Best Known (68, 68+16, s)-Nets in Base 7
(68, 68+16, 14709)-Net over F7 — Constructive and digital
Digital (68, 84, 14709)-net over F7, using
- net defined by OOA [i] based on linear OOA(784, 14709, F7, 16, 16) (dual of [(14709, 16), 235260, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(784, 117672, F7, 16) (dual of [117672, 117588, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(784, 117673, F7, 16) (dual of [117673, 117589, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- linear OA(779, 117649, F7, 16) (dual of [117649, 117570, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(74, 23, F7, 3) (dual of [23, 19, 4]-code or 23-cap in PG(3,7)), 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(15) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(784, 117673, F7, 16) (dual of [117673, 117589, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(784, 117672, F7, 16) (dual of [117672, 117588, 17]-code), using
(68, 68+16, 103156)-Net over F7 — Digital
Digital (68, 84, 103156)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(784, 103156, F7, 16) (dual of [103156, 103072, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(784, 117673, F7, 16) (dual of [117673, 117589, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- linear OA(779, 117649, F7, 16) (dual of [117649, 117570, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(74, 23, F7, 3) (dual of [23, 19, 4]-code or 23-cap in PG(3,7)), 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(15) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(784, 117673, F7, 16) (dual of [117673, 117589, 17]-code), using
(68, 68+16, large)-Net in Base 7 — Upper bound on s
There is no (68, 84, large)-net in base 7, because
- 14 times m-reduction [i] would yield (68, 70, large)-net in base 7, but