Best Known (206−34, 206, s)-Nets in Base 3
(206−34, 206, 1480)-Net over F3 — Constructive and digital
Digital (172, 206, 1480)-net over F3, using
- 2 times m-reduction [i] based on digital (172, 208, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 52, 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, 52, 370)-net over F81, using
(206−34, 206, 8536)-Net over F3 — Digital
Digital (172, 206, 8536)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3206, 8536, F3, 2, 34) (dual of [(8536, 2), 16866, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3206, 9855, F3, 2, 34) (dual of [(9855, 2), 19504, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3206, 19710, F3, 34) (dual of [19710, 19504, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [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(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(37, 27, F3, 4) (dual of [27, 20, 5]-code), using
- an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- OOA 2-folding [i] based on linear OA(3206, 19710, F3, 34) (dual of [19710, 19504, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(3206, 9855, F3, 2, 34) (dual of [(9855, 2), 19504, 35]-NRT-code), using
(206−34, 206, 2170195)-Net in Base 3 — Upper bound on s
There is no (172, 206, 2170196)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 193 633208 252582 078676 396497 024517 735803 982673 972827 208006 682748 634785 026168 217767 055884 703796 693929 > 3206 [i]