Best Known (90, 108, s)-Nets in Base 7
(90, 108, 91506)-Net over F7 — Constructive and digital
Digital (90, 108, 91506)-net over F7, using
- 71 times duplication [i] based on digital (89, 107, 91506)-net over F7, using
- net defined by OOA [i] based on linear OOA(7107, 91506, F7, 18, 18) (dual of [(91506, 18), 1647001, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(7107, 823554, F7, 18) (dual of [823554, 823447, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(7107, 823558, F7, 18) (dual of [823558, 823451, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(7106, 823543, F7, 18) (dual of [823543, 823437, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(792, 823543, F7, 16) (dual of [823543, 823451, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(71, 15, F7, 1) (dual of [15, 14, 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(17) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(7107, 823558, F7, 18) (dual of [823558, 823451, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(7107, 823554, F7, 18) (dual of [823554, 823447, 19]-code), using
- net defined by OOA [i] based on linear OOA(7107, 91506, F7, 18, 18) (dual of [(91506, 18), 1647001, 19]-NRT-code), using
(90, 108, 508127)-Net over F7 — Digital
Digital (90, 108, 508127)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7108, 508127, F7, 18) (dual of [508127, 508019, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(7108, 823560, F7, 18) (dual of [823560, 823452, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(15) ⊂ Ce(14) [i] based on
- linear OA(7106, 823543, F7, 18) (dual of [823543, 823437, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(792, 823543, F7, 16) (dual of [823543, 823451, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(785, 823543, F7, 15) (dual of [823543, 823458, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(71, 16, F7, 1) (dual of [16, 15, 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(17) ⊂ Ce(15) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(7108, 823560, F7, 18) (dual of [823560, 823452, 19]-code), using
(90, 108, large)-Net in Base 7 — Upper bound on s
There is no (90, 108, large)-net in base 7, because
- 16 times m-reduction [i] would yield (90, 92, large)-net in base 7, but