Best Known (65, 65+12, s)-Nets in Base 9
(65, 65+12, 797172)-Net over F9 — Constructive and digital
Digital (65, 77, 797172)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 10)-net over F9, using
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 0 and N(F) ≥ 10, using
- the rational function field F9(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- digital (59, 71, 797162)-net over F9, using
- net defined by OOA [i] based on linear OOA(971, 797162, F9, 12, 12) (dual of [(797162, 12), 9565873, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(971, 4782972, F9, 12) (dual of [4782972, 4782901, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(971, 4782976, F9, 12) (dual of [4782976, 4782905, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(971, 4782969, F9, 12) (dual of [4782969, 4782898, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(964, 4782969, F9, 11) (dual of [4782969, 4782905, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 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
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(971, 4782976, F9, 12) (dual of [4782976, 4782905, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(971, 4782972, F9, 12) (dual of [4782972, 4782901, 13]-code), using
- net defined by OOA [i] based on linear OOA(971, 797162, F9, 12, 12) (dual of [(797162, 12), 9565873, 13]-NRT-code), using
- digital (0, 6, 10)-net over F9, using
(65, 65+12, 4783003)-Net over F9 — Digital
Digital (65, 77, 4783003)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(977, 4783003, F9, 12) (dual of [4783003, 4782926, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- linear OA(971, 4782969, F9, 12) (dual of [4782969, 4782898, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(943, 4782969, F9, 7) (dual of [4782969, 4782926, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(96, 34, F9, 4) (dual of [34, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- 1 times truncation [i] based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
(65, 65+12, large)-Net in Base 9 — Upper bound on s
There is no (65, 77, large)-net in base 9, because
- 10 times m-reduction [i] would yield (65, 67, large)-net in base 9, but