Best Known (76, 76+17, s)-Nets in Base 7
(76, 76+17, 14714)-Net over F7 — Constructive and digital
Digital (76, 93, 14714)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- digital (68, 85, 14706)-net over F7, using
- net defined by OOA [i] based on linear OOA(785, 14706, F7, 17, 17) (dual of [(14706, 17), 249917, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(785, 117649, F7, 17) (dual of [117649, 117564, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(785, 117649, F7, 17) (dual of [117649, 117564, 18]-code), using
- net defined by OOA [i] based on linear OOA(785, 14706, F7, 17, 17) (dual of [(14706, 17), 249917, 18]-NRT-code), using
- digital (0, 8, 8)-net over F7, using
(76, 76+17, 117688)-Net over F7 — Digital
Digital (76, 93, 117688)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(793, 117688, F7, 17) (dual of [117688, 117595, 18]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(792, 117686, F7, 17) (dual of [117686, 117594, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(785, 117649, F7, 17) (dual of [117649, 117564, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(755, 117649, F7, 11) (dual of [117649, 117594, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(77, 37, F7, 5) (dual of [37, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(792, 117687, F7, 16) (dual of [117687, 117595, 17]-code), using Gilbert–Varšamov bound and bm = 792 > Vbs−1(k−1) = 4132 976211 996634 955889 500582 555088 850438 019210 954011 272925 744915 260734 160913 [i]
- linear OA(70, 1, F7, 0) (dual of [1, 1, 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
- linear OA(792, 117686, F7, 17) (dual of [117686, 117594, 18]-code), using
- construction X with Varšamov bound [i] based on
(76, 76+17, large)-Net in Base 7 — Upper bound on s
There is no (76, 93, large)-net in base 7, because
- 15 times m-reduction [i] would yield (76, 78, large)-net in base 7, but