Best Known (59−9, 59, s)-Nets in Base 7
(59−9, 59, 1441202)-Net over F7 — Constructive and digital
Digital (50, 59, 1441202)-net over F7, using
- 72 times duplication [i] based on digital (48, 57, 1441202)-net over F7, using
- net defined by OOA [i] based on linear OOA(757, 1441202, F7, 9, 9) (dual of [(1441202, 9), 12970761, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(757, 5764809, F7, 9) (dual of [5764809, 5764752, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(757, 5764801, F7, 9) (dual of [5764801, 5764744, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(70, 8, F7, 0) (dual of [8, 8, 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(8) ⊂ Ce(7) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(757, 5764809, F7, 9) (dual of [5764809, 5764752, 10]-code), using
- net defined by OOA [i] based on linear OOA(757, 1441202, F7, 9, 9) (dual of [(1441202, 9), 12970761, 10]-NRT-code), using
(59−9, 59, 5662497)-Net over F7 — Digital
Digital (50, 59, 5662497)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(759, 5662497, F7, 9) (dual of [5662497, 5662438, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(759, 5764812, F7, 9) (dual of [5764812, 5764753, 10]-code), using
- construction XX applied to Ce(8) ⊂ Ce(7) ⊂ Ce(5) [i] based on
- linear OA(757, 5764801, F7, 9) (dual of [5764801, 5764744, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(741, 5764801, F7, 6) (dual of [5764801, 5764760, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(70, 9, F7, 0) (dual of [9, 9, 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(71, 2, F7, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(8) ⊂ Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(759, 5764812, F7, 9) (dual of [5764812, 5764753, 10]-code), using
(59−9, 59, large)-Net in Base 7 — Upper bound on s
There is no (50, 59, large)-net in base 7, because
- 7 times m-reduction [i] would yield (50, 52, large)-net in base 7, but