Best Known (57, 80, s)-Nets in Base 7
(57, 80, 226)-Net over F7 — Constructive and digital
Digital (57, 80, 226)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (5, 12, 26)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 50)-net over F7, using
- net defined by OOA [i] based on linear OOA(74, 50, F7, 3, 3) (dual of [(50, 3), 146, 4]-NRT-code), using
- digital (1, 8, 13)-net over F7, using
- 5 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (1, 4, 50)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (11, 22, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 11, 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, 11, 50)-net over F49, using
- digital (23, 46, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 23, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49 (see above)
- trace code for nets [i] based on digital (0, 23, 50)-net over F49, using
- digital (5, 12, 26)-net over F7, using
(57, 80, 2173)-Net over F7 — Digital
Digital (57, 80, 2173)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(780, 2173, F7, 23) (dual of [2173, 2093, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(780, 2412, F7, 23) (dual of [2412, 2332, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(777, 2401, F7, 23) (dual of [2401, 2324, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(769, 2401, F7, 20) (dual of [2401, 2332, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(73, 11, F7, 2) (dual of [11, 8, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(780, 2412, F7, 23) (dual of [2412, 2332, 24]-code), using
(57, 80, 959843)-Net in Base 7 — Upper bound on s
There is no (57, 80, 959844)-net in base 7, because
- 1 times m-reduction [i] would yield (57, 79, 959844)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 5 790942 398881 411806 149643 587586 755800 716961 577613 781600 412250 239905 > 779 [i]