Best Known (55, 67, s)-Nets in Base 7
(55, 67, 19617)-Net over F7 — Constructive and digital
Digital (55, 67, 19617)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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 (49, 61, 19609)-net over F7, using
- net defined by OOA [i] based on linear OOA(761, 19609, F7, 12, 12) (dual of [(19609, 12), 235247, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(761, 117654, F7, 12) (dual of [117654, 117593, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(761, 117655, F7, 12) (dual of [117655, 117594, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(761, 117649, F7, 12) (dual of [117649, 117588, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(70, 6, F7, 0) (dual of [6, 6, 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 Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(761, 117655, F7, 12) (dual of [117655, 117594, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(761, 117654, F7, 12) (dual of [117654, 117593, 13]-code), using
- net defined by OOA [i] based on linear OOA(761, 19609, F7, 12, 12) (dual of [(19609, 12), 235247, 13]-NRT-code), using
- digital (0, 6, 8)-net over F7, using
(55, 67, 117681)-Net over F7 — Digital
Digital (55, 67, 117681)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(767, 117681, F7, 12) (dual of [117681, 117614, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(765, 117677, F7, 12) (dual of [117677, 117612, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(761, 117649, F7, 12) (dual of [117649, 117588, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(74, 28, F7, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,7)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(765, 117679, F7, 11) (dual of [117679, 117614, 12]-code), using Gilbert–Varšamov bound and bm = 765 > Vbs−1(k−1) = 8482 860616 671076 086097 826866 924808 270876 233137 478993 [i]
- linear OA(70, 2, F7, 0) (dual of [2, 2, 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(765, 117677, F7, 12) (dual of [117677, 117612, 13]-code), using
- construction X with Varšamov bound [i] based on
(55, 67, large)-Net in Base 7 — Upper bound on s
There is no (55, 67, large)-net in base 7, because
- 10 times m-reduction [i] would yield (55, 57, large)-net in base 7, but