Best Known (74−11, 74, s)-Nets in Base 7
(74−11, 74, 1152963)-Net over F7 — Constructive and digital
Digital (63, 74, 1152963)-net over F7, using
- net defined by OOA [i] based on linear OOA(774, 1152963, F7, 11, 11) (dual of [(1152963, 11), 12682519, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(774, 5764816, F7, 11) (dual of [5764816, 5764742, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(774, 5764818, F7, 11) (dual of [5764818, 5764744, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- 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(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(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(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(774, 5764818, F7, 11) (dual of [5764818, 5764744, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(774, 5764816, F7, 11) (dual of [5764816, 5764742, 12]-code), using
(74−11, 74, 4946339)-Net over F7 — Digital
Digital (63, 74, 4946339)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(774, 4946339, F7, 11) (dual of [4946339, 4946265, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(774, 5764818, F7, 11) (dual of [5764818, 5764744, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- 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(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(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(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(774, 5764818, F7, 11) (dual of [5764818, 5764744, 12]-code), using
(74−11, 74, large)-Net in Base 7 — Upper bound on s
There is no (63, 74, large)-net in base 7, because
- 9 times m-reduction [i] would yield (63, 65, large)-net in base 7, but