Best Known (81, 81+31, s)-Nets in Base 9
(81, 81+31, 768)-Net over F9 — Constructive and digital
Digital (81, 112, 768)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (3, 18, 28)-net over F9, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- the Hermitian function field over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- net from sequence [i] based on digital (3, 27)-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 (3, 18, 28)-net over F9, using
(81, 81+31, 6537)-Net over F9 — Digital
Digital (81, 112, 6537)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9112, 6537, F9, 31) (dual of [6537, 6425, 32]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(9112, 6576, F9, 31) (dual of [6576, 6464, 32]-code), using
(81, 81+31, large)-Net in Base 9 — Upper bound on s
There is no (81, 112, large)-net in base 9, because
- 29 times m-reduction [i] would yield (81, 83, large)-net in base 9, but