Best Known (107, 107+19, s)-Nets in Base 9
(107, 107+19, 531471)-Net over F9 — Constructive and digital
Digital (107, 126, 531471)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 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 (94, 113, 531441)-net over F9, using
- net defined by OOA [i] based on linear OOA(9113, 531441, F9, 19, 19) (dual of [(531441, 19), 10097266, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(9113, 4782970, F9, 19) (dual of [4782970, 4782857, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(9113, 4782970, F9, 19) (dual of [4782970, 4782857, 20]-code), using
- net defined by OOA [i] based on linear OOA(9113, 531441, F9, 19, 19) (dual of [(531441, 19), 10097266, 20]-NRT-code), using
- digital (4, 13, 30)-net over F9, using
(107, 107+19, 4783031)-Net over F9 — Digital
Digital (107, 126, 4783031)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9126, 4783031, F9, 19) (dual of [4783031, 4782905, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(10) [i] based on
- linear OA(9113, 4782969, F9, 19) (dual of [4782969, 4782856, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(913, 62, F9, 7) (dual of [62, 49, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(913, 80, F9, 7) (dual of [80, 67, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(913, 80, F9, 7) (dual of [80, 67, 8]-code), using
- construction X applied to Ce(18) ⊂ Ce(10) [i] based on
(107, 107+19, large)-Net in Base 9 — Upper bound on s
There is no (107, 126, large)-net in base 9, because
- 17 times m-reduction [i] would yield (107, 109, large)-net in base 9, but