Best Known (80, 80+20, s)-Nets in Base 3
(80, 80+20, 640)-Net over F3 — Constructive and digital
Digital (80, 100, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 25, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
(80, 80+20, 1573)-Net over F3 — Digital
Digital (80, 100, 1573)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3100, 1573, F3, 20) (dual of [1573, 1473, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3100, 2217, F3, 20) (dual of [2217, 2117, 21]-code), using
- construction XX applied to Ce(19) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(392, 2187, F3, 20) (dual of [2187, 2095, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(371, 2187, F3, 16) (dual of [2187, 2116, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(364, 2187, F3, 14) (dual of [2187, 2123, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(36, 28, F3, 3) (dual of [28, 22, 4]-code or 28-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(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(19) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3100, 2217, F3, 20) (dual of [2217, 2117, 21]-code), using
(80, 80+20, 133698)-Net in Base 3 — Upper bound on s
There is no (80, 100, 133699)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 515379 431190 280470 307568 433529 389619 174810 046421 > 3100 [i]