Best Known (12−5, 12, s)-Nets in Base 7
(12−5, 12, 108)-Net over F7 — Constructive and digital
Digital (7, 12, 108)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 2, 8)-net over F7, using
- digital (5, 10, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 5, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- trace code for nets [i] based on digital (0, 5, 50)-net over F49, using
(12−5, 12, 174)-Net over F7 — Digital
Digital (7, 12, 174)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(712, 174, F7, 5) (dual of [174, 162, 6]-code), using
- construction X applied to C([24,28]) ⊂ C([25,28]) [i] based on
- linear OA(712, 171, F7, 5) (dual of [171, 159, 6]-code), using the BCH-code C(I) with length 171 | 73−1, defining interval I = {24,25,26,27,28}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(79, 171, F7, 4) (dual of [171, 162, 5]-code), using the BCH-code C(I) with length 171 | 73−1, defining interval I = {25,26,27,28}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(70, 3, F7, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to C([24,28]) ⊂ C([25,28]) [i] based on
(12−5, 12, 10480)-Net in Base 7 — Upper bound on s
There is no (7, 12, 10481)-net in base 7, because
- 1 times m-reduction [i] would yield (7, 11, 10481)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 1977 638929 > 711 [i]