Best Known (78, 91, s)-Nets in Base 7
(78, 91, 960803)-Net over F7 — Constructive and digital
Digital (78, 91, 960803)-net over F7, using
- 71 times duplication [i] based on digital (77, 90, 960803)-net over F7, using
- net defined by OOA [i] based on linear OOA(790, 960803, F7, 13, 13) (dual of [(960803, 13), 12490349, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(790, 5764819, F7, 13) (dual of [5764819, 5764729, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(789, 5764801, F7, 13) (dual of [5764801, 5764712, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(773, 5764801, F7, 11) (dual of [5764801, 5764728, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(717, 18, F7, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,7)), using
- dual of repetition code with length 18 [i]
- linear OA(71, 18, F7, 1) (dual of [18, 17, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(790, 5764819, F7, 13) (dual of [5764819, 5764729, 14]-code), using
- net defined by OOA [i] based on linear OOA(790, 960803, F7, 13, 13) (dual of [(960803, 13), 12490349, 14]-NRT-code), using
(78, 91, 5764821)-Net over F7 — Digital
Digital (78, 91, 5764821)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(791, 5764821, F7, 13) (dual of [5764821, 5764730, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(790, 5764819, F7, 13) (dual of [5764819, 5764729, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(789, 5764801, F7, 13) (dual of [5764801, 5764712, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(773, 5764801, F7, 11) (dual of [5764801, 5764728, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(717, 18, F7, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,7)), using
- dual of repetition code with length 18 [i]
- linear OA(71, 18, F7, 1) (dual of [18, 17, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(790, 5764820, F7, 12) (dual of [5764820, 5764730, 13]-code), using Gilbert–Varšamov bound and bm = 790 > Vbs−1(k−1) = 2123 960660 218861 187732 971678 039600 392570 059293 777703 896304 444695 670364 218839 [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(790, 5764819, F7, 13) (dual of [5764819, 5764729, 14]-code), using
- construction X with Varšamov bound [i] based on
(78, 91, large)-Net in Base 7 — Upper bound on s
There is no (78, 91, large)-net in base 7, because
- 11 times m-reduction [i] would yield (78, 80, large)-net in base 7, but