Best Known (161, 161+32, s)-Nets in Base 3
(161, 161+32, 1480)-Net over F3 — Constructive and digital
Digital (161, 193, 1480)-net over F3, using
- 31 times duplication [i] based on digital (160, 192, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 48, 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, 48, 370)-net over F81, using
(161, 161+32, 8073)-Net over F3 — Digital
Digital (161, 193, 8073)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3193, 8073, F3, 2, 32) (dual of [(8073, 2), 15953, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3193, 9848, F3, 2, 32) (dual of [(9848, 2), 19503, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3193, 19696, F3, 32) (dual of [19696, 19503, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(28) [i] based on
- 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(3172, 19683, F3, 29) (dual of [19683, 19511, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(31) ⊂ Ce(28) [i] based on
- OOA 2-folding [i] based on linear OA(3193, 19696, F3, 32) (dual of [19696, 19503, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(3193, 9848, F3, 2, 32) (dual of [(9848, 2), 19503, 33]-NRT-code), using
(161, 161+32, 1935443)-Net in Base 3 — Upper bound on s
There is no (161, 193, 1935444)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 121 452300 384489 696641 601578 854813 675081 370213 705504 583876 555045 784589 029414 760215 408987 616385 > 3193 [i]