Best Known (78−12, 78, s)-Nets in Base 7
(78−12, 78, 137271)-Net over F7 — Constructive and digital
Digital (66, 78, 137271)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 13)-net over F7, using
- 6 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (59, 71, 137258)-net over F7, using
- net defined by OOA [i] based on linear OOA(771, 137258, F7, 12, 12) (dual of [(137258, 12), 1647025, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(771, 823548, F7, 12) (dual of [823548, 823477, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(771, 823550, F7, 12) (dual of [823550, 823479, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(771, 823543, F7, 12) (dual of [823543, 823472, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(764, 823543, F7, 11) (dual of [823543, 823479, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 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(771, 823550, F7, 12) (dual of [823550, 823479, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(771, 823548, F7, 12) (dual of [823548, 823477, 13]-code), using
- net defined by OOA [i] based on linear OOA(771, 137258, F7, 12, 12) (dual of [(137258, 12), 1647025, 13]-NRT-code), using
- digital (1, 7, 13)-net over F7, using
(78−12, 78, 823585)-Net over F7 — Digital
Digital (66, 78, 823585)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(778, 823585, F7, 12) (dual of [823585, 823507, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
- linear OA(771, 823543, F7, 12) (dual of [823543, 823472, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(736, 823543, F7, 6) (dual of [823543, 823507, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(77, 42, F7, 5) (dual of [42, 35, 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(11) ⊂ Ce(5) [i] based on
(78−12, 78, large)-Net in Base 7 — Upper bound on s
There is no (66, 78, large)-net in base 7, because
- 10 times m-reduction [i] would yield (66, 68, large)-net in base 7, but