Best Known (81, 81+20, s)-Nets in Base 3
(81, 81+20, 640)-Net over F3 — Constructive and digital
Digital (81, 101, 640)-net over F3, using
- 31 times duplication [i] based on 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
- trace code for nets [i] based on digital (5, 25, 160)-net over F81, using
(81, 81+20, 1673)-Net over F3 — Digital
Digital (81, 101, 1673)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3101, 1673, F3, 20) (dual of [1673, 1572, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3101, 2219, F3, 20) (dual of [2219, 2118, 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, 29, F3, 3) (dual of [29, 23, 4]-code or 29-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, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(19) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3101, 2219, F3, 20) (dual of [2219, 2118, 21]-code), using
(81, 81+20, 149225)-Net in Base 3 — Upper bound on s
There is no (81, 101, 149226)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 546174 000810 167761 447412 858966 358566 623482 496149 > 3101 [i]