, 200, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 50, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 50, 370)-net over F81, using
(167, 167+34, 7145)-Net over F3 — Digital
Digital (167, 201, 7145)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3201, 7145, F3, 2, 34) (dual of [(7145, 2), 14089, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3201, 9847, F3, 2, 34) (dual of [(9847, 2), 19493, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3201, 19694, F3, 34) (dual of [19694, 19493, 35]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3200, 19693, F3, 34) (dual of [19693, 19493, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3190, 19683, F3, 32) (dual of [19683, 19493, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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 X applied to Ce(33) ⊂ Ce(31) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3200, 19693, F3, 34) (dual of [19693, 19493, 35]-code), using
- OOA 2-folding [i] based on linear OA(3201, 19694, F3, 34) (dual of [19694, 19493, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(3201, 9847, F3, 2, 34) (dual of [(9847, 2), 19493, 35]-NRT-code), using
(167, 167+34, 1570969)-Net in Base 3 — Upper bound on s
There is no (167, 201, 1570970)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 796843 642726 787131 155973 548079 093858 748585 624173 851777 970493 852876 694589 517176 670402 031362 327445 > 3201 [i]