Best Known (82, 113, s)-Nets in Base 9
(82, 113, 770)-Net over F9 — Constructive and digital
Digital (82, 113, 770)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (4, 19, 30)-net over F9, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 4 and N(F) ≥ 30, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- digital (63, 94, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 47, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 47, 370)-net over F81, using
- digital (4, 19, 30)-net over F9, using
(82, 113, 6578)-Net over F9 — Digital
Digital (82, 113, 6578)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9113, 6578, F9, 31) (dual of [6578, 6465, 32]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9112, 6576, F9, 31) (dual of [6576, 6464, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(9109, 6561, F9, 31) (dual of [6561, 6452, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(997, 6561, F9, 28) (dual of [6561, 6464, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(93, 15, F9, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(9112, 6577, F9, 30) (dual of [6577, 6465, 31]-code), using Gilbert–Varšamov bound and bm = 9112 > Vbs−1(k−1) = 8656 684908 192711 126197 785130 684955 422093 514870 924331 053934 796746 499294 556463 215359 061977 838204 526148 196225 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(9112, 6576, F9, 31) (dual of [6576, 6464, 32]-code), using
- construction X with Varšamov bound [i] based on
(82, 113, large)-Net in Base 9 — Upper bound on s
There is no (82, 113, large)-net in base 9, because
- 29 times m-reduction [i] would yield (82, 84, large)-net in base 9, but