Best Known (60, 75, s)-Nets in Base 7
(60, 75, 16808)-Net over F7 — Constructive and digital
Digital (60, 75, 16808)-net over F7, using
- net defined by OOA [i] based on linear OOA(775, 16808, F7, 15, 15) (dual of [(16808, 15), 252045, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(775, 117657, F7, 15) (dual of [117657, 117582, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(775, 117658, F7, 15) (dual of [117658, 117583, 16]-code), using
- construction XX applied to Ce(14) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- linear OA(773, 117649, F7, 15) (dual of [117649, 117576, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(767, 117649, F7, 13) (dual of [117649, 117582, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(761, 117649, F7, 12) (dual of [117649, 117588, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(71, 8, F7, 1) (dual of [8, 7, 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
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(14) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(775, 117658, F7, 15) (dual of [117658, 117583, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(775, 117657, F7, 15) (dual of [117657, 117582, 16]-code), using
(60, 75, 61060)-Net over F7 — Digital
Digital (60, 75, 61060)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(775, 61060, F7, 15) (dual of [61060, 60985, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(775, 117658, F7, 15) (dual of [117658, 117583, 16]-code), using
- construction XX applied to Ce(14) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- linear OA(773, 117649, F7, 15) (dual of [117649, 117576, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(767, 117649, F7, 13) (dual of [117649, 117582, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(761, 117649, F7, 12) (dual of [117649, 117588, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(71, 8, F7, 1) (dual of [8, 7, 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
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(14) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(775, 117658, F7, 15) (dual of [117658, 117583, 16]-code), using
(60, 75, large)-Net in Base 7 — Upper bound on s
There is no (60, 75, large)-net in base 7, because
- 13 times m-reduction [i] would yield (60, 62, large)-net in base 7, but