Best Known (108, 108+25, s)-Nets in Base 3
(108, 108+25, 688)-Net over F3 — Constructive and digital
Digital (108, 133, 688)-net over F3, using
- 31 times duplication [i] based on digital (107, 132, 688)-net over F3, using
- t-expansion [i] based on digital (106, 132, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 33, 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, 33, 172)-net over F81, using
- t-expansion [i] based on digital (106, 132, 688)-net over F3, using
(108, 108+25, 3119)-Net over F3 — Digital
Digital (108, 133, 3119)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3133, 3119, F3, 2, 25) (dual of [(3119, 2), 6105, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3133, 3290, F3, 2, 25) (dual of [(3290, 2), 6447, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3133, 6580, F3, 25) (dual of [6580, 6447, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3133, 6581, F3, 25) (dual of [6581, 6448, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3129, 6561, F3, 25) (dual of [6561, 6432, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3113, 6561, F3, 22) (dual of [6561, 6448, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(34, 20, F3, 2) (dual of [20, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3133, 6581, F3, 25) (dual of [6581, 6448, 26]-code), using
- OOA 2-folding [i] based on linear OA(3133, 6580, F3, 25) (dual of [6580, 6447, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(3133, 3290, F3, 2, 25) (dual of [(3290, 2), 6447, 26]-NRT-code), using
(108, 108+25, 468440)-Net in Base 3 — Upper bound on s
There is no (108, 133, 468441)-net in base 3, because
- 1 times m-reduction [i] would yield (108, 132, 468441)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 955 013948 133533 740010 722850 995299 000922 683913 768064 374342 196273 > 3132 [i]