Best Known (67, 82, s)-Nets in Base 7
(67, 82, 16820)-Net over F7 — Constructive and digital
Digital (67, 82, 16820)-net over F7, using
- 71 times duplication [i] based on digital (66, 81, 16820)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 13)-net over F7, using
- 5 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (58, 73, 16807)-net over F7, using
- net defined by OOA [i] based on linear OOA(773, 16807, F7, 15, 15) (dual of [(16807, 15), 252032, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(773, 117650, F7, 15) (dual of [117650, 117577, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(773, 117650, F7, 15) (dual of [117650, 117577, 16]-code), using
- net defined by OOA [i] based on linear OOA(773, 16807, F7, 15, 15) (dual of [(16807, 15), 252032, 16]-NRT-code), using
- digital (1, 8, 13)-net over F7, using
- (u, u+v)-construction [i] based on
(67, 82, 117690)-Net over F7 — Digital
Digital (67, 82, 117690)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(782, 117690, F7, 15) (dual of [117690, 117608, 16]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(780, 117686, F7, 15) (dual of [117686, 117606, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(8) [i] based on
- linear OA(773, 117649, F7, 15) (dual of [117649, 117576, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(743, 117649, F7, 9) (dual of [117649, 117606, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(14) ⊂ Ce(8) [i] based on
- linear OA(780, 117688, F7, 14) (dual of [117688, 117608, 15]-code), using Gilbert–Varšamov bound and bm = 780 > Vbs−1(k−1) = 1 741311 565014 245928 585556 903158 309889 306255 037399 184975 583745 196151 [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(780, 117686, F7, 15) (dual of [117686, 117606, 16]-code), using
- construction X with Varšamov bound [i] based on
(67, 82, large)-Net in Base 7 — Upper bound on s
There is no (67, 82, large)-net in base 7, because
- 13 times m-reduction [i] would yield (67, 69, large)-net in base 7, but