Best Known (121−22, 121, s)-Nets in Base 3
(121−22, 121, 688)-Net over F3 — Constructive and digital
Digital (99, 121, 688)-net over F3, using
- 31 times duplication [i] based on digital (98, 120, 688)-net over F3, using
- t-expansion [i] based on digital (97, 120, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 30, 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, 30, 172)-net over F81, using
- t-expansion [i] based on digital (97, 120, 688)-net over F3, using
(121−22, 121, 3296)-Net over F3 — Digital
Digital (99, 121, 3296)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3121, 3296, F3, 2, 22) (dual of [(3296, 2), 6471, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3121, 6592, F3, 22) (dual of [6592, 6471, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3121, 6593, F3, 22) (dual of [6593, 6472, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- 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(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(38, 32, F3, 4) (dual of [32, 24, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3121, 6593, F3, 22) (dual of [6593, 6472, 23]-code), using
- OOA 2-folding [i] based on linear OA(3121, 6592, F3, 22) (dual of [6592, 6471, 23]-code), using
(121−22, 121, 434817)-Net in Base 3 — Upper bound on s
There is no (99, 121, 434818)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5391 036198 074366 700700 404401 212063 016124 853914 233725 854849 > 3121 [i]