Best Known (75, 75+18, s)-Nets in Base 7
(75, 75+18, 13073)-Net over F7 — Constructive and digital
Digital (75, 93, 13073)-net over F7, using
- 71 times duplication [i] based on digital (74, 92, 13073)-net over F7, using
- net defined by OOA [i] based on linear OOA(792, 13073, F7, 18, 18) (dual of [(13073, 18), 235222, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(792, 117657, F7, 18) (dual of [117657, 117565, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(792, 117662, F7, 18) (dual of [117662, 117570, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(791, 117649, F7, 18) (dual of [117649, 117558, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(71, 13, F7, 1) (dual of [13, 12, 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(792, 117662, F7, 18) (dual of [117662, 117570, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(792, 117657, F7, 18) (dual of [117657, 117565, 19]-code), using
- net defined by OOA [i] based on linear OOA(792, 13073, F7, 18, 18) (dual of [(13073, 18), 235222, 19]-NRT-code), using
(75, 75+18, 81969)-Net over F7 — Digital
Digital (75, 93, 81969)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(793, 81969, F7, 18) (dual of [81969, 81876, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(793, 117664, F7, 18) (dual of [117664, 117571, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(15) ⊂ Ce(14) [i] based on
- linear OA(791, 117649, F7, 18) (dual of [117649, 117558, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(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(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(17) ⊂ Ce(15) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(793, 117664, F7, 18) (dual of [117664, 117571, 19]-code), using
(75, 75+18, large)-Net in Base 7 — Upper bound on s
There is no (75, 93, large)-net in base 7, because
- 16 times m-reduction [i] would yield (75, 77, large)-net in base 7, but