Best Known (120−21, 120, s)-Nets in Base 3
(120−21, 120, 688)-Net over F3 — Constructive and digital
Digital (99, 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
(120−21, 120, 3841)-Net over F3 — Digital
Digital (99, 120, 3841)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3120, 3841, F3, 21) (dual of [3841, 3721, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3120, 6593, F3, 21) (dual of [6593, 6473, 22]-code), using
- construction XX applied to Ce(21) ⊂ Ce(16) ⊂ Ce(15) [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(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(36, 31, F3, 3) (dual of [31, 25, 4]-code or 31-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), 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(21) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3120, 6593, F3, 21) (dual of [6593, 6473, 22]-code), using
(120−21, 120, 1078165)-Net in Base 3 — Upper bound on s
There is no (99, 120, 1078166)-net in base 3, because
- 1 times m-reduction [i] would yield (99, 119, 1078166)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 599 006468 785718 923490 502359 181983 857875 933280 033758 428637 > 3119 [i]