Best Known (110−20, 110, s)-Nets in Base 3
(110−20, 110, 688)-Net over F3 — Constructive and digital
Digital (90, 110, 688)-net over F3, using
- 32 times duplication [i] based on digital (88, 108, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 27, 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, 27, 172)-net over F81, using
(110−20, 110, 3291)-Net over F3 — Digital
Digital (90, 110, 3291)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3110, 3291, F3, 2, 20) (dual of [(3291, 2), 6472, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3110, 6582, F3, 20) (dual of [6582, 6472, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3110, 6583, F3, 20) (dual of [6583, 6473, 21]-code), using
- construction XX applied to Ce(19) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(389, 6561, F3, 17) (dual of [6561, 6472, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(34, 21, F3, 2) (dual of [21, 17, 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
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(19) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3110, 6583, F3, 20) (dual of [6583, 6473, 21]-code), using
- OOA 2-folding [i] based on linear OA(3110, 6582, F3, 20) (dual of [6582, 6472, 21]-code), using
(110−20, 110, 401115)-Net in Base 3 — Upper bound on s
There is no (90, 110, 401116)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 30432 640605 421340 850705 606569 038616 221459 215581 798217 > 3110 [i]