Best Known (69, 69+29, s)-Nets in Base 7
(69, 69+29, 215)-Net over F7 — Constructive and digital
Digital (69, 98, 215)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (1, 10, 13)-net over F7, using
- 3 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (14, 28, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 14, 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, 14, 50)-net over F49, using
- digital (31, 60, 102)-net over F7, using
- trace code for nets [i] based on digital (1, 30, 51)-net over F49, using
- net from sequence [i] based on digital (1, 50)-sequence over F49, using
- trace code for nets [i] based on digital (1, 30, 51)-net over F49, using
- digital (1, 10, 13)-net over F7, using
(69, 69+29, 1962)-Net over F7 — Digital
Digital (69, 98, 1962)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(798, 1962, F7, 29) (dual of [1962, 1864, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(798, 2406, F7, 29) (dual of [2406, 2308, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(797, 2401, F7, 29) (dual of [2401, 2304, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(793, 2401, F7, 27) (dual of [2401, 2308, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(71, 5, F7, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(798, 2406, F7, 29) (dual of [2406, 2308, 30]-code), using
(69, 69+29, 722143)-Net in Base 7 — Upper bound on s
There is no (69, 98, 722144)-net in base 7, because
- 1 times m-reduction [i] would yield (69, 97, 722144)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 9430 111467 693694 215121 637635 082371 140736 820100 720858 478374 369556 549135 695474 748993 > 797 [i]