Best Known (83−12, 83, s)-Nets in Base 7
(83−12, 83, 960803)-Net over F7 — Constructive and digital
Digital (71, 83, 960803)-net over F7, using
- 71 times duplication [i] based on digital (70, 82, 960803)-net over F7, using
- net defined by OOA [i] based on linear OOA(782, 960803, F7, 12, 12) (dual of [(960803, 12), 11529554, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(782, 5764818, F7, 12) (dual of [5764818, 5764736, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(781, 5764801, F7, 12) (dual of [5764801, 5764720, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(765, 5764801, F7, 10) (dual of [5764801, 5764736, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(71, 17, F7, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- OA 6-folding and stacking [i] based on linear OA(782, 5764818, F7, 12) (dual of [5764818, 5764736, 13]-code), using
- net defined by OOA [i] based on linear OOA(782, 960803, F7, 12, 12) (dual of [(960803, 12), 11529554, 13]-NRT-code), using
(83−12, 83, 5764821)-Net over F7 — Digital
Digital (71, 83, 5764821)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(783, 5764821, F7, 12) (dual of [5764821, 5764738, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(782, 5764819, F7, 12) (dual of [5764819, 5764737, 13]-code), using
- construction X4 applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(781, 5764801, F7, 12) (dual of [5764801, 5764720, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(765, 5764801, F7, 10) (dual of [5764801, 5764736, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(11) ⊂ Ce(9) [i] based on
- linear OA(782, 5764820, F7, 11) (dual of [5764820, 5764738, 12]-code), using Gilbert–Varšamov bound and bm = 782 > Vbs−1(k−1) = 675 465182 712931 841096 394325 003319 589709 251023 228919 659940 837312 665047 [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(782, 5764819, F7, 12) (dual of [5764819, 5764737, 13]-code), using
- construction X with Varšamov bound [i] based on
(83−12, 83, large)-Net in Base 7 — Upper bound on s
There is no (71, 83, large)-net in base 7, because
- 10 times m-reduction [i] would yield (71, 73, large)-net in base 7, but