Best Known (75−14, 75, s)-Nets in Base 7
(75−14, 75, 16809)-Net over F7 — Constructive and digital
Digital (61, 75, 16809)-net over F7, using
- net defined by OOA [i] based on linear OOA(775, 16809, F7, 14, 14) (dual of [(16809, 14), 235251, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(775, 117663, F7, 14) (dual of [117663, 117588, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(775, 117664, F7, 14) (dual of [117664, 117589, 15]-code), using
- construction XX applied to Ce(14) ⊂ Ce(11) ⊂ Ce(10) [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(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(755, 117649, F7, 11) (dual of [117649, 117594, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(71, 14, F7, 1) (dual of [14, 13, 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(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(775, 117664, F7, 14) (dual of [117664, 117589, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(775, 117663, F7, 14) (dual of [117663, 117588, 15]-code), using
(75−14, 75, 117664)-Net over F7 — Digital
Digital (61, 75, 117664)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(775, 117664, F7, 14) (dual of [117664, 117589, 15]-code), using
- construction XX applied to Ce(14) ⊂ Ce(11) ⊂ Ce(10) [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(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(755, 117649, F7, 11) (dual of [117649, 117594, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(71, 14, F7, 1) (dual of [14, 13, 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(11) ⊂ Ce(10) [i] based on
(75−14, 75, large)-Net in Base 7 — Upper bound on s
There is no (61, 75, large)-net in base 7, because
- 12 times m-reduction [i] would yield (61, 63, large)-net in base 7, but