Best Known (78−12, 78, s)-Nets in Base 9
(78−12, 78, 797178)-Net over F9 — Constructive and digital
Digital (66, 78, 797178)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 16)-net over F9, using
- net from sequence [i] based on digital (1, 15)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 1 and N(F) ≥ 16, using
- net from sequence [i] based on digital (1, 15)-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 (1, 7, 16)-net over F9, using
(78−12, 78, 4783011)-Net over F9 — Digital
Digital (66, 78, 4783011)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(978, 4783011, F9, 12) (dual of [4783011, 4782933, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(5) [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(936, 4782969, F9, 6) (dual of [4782969, 4782933, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(97, 42, F9, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
(78−12, 78, large)-Net in Base 9 — Upper bound on s
There is no (66, 78, large)-net in base 9, because
- 10 times m-reduction [i] would yield (66, 68, large)-net in base 9, but